Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
�
��������� ������������������������� �� ���!�#"%$&�'$�(�)� �
*,+.-0/)13254 687963+:2;2<1>=?6A@CBED(F9G +.GIHAJLKLM'+.G 6N@PO�6Q2;2R +:2;2.6AS31TGE136AFVUW+.1QX)X 1Y+.G>ZTD(F9O>6QGE[L\^]W_T+.@a`<6AOC+:2 \b/(6L[c6R 13Fd\cOeD)[)1Q@f7g+:GE-hD(O>15i</j+.@a`<637h/)13F^=?6Q@k6l=mD(FhnAO>6QGm+.Gc6`<6A-97fD(Fd\Aohp3Bjq(7rps=t+u7h/l7h/)1V/)132:XtDj`L6v@h7w63791v@P-0/)Dj2.6QFw@r/j+:X^p8/)1=?6Q@^6LBj2<1a79DY687w7f13GE[T@P1A-fD(GE[)6QFd\T@f79q [(+.1Q@x+:GcUW+.1QX)X 1kqEGj79+:2HQJLyAJt6QGE[?7h/)1QGbS8Fw6L[jqE637915687z7h/)15{lG(+ R 1QFw@h+;7d\bDj`z|xDjq 1QGz]_t1'@f796QFw791A[N791Q6A-0/j+.GESe@P1A-fD(GE[)6QFd\&@P-0/)D)Dj2k+:G}HQJ(~3y ]��dG/j+:@�@0XE6QF91}79+.O>1�/)1�XEF913XE6QF91A[�`�D(Fe7h/)1��r6AS8F8�1QS86379+.D(Gx�i;7h/)15/j+.S3/)1Q@f7a-hD(O�X 1�79+;79+ R 1T10�Q6QO�+:G)6379+.D(Gm`�D(Fg@P1A-fD(GE[)6QFd\@P-0/)D)Dj237f136L-0/)13Fw@g+:Gb�LFw6AGE-f1�ofp�=m/j+.-0/?/)1VXE6Q@P@P1A[T+.GmHQJL�L�E]
�WX)6AF�7k`<F9D(O�7�=mD?\(1Q6QFw@V+:G*&6QFfFw6AnA1A-0/�p�* D(F9D)-h-fDi:`<F9D(O�HAJj~L~�79DIHQJ(~3J8ofp�/)1N/(6A@C632u=?6�\)@c7w6Qq S3/L7m+.G&7h/)1
@0q BjqEFhB(@gDj`E|xDjq 13GxpA�jFw6QGE-h1)]a_?1k+.G(+u79+.632:2 \l7w6Qq S3/L7�@h79q [)1QGj7w@gB)187�=m1Q1QG�7h/)1W6LS313@zDj`jHAM6AGE[�HAy(�jBjq(7�7h/)13GxpL`�D(Fg791QG?\(1Q6QFw@8pj/)15XEF913XE6QF91A[C@f79q [E13Gj79@k`�D(Fg7h/)1b�sB(6L-f-f632.63qEF8�13637r�Qp7h/)1�jF913GE-0/�@r-0/)D)Dj2.�P1h�A+u7t-f1QF�79+:� -f687f1)]N�3+.GE-f1'HAJAJ3� pk/)1�/(6Q@cBE1Q1QG7f136L-0/j+:GES>687T7h/)1qEGE[E13F9S8Fw6L[jqE63791e2.1 R 132�]�4 687963+:2;2<1j� @t@f79q [E13Gj79@t@0+u75-hD(O�X 1�79+;79+ R 1e13Gj7fFw6AGE-f1e10�Q6QO�+:G)6L�79+<D(G)@z7fD R 6AFw+<DjqE@a1QGES�+:GE1A13F9+:GESl@r-0/)D)Dj2:@Q]x�x/)1r\?6QF91YS313GE1QFw632:2 \t`<F9D(O�6�OeD)[)1Q@f7gB)6A-0nj�S8F9DjqEGE[xpQ6AGE[T7h/)1r\?@0X 1Q-f+.632:+u�01v+.GT7f1Q-0/(GEDj2<D)Sr\?[jqEF9+:GESv7h/)18+.F^@P1A-fD(GE[)6QFd\t@h79q [j+<13@Q]a_5+:@796Q@rnm+.@W7fD�B)Fw+.GEST7h/)1QO�7fDc7h/)1mF91Q�(q)+:F91A[�2.1 R 132 +.G�O>687h/)1QO�6379+.-f@3]k��/)1c-0/(632:2.1QGES31mDj`+.O�XEF9D R +:GESY7h/)18+.FknLGEDA=t2.1A[)S315+:@V�(q)+u7f1t1QGL�dD3\j6LBj2<1jp(@h+.GE-f1C�rO>D(@h7kDj`E7h/)13O�6QF91Y=m182:2.�OeDL79+ R 63791A[��Qp(@P6�\)@54 687963+:2;2<1)]_?1I/(6A@NBE1Q1QG�O�6AF9F9+.1A[�@h+.GE-f1�HAJj~AM)]�_5+:@e=t+:`�1 +:@6Q2:@rD�6IO>687h/!7f136L-0/)13Fi�/)DA=m1 R 13Frpg/)1e-f2:6Q+:O>@Y7h/(637T/)1QF�+.Gj791QF913@h7w@52:+.1�O>D(F91�+:G'7h/)1�F9136Q2:O�Dj`kB)DL796QGL\N6QGE[S86AF9[)1QG(+:GEStGEDA=?6L[(68\(@fo0]g�x/)1r\>/(6 R 157�=mD�-0/j+;2<[(F91QGz]_5+:@k/)D)BEBj+<13@�+.GE-f2;q [E1Y-f+.GE13O>6(pAF9136L[j+.GES�i:+.GC�LF91QGE-0/mD(Fa�jGES�2:+:@r/AofpL-hF9D(@0@f=mD(F9[(@8p2:+:@h791QG(+:GESl7fDmO�qE@h+<-Wi:+.GE[j+<1TF9D)-0n?6QGE[C182<1Q-97fF9D(G(+<-968ofpL6AGE[?=?632<nQ+.GES(]vUWqEF9+:GESl7h/)1c� JL Q@8p/j+.@?2.13+:@0qEF91�79+:Oe1b=?6A@m@0X 13Gj75+:O>X)F9D R +.GESe/j+:@C�jGES�2:+:@r/�pg2:+:GES�q)+.@f79+<-9@8p�6QGE[&Bj+:2;2:+:6AF9[(@Q]* D(@h7vDj`E7h/)1Q@P1?6A-979+ R +;79+.1Q@V/(6 R 1tGEDA=�BE1Q1QG�F91QX(2.6A-h1Q[>BQ\�X)F9D)B(2.1QO�@rDj2 R +:GES(]_5+:@z�EFw@h7xX)q Bj2:+:@r/)1Q[5@rDj2;q(79+<D(G56QXEXE1Q6QF91A[T+.Gb¡ ¢V£j¤�¥v¦�§©¨?ª�«V¬xT®Eª¯+.Gc*&6AF9-0/HAJAJLJE]��3+.GE-f17h/)13GxpY/)1�/(6A@>[E1 R 182<D(XE1A[}6,F9136Q2v796Q@h791`�D(FbX)F9D)B(2.1QO#@rDj2 R +:GES6QGE[/(6A@m-hD(Gj7fFw+<Bjq(7f1Q[,O>D(F91>6AGE[�O>D(F91e@PDj2:q(79+.D(G)@Q],4�1Q@h+<[)1Q@C¡ ¢V£j¤�¥v¦�§©¨,ª�«V¬xT®Eª�p4 637w6Q+;2:2.1�/(6A@tBE1Q-hD(O>1�6>F91QS�q)2.6QFY@rDj2 R 1QF�6QGE[X D(@P1QFl+.G@P1 R 1QFw632�X)F9D)B(2.1QO°@P1A-979+.D(G)@i.+:Gc7h/)1c*N���&X)q Bj2:+.-f6879+<D(G)@�pA7h/)1c*&687h/)1QO�6379+.-f632 ±�63�01�7w7f1jpj6QGE[m7h/)1b4Eq)2:2.1879+:GCDj`E7h/)1�W@P@PD)-f+.6879+<D(G�Dj`z�jF913GE-0/'*&687h/C�)1Q6A-0/)1QFw@9oP]
�jF9D(O²6AGb�([(+u7fD(FP� @�XEDj+.Gj7gDj` R +.1�=CpQ/(6 R +.GESV6�F91AS�q)2:6AFz-hD(Gj7fFw+<Bjq(7fD(F^2;+<nA15*,+.-0/)1324 637w6Q+;2:2.1�+:@?6eF91Q632z�dDQ\^]�_T+.@t@rDj2;q(79+<D(G)@t6AF91>6Q2 =?68\(@m-fD(FfF91Q-9756QGE['=m182:2.�PD(F9S86QG(+;�01Q[z]ZTDL7aD(G(2u\?7h/(687rpjB(q(7a/)1T@0q B(O�+u79@WX)F9D(X D(@06Q2:@vD(Gc65F91QS�q)2.6QFkB(6A@h+.@�p3=m/j+.-0/c6QF91T6Q2 =?68\(@=m132;287h/)Djq S3/L7xDjq(7�6AGE[51h�LX(2.63+.GE1Q[xpP=t+;7h/V�<qE@f7 7h/)1W6AX)XEF9D(X)F9+:63791v6AO>DjqEGj7^Dj`L[)187w6Q+;2w]��<7/(6A@gB)1A13G56vF91Q632QX(2<136A@hqEF91W`�D(FE7h/j+:@a�([(+u7fD(Ff�h+:GE�P³x/j+<18`37fDT/(6 R 1V[E1 R 182<D(XE1A[T6QGtD(GES3Dj+:GES`<F9+.1QGE[(@r/j+:Xb=t+;7h/C@hq -0/C65=?6QFfO´/jqEO>6QGeB)13+:GESt6QGE[�@hq -0/�6mS3D)D)[CO>687h/)1QO�6379+.-f+.6QGz]
�
������������ ��
��������������� �
!#"%$'&)(+*,(.-+/10324$5(768&�9�:<;=619>6@?�?�0@68(�24$=$'&�ABAC086D*)0@AFE 6G2�2H;'-IEB2C-7J>0�K)0@E<?L0@:4-768(I( M24$5$'&�AN(7$�9�O8P42C-IJQ03AC0@61*�08AFE@RTS�;)0VU�619)61*�-I619XWY6G2H;)0@J=682C-7:468(LZ1$):4-+0[2\M,]+U^W_Z[`4K1"%;'-7:<;?)&La'(I-IEb;)08EQcLdfe'gihBjlk\m_npoBqsr3t)niK^;�6@E�61E�/101*u$'&�AV01*'-.2C$�AC-I6@(va�$�6@AC*w24$x?)AC$@y'-+*)06>(.-7E42z$'{^{+6Gy�$'&�AF-.2C0>?�AC$)a'(+08J>Ez{+AC$�J|?)61E42�-7E<E<&L08EV$'{}cD~[�[��n��Dk\mL�[�,�Dklj����@~��8��6@9�*cLdfe'g�hBj+k�m=npofq#r3t�n�R�Z8-79�:40VJ=619DM=$�9=$'&�Av01*'-.2C$�AC-I6@(La�$�6@AC*>;�6[y�0�9�$D2Ba)01089=61E�PEb$):4-I682C01*X"�-�2H;xU^���B��{l$�A361E�(7$�9�O=6@EVEb$�J>0Q$'{v$'&�AzAC086D*)0@AFE[Ks"%0Q*)01:4-7*�0@*�2H;�6G2z6� �#0@61*�08AFEG�1U#;)$'-+:40��BEb0G(+0@:C2C-+$�9%$'{){+6[y�$'&�AC-�240�?)AC$)a�(70@J=EN{+AC$�J�$@y�08As2H;)0fM�0@6@AFET"%$'&)(+*(I-7/108( M5a)03J>$�AC03?)AC$)*�&L:C2C- y�0�KD6@E^"%0G(I(�6@Eva�0G-79�O}JQ$�AC0�JQ08619�-I9�O[{7&)(FRT��0�6@AC0�(+$)$)/@-79�O0@E<?L0@:4-768(I( M5{l$�A�{+6[y�$'&�AC-�2408EB{+AC$�J�2H;)0%0861AF(I-70@ATy�$'(.&�JQ08E@R
!v��AFEH2�J>0@9'2C-7$�9�01*X2H;'-IE�?)AC$1��01:C2}-79�J5M>�T0@6@A��'9�*u�D-I9)6@(70,-I9X2H;)0��f01:40@J>a�08A�1�1�D -7E<E<&L0'K'619�*Q!�"V6@9'2N2C$5AC08-�2408AF682C0z-.2v;)08AC0�Rv!l{�M�$'&>;�6Gy�0�E�$�JQ03{+6Gy�$'&�AF-.2C0�?�AC$)a)P(+08J>E>{+AC$�J¡2H;)0u?�61O80@EX$'{�c'~��[�¢n��Dk\mL�[�,�Dklj����@~��8�£$�A¤cLdfe'g¥hBjlk\mpnpofq#r3t�niK?)(70@6@Eb0z{l$�A\"V61AC*�2H;)$�Eb0z?�AC$)a'(+08J¦9�&�J>a�08AFEB24$�&�E8Rv!�"%$'&)(+*,68(7E�$561?)?�AC0@:4-76G240z-I{)M�$'&:H$'&)(7*=?)AC$@y'-+*)0z2H;)0VAC0G{l0@AC089�:H0z2C$�2H;)03y�$'(.&�JQ0'K'-7E<E<&L0'K'619�*=?)6DO80�9�&�JQa)0@AFE[K�-I9>:46@Eb06Q2\M�?L$u:HAC0108?�EV-79'2C$w2H;)0Q(.-7E42V$'{B?�&�AC0¤?)AC$)a�(70@J§9�&�J>a�08AFE@RuS�;�619�/QM�$'&TRu��0Q61AC0(+$)$)/@-79�O�{l$�A}Eb$�J>0["%;)08AC0>a)0G2�"%0@0@9 � �1� 619�* �@�D� ?�AC$)a'(+08J>E[Ks619�*Q"%0,61AC0>;)$�?�-79�O24$XAC01:408- y�0�M�$'&�A (I-IEH2}$'{^{+6[y�$'&�AC-�2408E�a@Mu�'01a�AC&�6@A�M �1¨ K �1�1��© R,ªf9�:40�"%0=;�6[y�0%M�$'&�A{+6Gy�$'&�AF-.2C05?)AC$)a�(70@J=EGK�2H;)05U^W_Z�-IE�?)(I619)9�-I9�OV2C$QJ=6D/10V2H;)0@J«6Gy'68-I(I6Da'(+05$�9¤U^��]l$�A?L08AH;�6@?�E 6�¬L6@Eb;5JQ08JQ$�A�M=E42C-+:</G`�R
ªB&�A�E<-I9�:H08AC032H;�6@9�/DE O8$,$'&�2^24$,Z1;�6["V9>N$)*�-I9#K1"%;)$,;�6@E a)0108952H;)05WY6[M�;)08J�)*'-.2C$�Af{l$�Av2H;)0�?�6@EH2fE<- ®%M�0@6@AFE@R3Z1;�6G"V9¤;�61E�J>$'(+*)01*=2H;�682 E�01:C2C-7$�9X$'{^$'&�A���$'&�A49)68(6@(IJQ$�E42TEH-79�O[(701P<;�619�*)01*'(�M�-79'2C$V6@9�0<®1:408(.(+089'2�{l08682C&�AC0 {l$�A^;'-7O8;VEb:<;)$)$'(DEH2C&L*)0@9'2FE�6@9�*24086D:<;)08AFE@R¤¯�$1"%0by�0@A�KT;)0Q;�6@EVa�0@0@9X�+&LO8O[(I-I9�O=J=619DMw*�&�2C-70@Ez{l$�A3°�&)-�240=Eb$�J>052C-IJQ09�$1"±619�*,{l0G(.2N-.2^"V6@Ev2C-7J>0324$,J>$@y�0V$�9�R
¯�6[y�-I9�O�E<6@-7*>2H;�6G2bKL2H;)0Q²�$�61AC*¤$'{Bc�d e'g³hBj+k�m�npoBqsr3t)n´0H®@2C0@9�*�E36%"V6@A4J"%08(7:H$�J>0=2C$¤µ�0G¶p¯�$)$�?L08Az61Ez2H;)0Q9�0["·WY6GM);)0@J¸�)*'-.2C$�A[RQµ�08¶¢;�6@E%61:C240@*x61E�2H;)0¹ E�EH-7E42C6@9'2TWY6[M�;)08Ji��*�-�24$�AL{l$�A)2H;)0B?�6@EH2)M�0@6@AbKGE�$ 2H;)0�24AF619)EH-.2C-7$�93Eb;)$'&)(7*za�0B68(7J>$�EH2Eb0861J,(+08E�E8R
�T$'&>"�-.(I(s;�6[y�059�$D2C-7:H0@*>2H;�6G2v"%0%-I9'24AC$)*�&L:401*X659�0�"i{l08682C&�AC0%2C$wcLdfe'g³hBj+k�mnpofq#r3t�nº2H;'-IE�?�6@EH2�M�0@6@AbKL9)61J>08( M=2H;)0 � U#$�9'24AC-7a�&�2C$�A�»�AC$'�)(+08EG�zE�01:C2C-7$�9�R�!+2 *�-7*9�$D2}6@?�?�0@6@Af-79¤0by�0@A�MQ-IE�EH&L0�K�a�&�2�$�9�( M="%;)089Q2H;)08AC0V"V61E�E�?)6D:4056[y'6@-.(761a�(70�RV!v;�6[y�0AC01:408- y�01*5(I-IJ=-�240@*5{l0@01*)a)61:</�$�9�2H;'-7EB9�0["�{l0@6G2C&�AC0�K�a�&�2�6@(.()2H;)0z{l0@01*)a)61:</�;�61Efa�0@0@9?L$�EH-.2C- y�0�RV!N?L08AFEb$�9)68(I( M¤;�6[y�056@?�?)AC01:4-I682C01*¤O80[2F2C-79�O�24$X/19�$1"�65(.-.2F2C(70�J>$�AC056Da)$'&�2J>6@9DM>$'{s$'&�A�AC01O[&)(I61AB:4$�9'24AC-7a�&�2C$�AFE 619�*%"%;�682N2H;)0bM,(7$)$)/V(.-+/10)R
!l{�M�$'&=:C619V2H;'-79�/V$'{�"V6[M)EN{l$�AT&�EN24$%?)AC$)*�&L:40z6�a)0G2F2C0@A^J>61OG68¼4-I9�0�K'a)03E<&�AC0}2C$Eb089�*=JQ032H;)$�Eb0�EH&LO8O80@E42C-+$�9)E8RT��0�61AC0�6@( "V6GM�EB-79'2C0@AC08EH2C01*5-79�"V6[M)Ev2C$5-7J=?�AC$8y�0�R
�
��� �����T �� �������¤�� 5���D���5���H�b�����}�
»)(+0861E�03E�0@9�*�M�$'&�ANEb$'(.&�2C-+$�9)E^24$�2H;)0z?)AC$)a�(70@J=Ev-I9%2H;'-7EB0@*�-�2C-+$�95a@M��fo �������@k���! " $# R ¹ :4$�?1M¢$'{&%('�)$*,+-%('�)$.�/!'�01%('324*&+5%7698;:=<>�"�-I(.(}a)0w?�AC08Eb089'240@*_2C$$�9�0=?)AC01P4&�9�-�y�08AFE<-�2\M�AC086D*)0@Af"%;)$XEb089�*)E3-79�E�$'(I&�2C-7$�9)E�a)08{l$�AC052H;)0=*�086D*'(I-I9�0�R�S�;)0*�0@:4-7EH-+$�9>$'{�2H;)0%0@*�-�24$�AN-7EB��9)6@(�R
ªB&�A�?�AC$)a'(+08J>E�2H;'-7EsJQ$�9'2H;3:H$�J>0v{+AC$�Jp2H;)0fUs$'(.(@?0@O80�W_$�9'24J>$�AC0@9�:\MzUs$�9'2C0@E42bK�1�1��ACB'�@�D�;D Rw��0=2H;�619�/ ¹ 9�*)A�E0GF861a�0G(I(70�KN$'{BUs$'(.(@?0@O80wW_$�9'24J>$�AC0@9�:\M#K�"%;)$�(+$)$)/1E6@{I2C0@AN2H;'-IE :H$�9'2C0@E42B*�08E<-7OG9�01*5{l$�AvE�01:4$�9�*)6@A�M=E42C&L*�089'2CEf{+AC$�J�2H;)0HF86[y'6@(LAC01O[-7$�9�R
IKJMLON�PQJSR-T�LSUWVYXHJSLN�T�Z[N]\�^&^O_�`&^SabCcCd 64e-fO8Cg c�h�i$c�j,kCl;l�m
n R�S�;)0xO8$'(+*)0@9 AF682C-7$N = 1+
√5
2≈ 1.618033989 . . .
;�6@E52H;)0wAC08J>6@AH/@61a�(70?�AC$�?�0@A�2\Mp2H;�6G2Q-�2CE¤J=&)(�2C-7?�(I-7:46G2C-�y�0_-I9Dy�0@AFE�0
1/N-IEw0@°�&�68(324$�-.2FEw*�0@:4-7J=6@(z?�6@AF2
0.618033989 . . .Rv�1-79�*,619�$D2H;)08Av9�&�JQa)0@A^"�-�2H;�2H;'-7Ef?)AC$�?L08AF2�M�R
\ RNUs$�9)EH-+*)0@A�6�°'&�61A�2408AB:4-7AC:4(70V$'{#AF6D*'-I&�E1R
]76G`Q�D-I9�*p6 AC0@:C2C6@9�O[(+0 ;�6[y�-I9�Ox2H;)0_E<61J>0_6@AC0@6_619�*¢2H;)0_E<61J>0?L08AC-IJQ0[2408Av6@Ev2H;)0%°'&�61A�2408Af:4-IAC:4(+0)R
]la1`X��$�Az6¤:4$�J>?�(+0[240X:4-7AC:4(70X$'{BAF6D*'-I&�E1K^-IEV-.2z?L$�E<E<-7a�(70=24$��L9�*
E<&L:<;Q6,AC0@:C2C6@9�O[(+0'Ks;�6[y�-I9�O�619X61AC086,6@9�*¤65?L08AC-IJQ0[2408A}01°'&�6@(24$%2H;�682B$'{L2H;)0V:4-7AC:4(70"o�µH&�E42C-I{.M%M�$'&�Av6@9)E4"%08A[R
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
..
..
.
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.......................................................................
........................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
..
.
..
.
1
p R ¹ a�61ACAC08('-7EN�)(I(701*z"�-�2H;z"V6G2408A[R���0308J>?'2\M�;�6@(.{L$'{)-�2CEv:4$�9'24089'2CE�619�*�2H;)0@9�6D*)*�6(I-�24AC03$'{1"V682C0@A�R ¹ {I2C0@AT*�$'-I9�OB2H;'-7E�$�?�0@AF6G2C-+$�9%E�0by�0@9�:4$�9)Eb0@:4&�2C-�y�0f2C-7J>0@E[K["%0}61AC0 (+0G{I2"�-.2H;,2H;�AC010%(I-�24AC0@E�$'{#"V6G2408AB-I9>2H;)05a)6@A4AC0G(FR�¯�$1"iJ>6@9DMQ(.-.24AC08Ef"%08AC0%-79=2H;)05a)6@A4AC0G(682N2H;)0Va�0@O[-79)9�-I9�O�o_ RTS�;)0z6@AC0@6@EB$'{)2H;�AC010�{+6D:40@EB$'{L6zAC0@:C2C6@9�O[&)(76@AN?�6@AF6@(7P(+0G(+08?)-I?L0@*=6@AC0
18:HJ 2 K 40 :HJ 2 619�* 80
:HJ 2 Rv�D-I9�*Wq]+6[`T-.2FE�y�$'(I&�J>0$rw]la1`�2H;)0�(70@9�O�2H;=$'{�-.2FE *�-I6DO8$�9)68(PQ
R.....................................................................................................................................................................................
.
..
.
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
.
..
.............................................................................................................................................................................................................................................................................................................................................................................
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
....................................................................................................................................................................................
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
...
P
Q
a Rv�1y'68(I&�6G240 √6 +
√
6 +√
6 + · · · Rs RtF@0G2
AKBKCK�6@9�*
Da�0>:4$'(I(.-79�0861A}?L$'-I9'2CE�EH&L:<;
2H;�682AB = BC = CD = 1
R Us$�9)EH-+*)0@AV2H;�AC010Eb08J=-7P�:4-IAC:4(+08Ev$'{�AC0@E<?L0@:C2C-�y�0z*'-76@JQ0[2408AFE
ACKBD
619�*AD
ROF@0[2Ea�0�2H;)0z-I9'2408AFEb0@:C2C-+$�9,$'{�2H;)0zEb08J=-7P�:4-IAC:4(+08E
"�-.2H;=:H089'24AC0@EB6@9�*
CR �f0[2408A4J,-79�032H;)0V61AC086�$'{L2H;)0
:4&�A�y�-.(I-I9�0@6@AL24AC-I619�O[(70AED
]7Eb;�61*�0@*3-79�2H;)0f*)AF6["�-79�Ob`<R ..............................................................................................................................................................................................................
................................................................................................................................................................................................................... ..
..
..
.
..
..
..
.
..
..
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...................................................................
............................................................................................................................................... ..
..
..
.
..
..
..
.
..
..
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.....................................................................
...............................................................................................................................................
A B C D
E
q q q q
q
A
� R�S#;)0X$�E�:4-I(.(76G2C-+$�9w?L08AC-7$)*u$'{B6X?�0@9�*'&)(I&�J�-IE%?)AC$)P?L$�A�2C-+$�9)68(#24$=2H;)05Eb°'&�61AC05AC$)$D2�$'{�-�2CE�(70@9�O�2H;¤]I{l$�A}0<®DP61J=?)(70�K�24$=24AF-7?�(+0V2H;)0,$�Eb:4-.(I(I682C-7$�9X?�0@AF-+$)*�K)"%05J,&)(.2C-7P?)( M%2H;)03(70@9�O�2H;5a@M59�-79�0�`�RTS'"%$�?�0@9�*'&)(I&�J=E $'{�*'-I¶s0@A4P0@9'2}(+089�O�2H;�E�6@AC0=AC0G(+0861E�01*X{+AC$�J 2H;)05-79�-�2C-768(T?L$�EH-.2C-7$�9Eb;)$1"V9�R}S#;)0�E�;)$�AF2C0@A $�9�0�JQ0861EH&�AC0@E
25:HJ¤K�6@9�*Q-�2CE
$�Eb:4-.(I(I682C-7$�9z?�0@AF-+$)*�-7E1Eb0@:H$�9�*sRsS#;)0v2�"%$3?�0@9�*'&)(I&�J=E
61AC0z68(I-7OG9�01*561OG6@-I95{l$�A�2H;)03�LAFE42T2C-7J>0z68{I2408A7Eb0@:H$�9�*�E
-79 2H;)0G-7A�-79�-�2C-768(}?�$�E<-�2C-+$�9�R��1-79�*_2H;)0w(+089�O�2H;¢$'{}2H;)0(+$�9�O808Av?�0@9�*'&)(I&�JwRv] ¹ -7A�AC08E<-IEH2F619�:40�-IEf9�01O[(701:C2C01*sR `
d
d
..................................................................................................................................................................................................
� ��� � � �� �� � � � �
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
......
...... ............
.............
......
......
......
......
............
............
.......
......
......
......
......
...... ..................
......
......
......
......
..
............
......
......
......
......
......
......
..
*Y
...... ............
......
......
...
............
......
......
......
...
............
.......
......
......
............
.......
......
......
*Y
� RQ!�9w6=AC08��9�0@A�M�K�6Q:\M�(.-79�*�AC-7:468(^E424$�AF61O8052C6@9�/X;�6@E�6>E<?)-IAF6@(TEH2F6@-IAC:46@Eb0=$�9�0=J>0G2C0@A"�-+*)0V682F2F6D:<;)0@*,2C$,-�2CE 0<®82408AC-7$�A[RNS�;)0VEH2F6@-IAC:46@Eb0%O8$)08Ef{+AC$�J�2H;)0%a�$D2F2C$�J�24$�2H;)0z24$�?"%;'-I(70�J>61/1-I9�O�0<®@61:C2C(�M
2:H$�J=?)(70G2C0�AC0by�$'(I&�2C-7$�9)E@R�!l{s2H;)0V2C6@9�/>;�6@E�6,;)08-7O8;D2}$'{
10Ji619�*%6z*�-I61J>0G2C0@A^$'{8J�K@��9�*V2H;)0}(70@9�O�2H;�$'{'2H;)030<®82408AC-7$�A^0@*�O803$'{�2H;)0�E42C68-7AC:C61E�0�R
XHJML U J� �R5Z IKJSLN9PQJMR[TOL UWV \�^&^O_�`&^Sab�cCd 64e��!8Cg c�h�i;j�c k"l"l�m
n R�F@0p9�$�J>a)AC0�*�� $�AN = 1+
√5
2≈ 1, 618033989 . . .
6p(76pAC0@J=61AC°'&�6Da'(+0?�AC$�?)AC-@E0G2�E0u°'&L0xE�$�9 -79Dy�08AFEb0wJ=&)(�2C-7?�(I-7:46G2C-I{
1/N08EH2tE01OG68( ?6xE<6x?�6@AF2C-70u*9E01:4-IJ>68(+0
0, 618033989 . . .R'S�AC$'&'y�0@A�&�9=6@&�24AC0V9�$�J>a)AC0V6[M'619'2B:40G2F2C0V?)AC$�?�AF-@E0[2�E0�R
\ R�Us$�9)EH-+*9E0@AC$�9)EB&�9>°�&�6@AF2B*�0V:H08AC:4(+0V*�0�AF6GM�$�91R
]76G`5S�AC$'&'y�0@Az(+0XAC0@:C2C6@9�O[(+0¤°�&)-v6 ?6X(I6X{l$'-7EV(76¤J��0@J>0¤68-7AC0¤0[2�(70J��08JQ0�? E08AC-IJG?0[24AC0%°'&L0%:H0V°�&�6@AF2B*)0%:H08AC:4(+0)R
]la1`X��E42HPH-.(@?L$�E<E<-7a�(70 °'&L0 ?�$'&�A#&�9V:40@AC:4(70 :H$�J=?)(70G2s*�0fAF6[M�$�91K8$�9
?)&)-IE�E�0�24AC$'&'y�08A}&�9X2C08(TAC0@:C2C6@9�O[(+0,6GM'6@9'2�?6,(I6,{l$'-7E�(76=J��0@J>06@-IAC0%0G2�(+0VJ��0@J>0V? E08AC-IJG?0[24AC0�°'&L0�:40�:40@AC:4(70 oBµH&�EH2C-.��08A^y�$D24AC0A�E08?L$�9)E�0�R
..
.
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
..
.
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.........................................................................
......................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
1
p RD� 95a�61AF-I(�0@E42TAC0@J=?)(.-)*�� 086@&TRDªf95089�y'-+*)0}(I63J>$'-.2C-@E0�0G2^$�9%6[��$'&�240}&�9V(I-�24AC03*�� 086@&TR¹ ?�A�?0@EN6[y�$'-7A�08¶s01:C2C& E0�:40G2F2C0�$�? E08AF682C-7$�9%E�0@?'2s{l$'-7E[KG-I('0@9VAC0@E4240f24AC$'-IE^(.-.24AC08E@R@Us$�J>a�-70@9M=6[y'6@-�2HPH-.(�*�0z(I-�24AC0@E *�� 086@&Q*)6@9)Ef(70%a�61AF-I(L68&Q*9E08?�6@AF25o_ R F@0@E�68-7AC08ET*�0�24AC$'-7ET*)0@Es{+61:H08E^*�� &�9�?)61AF68(I(@E08(@E0@?�-7? ?01*)0AC01:C2F619�O[(70³Eb$�9'2Y*)0
18:HJ 2 K 40
:HJ 2 0[280
:HJ 2 RS�AC$'&'y�0@A qz]+6[`Q(+0 y�$'(I&�J>0Or�]+aD`Q(76�(+$�9�O[&L0G&�Au*)0�(I6OGAF619�*)0�*'-761O8$�9)6@(70
PQR
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
...........................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
.
..
.
..
.
..
.
.
....................................................................................................................................................................................
......................................................................................................................................
......................................................................................................................................
......................................................................................................................................
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
........
...
P
Q
a R E�1y'68(I&L08A √6 +
√
6 +√
6 + · · · Rs RsZ1$'-�2AKBKC0G2
D*�08E3?L$'-I9'2CE�6@(.-+OG99E08E}240G(7E3°�&L0
AB = BC = CD = 1RLªf9X:4$�9)E<-7*9?08AC0�24AC$'-IE}*�08J=-7P
:H08AC:4(+08E�*)0X*'-76@JG?0[24AC0@EVAC0@E<?L0@:C2C-I{+EAC
KBD
0[2AD
RZ1$'-.2
E(\� -79'2C0@AFE�01:C2C-7$�9u*�08E�*)0@J,-+P<:H08AC:4(+08E�*�0Q:H089'24AC0@E
B0G2
CRf� E0G2C0@ACJ=-I9�0@A�(�� 68-7AC0¤*�&u24AC-I619�O[(70�:4&�A�y'-I(.-+OG9�0
AED]7?�6@AF2C-70%;�6D:<;'&�A�E010VEH&�A�(70V*�08E�EH-79D`<R ..
.
..
.
..
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
.
..
.
..
.
..
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
....................................................................................................
................................................................................................................................................................................................................... ..
..
..
.
..
..
..
.
..
..
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...................................................................
............................................................................................................................................... ..
..
..
.
..
..
..
.
..
..
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.....................................................................
...............................................................................................................................................
A B C D
E
q q q q
q
D
� Rsªf9¤E�68-.2}°'&L05(76,? E0@AF-+$)*)0=*�� $�E�:4-I(.(76G2C-+$�9¤*#� &�9�?�0@9�P*�&)(70w08EH2V?)AC$�?L$�A�2C-+$�9)9�0G(I(70 ?6¤(76¤AF6D:4-I9�0w:C61ACA�E0@0�*�0X(I6(+$�9�O[&L0G&�A�*)0,:H0G(I&)-7P�:4-�]+?)61A}0H®10@J=?)(70�KL?L$'&�AB24AF-7?�(+08Af(I6? E08AC-7$)*�0>*#� $�Eb:4-.(I(I682C-7$�9w*#� &�9x?L089�*�&)(70�K#-I(�{+6@&�23J,&)(.2C-7P?)(.-+08AB?)61AB9�0G&){�E<6%(7$�9�O[&L08&�A\`�Rs�'9=( �61:<;�619'2f*�0G&D®5?�0@9�P*�&)(70@ET*�0v(7$�9�O[&L08&�AFET*�-.¶E08AC0@9'2C0@E�?6B(76B?L$�EH-.2C-7$�9�-79�-�2C-768(+0-79�*'-+°'& E01050[2 0@9Q99E0@O[(I-7O80@6@9'2f(I6�A�E08E<-IEH2F619�:405*)0%(�� 68-7A�K�E<-(+0�?�(I&�E3:4$'&�AF2 J>0@EH&�AC0
25:HJ¤K�°'&L05E<65? E08AC-7$)*�05*�� $�E�P
:4-I(.(76G2C-+$�9w0@E42z*�0,&�9�0>Eb0@:H$�9�*)0Q0G2z°'&L0=(+$�AFE�°�&s� $�9w(70@EAC0G24AC$'&'y�089'2�?6�9�$'&'y�086@&X?)61AF68(I(@?08(70@E ?�$'&�Av(76%?�AC08J=-@?0@AC0{l$'-7E,68&¢a)$'&�25*�0�Eb08?�2%Eb0@:H$�9�*)0@E[KN-.(7E,$):H:4&�?�0@9'2%(+0G&�A?L$�EH-.2C-7$�9V-I9�-.2C-I6@(70�K5E0�y'6@(.&L0@As(76}(7$�9�O[&L08&�AT*�&�?)(.&�E^(+$�9�O�R
d
d
..................................................................................................................................................................................................
� � � � � � � ��� � � �
� �
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
...........
......
...... ............
.............
......
......
......
......
............
............
.......
......
......
......
......
............
............
......
......
......
......
..
............
............
......
......
......
......
..
*Y
............
......
......
......
...
............
......
......
......
...
............
.......
......
......
............
.......
......
......
*Y
� R��B619)Ef&�9�0%AF6��Q9�08AC-70�K�&�9QA�E0@E�0@A�y�$'-IAf*�� 08E�E�0@9�:40�:�M'(I-I9�*)AF-+°'&L0�0@E42BJ,&�9�-s*#� &�9X0@E�P:468(I-70@Af*#� &�9¤JG?0G24AC0�*�0%(I61ACO805°'&)-#(7$�9�O805*�05a�61E�0@9X;�68&�2f(76�?)61AC$'-�0H®@2�E08AC-708&�AC05*�619)E&�9�0zE<?)-IAF6@(70z0G2N0@9%{+6@-�2�0H®@6D:C2C0@J>0@9'2N*�0G&D®z{l$'-7EN(+0}2C$'&�A[R'Z@6D:<;�6@9'2�°�&L0�(+0�A�E08Eb08A�y�$'-7A^610JG?0G24AC08E *�0V;�6@&�2B0[2
8JG?0[24AC0@E *�0V*�-I61JG?0G24AC0'KD24AC$'&'y�08A�(76�(7$�9�O[&L08&�Av*�0z(76VAF6@J>?�0
0<®82�E0@AF-+0G&�AC0%*�0z(\� 0@E�:468(I-70@A�R
� 0H®@2T"%0VO[- y�032H;)0�Eb$'(.&�2C-+$�9)E�2C$�2H;)0 �@�D�;D ²�UXUs$'(.(+0@O80@EfZ10@9�-7$�Af¯z-7O8;=Z1:<;)$)$'(WY682H;)08J>6G2C-+:CE}U#$�9'2408EH2 �1-79)68(T�#$'&�9�*w»'6@AF2�²�� �@�D�1 q �� � B �� D� Rn R5F@0@E�:<;'-.¶sAC08E
1K2K3K40[2
5Eb$�9'2#24$'&�E^&�2C-I(.-7E"E0@E^&�9�0}{l$'-IEN?L$'&�A�E0@:HAC-IAC0}&�9%9�$�JQa�AC0?6z:4-79�°5:<;'-.¶sAC08E
abcde2C08()°�&L0�(+0�9�$�JQa�AC0H?6�24AC$'-IEB:<;'-.¶sAC08E
abc0@E42N*�- y�-IE<-7a�(703?)61A
4K
bcd08EH2v*�- y�-IE<-7a�(70�?�6@A
5Kcde
08EH2v*�- y�-IE<-7a�(70�?�6@A3R �B&L0G(�0@E42�(+0V:<;'-.¶sAC0
ao
������������������ �"!�#$� ��%'&(��)*�,+.-/�0����#$132#��4�5+ �6��7�6198:2#*;�#'�<+ #=&:�*�>?>@����+ )*�A�B�,#*1&:�*�>?>@����+ )*�A�B�,#*1DC:87E
»)&)-7E�°�&L05*'-�y'-7E�0
bcdKT9�$'&�E%*)0by�$�9)E%6[y�$'-7A
d*�- y�-IE<-7a�(70>?�6@A
50G2�*�619)E�(+08E
:<;)$'- ®�?�$�E�EH-+a'(+08E^*�0}:<;'-I¶#AC0@E[K@*)$�9�:d = 5
R�»)&)-7E�°�&L04*'-�y'-7E�0
abcK868(+$�AFE^:H0 9�$�JQa�AC0
Eb0�2C0@ACJ=-I9�0%?�6@A2$'&
4K)Eb0G(+$�9Q°�&L0
b08EH2B?)6@-IAf$'&Q-IJ>?)6@-IA[R�»)&)-7E�°�&L0
3*'-�y'-7E�0
cdeK
6@(7$�AFE3*'-�y'-7E�0V(76�Eb$�J=JQ0
c + d + eKL0[2 *�$�9�:
c + e ≡ 1 (mod 3)]+:46@A
d = 5`<R
�f$�9�:f*�0G&D®�?L$�E<E<-7a�-.(I-�2�E0@EBE[� $'¶#AC0@9'2M?6V9�$'&�ESqZ8-
c = 2K86@(7$�AFE
e ≡ 2 (mod 3)6[y�01:
e ∈ {1K3K4} K@:H0}°�&)-'0@E42s-7J=?L$�E<E<-7a�(70�RZ8-
c = 4K�6@(7$�AFE
e ≡ 0 (mod 3)6Gy�0@:
e ∈ {1K2K3} K�*)$�9�: e = 3
R�f$�9�:
c = 4K1:H0�°�&)-'-IJ>?�(I-7°�&L0�°�&L0
b0@E42T?)6@-IAT6Gy�0@:
b ∈ {1K2} R��f$�9�: b
y'6@&�22K):H0V°�&)-�(768-7E<Eb0
a = 1Rs�f$�9�:�K
abcde = 12453K�0G2
a = 1R
FHGJIBKHL MBGONQPR:S KOT�TUM�V SOWYX I R N'M RZW\[$]0^ R I I_G*` ^ K WAa K R L bcPR L6K'dOM S N�L�eHLAGfd Rhg FHi�b7j_NHTZL M_L6K'LAGfk RZW6lmS `N�GOTonop g I q�Ksr SOW b [ G aOW G�tN�GJb g T6V S4a N R�u5v�S L S I M S n R T�r4bDPR L6K'dOM S N�L b5FswyxYz S'{ORZW MBKOT}|Zb�x S I RZl:~sS N a b [ p j N`d*GONQPR TUM R�u$R LO��I R��o��RZl G W G { b0PR L6K'dOM S N�L b �7M I I M SOl@[ rsGON�w SH�A�*R Ns�6M R�� GJI I R�a M S L R j N*TZL MBL�KHL R b�OG W GON�LAGJb�i vD]\ R ¹ 9,&�A49>:H$�9'2F6@-I9)EB2H;�AC0@03"%;'-�240'K�E<- ®%AC01*�K�619�*,{l$'&�Ava�(I6D:</�a)68(I(IE@R]76G`Q!l{N$�9�0,a)68(I(s-IE3Eb0G(+0@:C240@*X682 AF619�*)$�J�K�"%;�682f-7E}2H;)05?�AC$)a�6Da'-I(.-.2�MQ2H;�6G2f2H;)0,a)68(I(
Eb0G(+0@:C240@*5-7EfAC0@*5o
]la1`X!l{^2�"%$wa�6@(.(7E�61AC0=Eb0G(+0@:C240@*�6823AF6@9�*�$�J¤K#"%;�682�-IE32H;)0>?�AC$)a�6Da'-I(.-.2�M¤2H;�6G2}2H;)0bM61AC0Va�$D2H;=a'(761:</ o
]l:F`Q!l{^2�"%$wa�6@(.(7E�61AC0=Eb0G(+0@:C240@*�6823AF6@9�*�$�J¤K#"%;�682�-IE32H;)0>?�AC$)a�6Da'-I(.-.2�M¤2H;�6G2}2H;)0bM61AC0Va�$D2H;=a'(761:</�K)O[- y�0@952H;�6G2N2H;)0bM=61AC032H;)0VE<61J>0V:H$'(7$'&�A o
��������������������Q�,#$����#�����E�}#H�A��� %����Z;���# �61����4��+ #$�7�� �,� + #��5����1����������,�4�����-/#��J>?������ &�� ����7������41��5� ; �O����Q� 17�$�������}E
]+6[`P (AC0@*
) =� �"!$#&%('
) � )+*(, � �"!-�". )0/ �21 %3 =6
3 + 6 + 4=
6
13
R
]la1`P (2
a�(I6D:</) =
(
42
)/(
132
)
=6
78=
1
13
R]l:F`T��0V;�6[y�0
P (2a'(761:</ |2 E�6@JQ0 ) =
� �"!�4 , *"/65879*(: #�3� �"!-4 , *"/;5879*(: #�3=<>� �"!$#&%(' 79*(: #�3?<@� �"!�A8B : ) % 72*(: #C3
=
(
42
)
(
42
)
+
(
62
)
+
(
32
) =1
4
R
��I TZG\TZGJI {OR d ~ rED RZS N`Un S'{ M_d:koG*KJI R b�TZL6K'd R N�L b � PR�aHR V d R n W K lml GON�d { M I I R b�n W K lml GON�d { M I I R bF �su*v�S L S I M S n R T�r4bHTZL6K'd R N�L b*Fswyx7z SH{JRZW MAKJT�|�b�x S I RZl:~sS N a b�j_N�d*GON R TUM S4u*S N�d���I R��}��RZl G W G { bHTZL�KHd R N�L b�}M I I M SJl [ rsGON(w S'���HR N'�6M R�� GJI I R�a M S L R j_NHTZL M_L6K'L R b��JG W GON�LBGJb5i vD]p R�!�9Q2H;)0,*'-761OGAF61J¤K
ABC-7E�6,AF-+O8;D2HPC24AC-I619�O[(70
"�-.2H;AB = 3
619�*AC = 4
R��D&�A�2H;)0@ACJQ$�AC0'K0@61:<;X(I-I9�0=E�01OGJ>0@9'2
AiBi
-7Ez?�0@AC?L089�*�-7:4&)(76@A}2C$AC
KA1
a�-IEb0@:C2CEAC
K�619�*Ai+1
a�-IEb0@:C2CEAiC
R�D-I9�*>2H;)0%24$D2F6@(#(+089�O�2H;¤$'{s2H;)05Eb0@°�&L089�:H0,$'{s2H;)0*�-I6DO8$�9)68(�E�01OGJ>0@9'2FE�q
BA1 + B1A2 + B2A3 + · · ·��GIH��6���J��������������}E ...........................................................................................................................................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
......................................................................................................................................................................
A A1 A2 A3
B
B1
B2
B3
CZ8-79�:40
A1a�-IEb0@:C2CEz(.-79�0=E�01OGJ>0@9'2
ACK#2H;)0,(+089�O�2H;w$'{^2H;)0=Eb0@OGJQ089'2
AA1-IE
AA1 = 26@9�*=2H;)0%(70@9�O�2H;Q$'{#2H;)0%Eb0@OGJQ089'2
B1A1-IE
B1A1 = 32
RfS�;)0%(+089�O�2H;Q$'{2H;)0VEb0@OGJQ089'2
BA1-IE O[-�y�089Qa@M�2H;)0�S#;)0@$�AC0@J¦$'{N»'M'2H;�61O8$�AF61Ef6@E
BA1 =
√
BA2+ AA
2
1 =√
32 + 22 =√
13R
��61:<; $'{f2H;)0¤24AF-76@9�O[(+08EBiAiAi+1
-7E,EH-7J,-I(I61Az24$x2H;)0¤24AF-76@9�O[(+0BAA1
K�"�-�2H;_6@(.(*�-IJQ089)E<-7$�9)EzAC0@*�&L:401*�a@M¤65{+6D:C2C$�A�$'{ 1
2
6G2�0@61:<;XEH2C0@?�R�S#;'&�E[K�2H;)0%24$D2F6@(s(70@9�O�2H;¤$'{
©
2H;)0VEb0@°�&L089�:H0%$'{#*�-I6DO8$�9)68(�E�01OGJ>0@9'2FEf-IE√
13 +
√13
2+
√13
22+ · · · +
√13
2n+ · · ·
=√
13(
1 +1
2+ · · · +
1
2i+ · · ·
)
=√
13
(
1
1 − 1
2
)
= 2√
13R
��I TZG\TZGJI {OR d ~ rED RZS N`Un S'{ M_d:koG*KJI R b�TZL6K'd R N�L b � PR�aHR V d R n W K lml GON�d { M I I R b�n W K lml GON�d { M I I R bF �su*v�S L S I M S n R T�r4bHTZL6K'd R N�L b*Fswyx7z SH{JRZW MAKJT�|�b�x S I RZl:~sS N a b�j_N�d*GON R TUM S4u*S N�d���I R��}��RZl G W G { bHTZL�KHd R N�L b�}M I I M SJl [ rsGON(w S'���HR N'�6M R�� GJI I R�a M S L R j_NHTZL M_L6K'L R b��JG W GON�LBGJb5i vD]_ RTS#;)0%0@°�&�6G2C-+$�9
x2 − 3x + q = 0;�61E^2�"%$%AC086@(�AC$)$D2FEGKα6@9�*
βR���9�$1"�-I9�Of2H;�682
α3 +β3 = 81K1��9�*V2H;)0fy'6@(.&L0z$'{
qR
-h�6����� !+2�-7E a)0@E42v9�$D2N24$�&�Eb032H;)0%°'&�6D*�AF682C-7:B{l$�ACJ=&)(I6LR��������������I����� ���6���A��� & #(��1J���4�5+ #����617�����Y��)#$�Z�Z�2���$1 �����,#$>I��� ��;1 �U��+ ���Q# �O�6��E
��0w;�6Gy�0α + β = 3
619�*αβ = q
R�Z8-79�:40α3 + β3 = 81
K�"%0w;�6[y�0(α + β)3 − 3αβ(α + β) = 81
$�A27 − 9q = 81
K1"%;'-7:<;>O[- y�0@Eq = −6
R��I TZGmTZGJI {JR d ~ r v�S L S I M S n R T�r4b�TZL�KHd R N�L b�Fswyx(z SH{JRZW MAKJT�|�b0x S I RZl:~sS N a b0j N�d*GON R TUM S
(T RO� GON�dTZGJIAK'L M_GON
)u D RZS N�`Un SH{ MBdokoG*KJI R b*TZL�KHd R N�L b � PR�a'R VYd R n W K lml GON�d { M I I R b�n W K lml GON�d { M I I R b2F �su'S N�d0�5I R��
��RZl G W G { b�TZL�KHd R N�L b��7M I I M SOl [ r'GONYw SH�A�*R Ns�6M R�� GJI I R�a M S L R j N*TZL MBL�KHL R b��OG W GON�LAGJb5i vD]a R ¹ {l$'&�A4P�*'-+O[-�2T9�&�J>a�08As"%;'-7:<;�-7EN6�?�0@AF{l01:C2TEb°'&�61AC0 -7E�:HAC08682C01*%a@Mz"VAC-�2C-79�O ¹ 9)9�0�� E6DO80}-I9�M�0861AFE^{l$'(I(7$1"%01*%a@MVS�$�J¤� E�61O80}-I9�M�0861AFE8R^Z8-7J,-I(I61AF(�M�K@-I9
31M�0@6@AFEGK82H;)0G-7AT61O80@E
-79%2H;)0zE�6@JQ0z$�AC*�08A�"�-I(.()6DOG68-79�{l$�A4J�63{l$'&�AHP<*�-7O[-.2N?L08AC{l0@:C2�E�°�&�6@AC0�Rf�f0[2408A4J,-79�0�2H;)0?�AC08Eb089'2B61O80@E $'{ ¹ 9)9�0�619�*,S)$�JxR�����)# + �����\��#�����#$>@���s�5)�1����4��+ #$�7�61�� �A�B�A��� > ������������H���#����J�,# 8Q��B�,#*;�������#���������,�4����#$1�����O���7��� 1 ��� R
F@0G2 ¹ 9)9�0�� E�61O80�a�0a619�*=S)$�J�� E�61O80�a�0
tR&F@0G2
a;�6[y�0
m*'-+O[-�2CE 619�*=(+0[2
t;�6Gy�0n*'-+O[-�2CE8Rf��0�/D9�$1"±2H;�6G2
m + n = 4R3!l{
m < 2K)2H;)089
n ≥ 3R�²�&�2B-79
31M�0@6@AFEGKaJ,&�EH2fa)0%6
2B *�-7O[-.2B9�&�JQa)0@A�RfS�;'&�EGK�2H;)0%E�01:4$�9�*
4B *�-7O[-.2B9�&�JQa)0@AN"%$'&)(+*
;�6Gy�0�682N(70@6@EH25*'-+O[-�2CE[K�6@9,-IJ>?�$�E�EH-+a'-I(.-.2�M�Rv²�M,EFM)J=JQ0[24A�M#K@"%0V:46@9)9�$D2B;�6[y�0
n(70@E<E
2H;�6192K1"%;'-7:<;,JQ08619)E
m = n = 2RF@0G2)2H;)0BE�°�&�6@AC0@E�a�0
x2 619�* y2 -I9}2H;�682�$�AC*�08AbK�"%;)0@AC0 0 < x < yR�Z8-I9�:H0fa)$D2H;
Eb°'&�61AC08E�;�6[y�04*�-7O[-.2FEGK�"%0};�6Gy�0
y2 = x2+3131KD$�A
(y−x)(y+x) = 31×101R
Z8-79�:400 < x < y
KB"%0xO80G2y − x = 31
619�*y + x = 101
RS�;'-7E5M'-+0G(+*�E(x, y) = (35, 66)
Rw¯V089�:H0'Kx2 = 1225
619�*y2 = 4356
Ks"%;'-+:<;¤-7J=?)(.-+08Ez2H;�6G2¹ 9)9�0�-7E
12619�*5S�$�J�-7E
25R
��I TZG�TZGJI {JR d ~ r v�S L S I M S n R T�rJb*TZL�KHd R N�L b$Fswyx}z SH{JRZW MAKJT |�b�x S I RZl:~sS N a b�j_N�d*GON R TUM S4u$S N�d�D RZS N�`n SH{ MBdmk�G*K4I R b�TZL6K'd R N�L b � PR�a'R Vhd R n W K lml GON�d { M I I R b n W K lml GON�d { M I I R b�F �]
S#;�6G2Va)AF-79�OGE%&�E�24$¤2H;)0¤0@9�*u$'{B619�$D2H;)08Az-IE�EH&L0�RwS�;'-7E%JQ$�9'2H;�� E�"�-I9)9�0@AFE�$'{6 ?�6@EH2��N$'(.&�JQ0 $'{%n��Dk\mL�[�,�Dklj��<� ��n���!@mL��� 61AC0xµ�08619�P[�B6Gy'-+*³¯V$'&)(70�K � 6G2C68(I-I6�f0@EFM�K^6@9�* ¹ (70<®u�#08JQ$�AC$8y�R¤Us$�9�OGAF6G2C&)(76G2C-+$�9)E[Ksµ�08619�P[�B6Gy'-+*�K � 682F6@(.-76�KT619�* ¹ (+0H®�RUs$�9'2C-I9�&L0VEb089�*�-I9�Oz-I9�M�$'&�AB:4$�9'2408EH2FE 6@9�*=E�$'(I&�2C-7$�9)E@R
¨
� ��� � � ���� � �� � ���� � �
WY682H;)08J>6G2C-+:C6@(^WY6[M�;)08J a)01OG6@9>-79 � � ¨D¨ 61E���%���� c�h �� � d ��:�� 8�� j ����:�� 8 j � ��� i��*����� bCd $8$8":�� ����� ��� g c�j�� � � ��b ����� c ��� � Rf!+2f:4$�9'2C-79�&L08EGKD"�-�2H;52H;)0VE�6@JQ0%08J>?�;�61EH-7E[K61Ef619,-I9'240@OGAF6@(L?�6@AF2B$'{NcD~[�[�,n��Dk\mL�[�,�Dklj����@~��8�¥hBj+k�m>n��Dk�mL���,�Dk�jl�<� �sn���!8mL�[��R
S#;)05WY6GM);)0@J«��*�-�24$�AN-IE�µ�0G¶x¯�$)$�?L08Av] ¹ :C6D*'-76%� 9�- y�[email protected]�MD`<RTS#;)0V$D2H;)0@A�E42C68¶JQ08JQa)0@AFEQ6@AC0¤µ�$);�9�NAF6@9'2¤W_: F@$'&LO8;'(.-79�]+� 9�-�y�08AFE<-�2\M¢$'{ � 0["£²LAF&�9)E4"�-7:</8`4Kz!�619�^619�*)0@Ab²�&�ACO8;x]l� 9�- y�[email protected]�Mw$'{N�¤6G2408AC(7$)$1`HK F861ACA�M_�#-7:H0>]+� 9�-�y�08AFE<-�2\Mw$'{N�¤6G2408AC(7$)$1`HK619�*±�#$�9 F86@9�:46@EH2C0@A>]l� 9�- y�[email protected]�M�$'{3S�$�AC$�9'2C$1`HK%�'AC-7:x�#$)a)0@A�2X] F@01$�¯�6[M�0@Ew¯3-+O8;Z1:<;)$)$'(\K@�'AC0@*�08AC-7:C24$�9D`4KDW_$�9�-7/@6 ��;)a�0G-7E�]\�D682H;)08ATWu-7:<;�6D0G(@�$)0[2F¼#Z101:4$�9�*)6@A�MzZ1:<;)$)$'(\KWu-7E<E<-IE�E<6@&LOG6[`HKNWY61A4/u²LAC0@*�-I9x]+ZG2[R�µ�$);�9#� E�PG��6[y�0@9)E�:H$'&�A�2�Z1:<;)$)$'(�Ks�X-I9)9�-7?�01Ob`4KT6@9�*�#$�9 F86@9�:46@EH2C0@Av]l� 9�- y�[email protected]�M>$'{�S)$�AC$�9'24$1`<R
!#"%$'&�( )+*�,.-0/�&�(21
3�465�78�598�5 :�;=<?>�@AB8�>C4D@?EGF6>H:�8IED>JELK�5NM�AO>�P465 QR8SFG:TELK?F�8U5;HF6EGF6>H:TPH<WV#X6Y�Z?[=\C]H]?^�_` >C4D@�EGF6>H:?8IAO5ab5�F6cC5;d7e6ED5 AUELK?F�8#;f7�ED5=ghF 4 4�>H:?4G<TP�5=ab>H:�8bFL;i5�AO5C;JF�eBELK�5 AO5IF�8jEGFGQU5=P�5�eD>HAO5Mk@�P4 F6a67�EGF6>H:U>elELK�5m8 >C4D@�EGF6>H:?8_
nf7�aoKpM�AO>�P465�QqF�8srCF6cC5�:SFG:Un?:frC4 F�8tKR7H:i;UuHAO5�:�avKfwxELK�5B>yIaDF�7C4i4�7H:fr�@7Cr�5�8�>ekz�7H:?7?;f7i_{L:WF�8|8�@�5�89}�w�~�wm��w�7H:�;0�|w�nf:?rC4 F�8|KdghF 4 4kM�AO5ab5;i5UufAO5 :�aoK?w�7H:i;#FG:WF�8O8C@?5 8S��w��w���w�7H:i;=� wufAO5 :�aoKUghF 4 4�M�AO5Hab5C;i5�n?:frC4 F�8tK�_
�K�5�5C;HF6ED>HA�ELK�7H:��H8��C5 7H:��|��7HAOas�5�AvAoF65 A�7H:i;R��7HA|EGFG:N�k>C4L;f8�E65�FG:R>e�ELK�5p��:fF6c5 AD8OF6EG<p>e�0>H:�ELAO5�7C4�e6>HA�ELAD7H:�8b4�7�EGF6>H:�8�>elELK�5�M�AO>�P465 QR8C_
I \ � s R �0�O���o����#+ ��� �Q���� �c����#���;����0�1D8-����� � �,+5��1 �7�s##�HJ# R!�9XAC0@:C2C6@9�O[(+0
ABCDK�?�$'-79'2FE
E6@9�*
F*�- y�-7*�0%EH-+*)0
DC-79'2C$,2H;�AC0@0501°'&�6@(
?�6@AF2FEDE = EF = FC
619�*%?�$'-79'2FEG619�*
H*�- y�-7*�0}E<-7*�0
BC-I9'24$z2H;�AC0@0301°'&�6@(
?�6@AF2FEBG = GH = HC
R^S�;)0�(I-I9�0AE
:4&�2CEB2H;)0z(I-I9�0@EDG
6@9�*DH
6G2v?L$'-I9'2CEK6@9�*
LK'AC0@E<?L0@:C2C-�y�0G(�MsRNZ8-7J,-I(I61AF(�M�K12H;)0z(.-79�0
AF:4&�2FEv2H;)0z(.-79�08E
DG619�*
DH6G2
?L$'-I9'2CEM
619�*NK�AC0@E<?L0@:C2C-�y�0G(�MsRNZ1;)$1"�2H;�682
KN ‖ CDR
I \ � � R �0�O���o����#+I���>� +��:��s+/!E��Q� � ��#$� ��1R�}�5�,)#$�6�O����� ��� ����O���7��� 1 ����O���7��� 1��� R
F@0G2f(n, k)
a)0V2H;)0�9�&�JQa)0@A $'{�"V6GM�E3$'{T*'-7E424AC-7a�&�2C-I9�Ok:C619�*'-+08E 24$
n:<;'-.(+P
*)AC089>E�$�2H;�6G2B0@61:<;=:<;'-I(7*=AC0@:H0G-�y�08E 6G2vJ>$�EH2^2�"%$=:C619�*'-+08E@Rv��$�Av0H®@61J=?)(70�K'-.{n = 3
K2H;)0@9
f(3, 7) = 0Kf(3, 6) = 1
K�619�*f(3, 4) = 6
R �f0[2408A4J,-79�032H;)03y'68(I&L0V$'{f(2007, 1) + f(2007, 4) + f(2007, 7) + · · · + f(2007, 4012)
RI \ � � R �0�O���o����#+ ���h!E�� ������#��=��� �JH;��1 �h�C��#$�-�s1 � �m1��o� � R
�D-I9�*>EH- ®8240@0@916B *'-+O[-�2f?)6@(.-79�*�AC$�JQ08EGK�-I9Q0@61:<;>$'{L"%;'-+:<;52H;)0V?�AC$)*'&L:C2 $'{�2H;)0
9�$�9�P4¼<08AC$�*'-+O[-�2CE3619�*Q2H;)0,E<&�J·$'{�2H;)0=*�-7O[-.2FEz6@AC0=a�$D2H;¤01°'&�6@(s24$16R=¯�$1"�J>6@9DM
E<&L:<;,9�&�JQa)0@AFE 6@AC0z2H;)08AC0"o
�
I \ ��� R �0�O���o����#+ �����\E �mEA��E0� �2���U�;�1��Q� ��; ���,���O#$1����Q+ �6� R�f0G2C0@ACJ=-I9�0w6@9 -I9��L9�-�240�Eb0[25$'{ AF682C-7$�9)6@(f9�&�JQa)0@A�E�$'(I&�2C-7$�9)E
(α, β)24$u2H;)0
01°'&�682C-7$�9α2 + β2 = α3 + β3 R
I \ �&^ R �0�O���o����#+ ��� �C�Q# �����9��#$> ������� R¹ 9%01°'&)-I(I682C0@AF68(124AF-76@9�O[(+0f(.-+08ET-79z2H;)0 ?)(I619�0v"�-.2H;�2�"%$V$'{D-.2FEsy�0@A�2C-+:40@E^682s?L$'-I9'2CE
(0, 0)619�*
(0, n)R��f0G2C0@ACJ=-I9�0V2H;)0�9�&�J>a�08A $'{�?�$'-79'2FE
(x, y)"�-�2H;=-I9'240@O80@A :4$)$�AHP
*�-I9)682C0@E�"%;'-+:<;5(I-70z-7952H;)0z-79'2C0@AF-+$�Av$'{L2H;)0324AC-I619�O[(70�RI \ � n R �0�s���o���$# + ��� �Q�*��HJ# �9� ���=��#��41�� #�>@���Z�����S�7����)#��0�s�,�C� ����� #������0���+0%�6���+�17���4E !�������� �s1 ����E
¹ Eb°'&�61AC0�$'{�E<-7*�03(70@9�O�2H;s-7Ev-I9)Eb:HAF-+a)01*,EFM�J>J>0G24AF-+:C6@(.(�M�-I9)E<-7*�0z6�Eb0@:C24$�A�$'{�6
:4-7AC:4(70�"�-�2H;5AF6D*'-I&�Ef$'{L(+089�O�2H;r619�*=:40@9'24AF68(L6@9�O[(+0V$'{
60◦ K'E<&L:<;%2H;�682^2�"%$�y�0@A�2C-+:40@E(I-70�$�9,2H;)0%EH24AF68-+O8;D2BEH-+*)0@E}$'{�2H;)0%Eb0@:C24$�Av619�*,2�"%$�y�08AF2C-7:H08E (.-+0%$�9,2H;)0�:4-7AC:4&)(I61AB6@AC:$'{L2H;)0VE�01:C2C$�A[Rf�f0[2408A4J,-79�032H;)0%0H®@6D:C2^y'68(I&L0V$'{
s/rRR R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R RI \ � s R �0�O���o����2#:��� �=�Q���� �c����#���;����0�1D8-����� � �,+5��1 �7���#�HJ# R
�B619)EB&�9=AC0@:C2C6@9�O[(+0ABCD
$�9>*'-�y'-7E�0z(70V: �$D2�E0DC
0@9524AC$'-IEf?)61A�2C-+08E�E01OG68(+08EDE = EF = FC
0G2v(+0%: �$D2�E0BC
089=24AC$'-7E ?�6@AF2C-70@E E01OG68(+08EBG = GH = HC
RF86w*�AC$'-.2C0AE
:4$'&�?L0X(70@E5*�AC$'-.2C0@EDG
0G2DH
6@&D®�?L$'-I9'2CE�AC0@E<?L0@:C2C-I{+EK0G2
LR
�f0VJ=619�-@?0@AC0�619)68(+$)O[&L0'K'(76%*�AC$'-.2C0AF
:H$'&�?�0�(70@E *�AC$'-.2C0@EDG
0[2DH
68&D®%?L$'-I9'2CEAC0@E<?L0@:C2C-I{+E
M0G2
NR#W_$�9'24AC08AB°�&L0
KN ‖ CDR
I \ � � R �0�O���o����2#f���� � +��:��s+y!E �Q�� ��#����1R�}�5�,)#$�6�O����2# + # �����s�����,� 1 ����O���7��� 1��� R
Z1$'-.2f(n, k)
(70V9�$�J>a)AC0%*)0V?L$�E<E<-7a�-.(I-�2�E0@Ef*)0%*�-IEH24AF-+a'&L0@Aka�$�9�a)$�9)E ?6
n0@9�P
{+619'2FEGK�*)0>E�$�AF2C0>°'&L0Q:<;�61°�&L0Q089�{+619'2z089wAC0�:<$'-�y�0=6@&x?�(I&�EV*)08&D®LRv»'61A30H®10@J=?)(70�KsEH-n = 3
K@68(+$�AFEf(3, 7) = 0
Kf(3, 6) = 1
0G2f(3, 4) = 6
R)� E0G2C0@ACJ=-I9�0@As(76By'6@(708&�AT*)0f(2007, 1) + f(2007, 4) + f(2007, 7) + · · · + f(2007, 4012)
RI \ � � R �0�O���o����2#:��� �}!E�� ������#��=��� �JH;��1 �h�C��#$�-�s1 � �m1��o� � R
S�AC$'&'y�0@AzE�08-�¼<0>?)6@(.-79�*�AC$�JQ08EGKT:<;�61:4&�9u:H$�J=?�AC089)619'2zEb0G-.¼<0>:<;'-I¶#AC0@E[KT*�0,240G(I(70Eb$�A�240�°'&L0z(70z?�AC$)*'&)-.2�*�08Ef:<;'-I¶#AC0@EB9�$�9,9�&)(IEf0G2^(76�E�$�J>J>0�*)0@Ef:<;'-.¶sAC08Ef*�0�:<;�6D°'&L0?�68(I-I9�*)AC$�J>0 E�$'-+089'2L2C$'&�E^*)08&D® E01OG68&D® ?6fEb0G-.¼<0)R8U#$�JQa'-+089�Mz61P42HP4-I(1*)0v240G(7E�9�$�JQa�AC0@E[oI \ ��� R �0�O���o����2#:��� ���\E �mEA��E0� �2���U�;�1��Q� ��; ���,���O#$1����Q+ # R
� E0G2C0@ACJ=-I9�0@A�&�9z0@9)E�0@J>a�(70v-I9��L9�-@*)0BE�$'(I&�2C-7$�9)Es0@939�$�J>a)AC08E�AF6G2C-+$�9)9�0G(7E(α, β)*�0�(���E0@°�&�6G2C-+$�9
α2 + β2 = α3 + β3 RI \ �&^ R �0�O���o����2#:��� �D�;� 2��Q���_�o# + # � �����Q#�> R
�B619)Ev(+0z?)(I619#KD&�9524AF-76@9�O[(+0�E0@°�&)-.(76G2�E0@AF68(L6V*�0G&D®�*�0zEb08EfE�$�J>J>0G2FEB6@&D®V?L$'-I9'2CE(0, 0)
0G2(0, n)
RD� E0G2C0@ACJ=-I9�0@A�(+0v9�$�JQa�AC0f*)0B?�$'-79'2FE(x, y)
?6f:4$)$�AC*�$�9)99E0@0@E�0@9'2C-@?0@AC08EE<-�2C& E0@E�?6�(\� -79'2�E08AC-708&�AB*'&,24AC-I619�O[(70�R
� �
I \ � n R �0�O���o����2#D��� � �Q�*�$HO#:�9� ���=��#��41��}�5�,)#$�6�O����2# � #�>@���Z����� + # ��#��J�O#�% � #5�5)#$1���4E !�������� �s1 ����E
� 95:46@A4A�E0z*)0z: �$D2�E0s08EH2�-79)E�:HAC-�2^EFM�JGE0[24AC-7°�&L08JQ089'2�*�619)EN&�95E�01:C2C08&�A^*�03:H08AC:4(+0
*�0fAF6[M�$�9r0G2#*#� 619�O[(70 68&V:H089'24AC0 *�0
60◦ KG*�0�240G(I(70fE�$�AF2C0f°�&L0 *)08&D®}*�0fE�0@E�E�$�J>J>0G2FEEb$'-70@9'2VE<&�A�(70@E5?)61A�2C-+08E5AC01:C2C-.(I-7OG9�0@E,*'& Eb0@:C240G&�A%0G2V(70@E5*�0G&D®x68&�24AC0@E5EH&�AV(�� 6@AC:5*�0:H08AC:4(+0)Rs� E0G2C0@ACJ=-I9�0@A�(I63y'6@(708&�Av0<®@61:C240V*�0
s/rR
!W" $'&�( �j,0/������?,��'1
I \&\�s R �0�O���o����#+ ���h!������ 85���Z�6����A1 ���> �,�����J�s1���8 R¹ 9'24$�9�-I9�$Q;�6@E}6,*�AF6G"%08Av{7&)(I(s$'{s-7*�089'2C-+:C6@(sa'(761:</5Eb$):</1E}6@9�*>-7*�089'2C-+:C6@(�"%;'-�240
Eb$):</1E@Rz!l{^;)0�"%0@AC0V24$>E�08(701:C2v2�"%$>Eb$):</1E�6G2 AF6@9�*�$�J¥{+AC$�J ;'-7E3*�AF6G"%08AbK)2H;)0�?�AC$)a)P6Da'-I(.-.2�M=2H;�682v2H;)0bMXJ=682C:<;,"%$'&)(+*Xa)0 1
2
R�¯V$1"iJ=619DMX$'{^086D:<;Q:H$'(7$'&�A $'{TE�$):</=*)$)0@E¹ 9'24$�9�-I9�$=;�6Gy�0Co,]IS#;)08AC0V-IEfJQ$�AC032H;�619>$�9�0�619)EC"%0@A�R `�����)# +>�������&H;��� �O+ �4E7-/#(��s1 �7�� H;�Q� �����,���0#��O+ # �s1\8 �m1��}� � R
F@0G2 2H;)0@AC0>a�0ba�(I6D:</Q619�*
w"%;'-.2C0=Eb$):</1E@RtF@0[2
Pb
a)0,2H;)0=?�AC$)a�6Da'-I(.-.2�Mw$'{O80G2F2C-I9�O>6Xa'(761:</¤?)6@-IA�$'{vE�$):</DE[KT619�*
Pw
a�0,2H;)0Q?)AC$)a)61a�-.(I-�2\Mx$'{BO80G2F2C-I9�O=6="%;'-�240?�68-7A�K1"%;)0@952�"%$,E�$):</DEf6@AC0%*�AF6G"V9=6G2vAF619�*)$�JxRTS#;)089
Pb =b
b + w· b − 1
b + w − 1
619�*Pw =
w
b + w· w − 1
b + w − 1
RZ8-79�:40
Pb + Pw = 12
K1"%0V;�6[y�0b
b + w· b − 1
b + w − 1+
w
b + w· w − 1
b + w − 1=
1
2
K"%;'-+:<;,E<-IJ>?�(I-.��08Ev2C$
2[
b(b − 1) + w(w − 1)]
= (b + w)(b + w − 1)K
2b2 − 2b + 2w2 − 2w = b2 − b + w2 − w + 2bwK
b2 − 2bw + w2 = b + wK
(b − w)2 = b + wR
²�M_EFM�J>J>0G24A�M�K^"%0¤J>6[M_6@E�EH&�JQ0Q2H;�6G2b > w
R F@0[2d = b − w
R S#;)0¤6Da)$@y�001°'&�682C-7$�9XEH-7J=?)(.-I�L0@Ef2C$
d2 − d − 2w = 0K�$�A
w = 12d(d − 1)
K�"%;'-+:<;QO[-�y�08E}&�Eb = w + d = 1
2d(d + 1)
R� ��+ � S�;)0�E�$'(I&�2C-7$�9)Ep{l$�A
(b, w)6@AC0�?)6@-IAFE³$'{u:4$�9)Eb0@:4&�2C-�y�0±24AF-76@9�O[&)(76@A
9�&�JQa)0@AFE r�2H;�6G2=-IEGK}2H;)0bM�6@AC0Y?�68-7AFE¤$'{�:H$�9)E�01:4&�2C- y�0x2408A4J=EQ{+AC$�J£2H;)0_E�01°'&L0@9�:40{
n(n + 1)
2
}∞
n=1
= 1K3K6K10K15K. . .
R ��I TZGoTZGJI {JR d ~ rDn S N'M R I5�ZT S M b�TZL6K'd R N�L b�� S M V R M'� l:RZW M �AS NmF � q�GHGJI b � S M V R M b�� S M � S N ]
�A�
I \&\ � R �0�s���o���$# + ��� �o#$� �Q#��C����E � �B�A�B�6�> �'1f8���Z�,# �,��� �7����)#��0�s�,�C��1 �(�Z�����:��1��� R
F@0G2Na�0,6=?�$�E<-�2C-�y�05-I9'240@O80@A}EH&L:<;Q2H;�682
N(+086Gy�08Ez6=AC0@J=6@-I9�*�08A3$'{
2$�A
4"%;)0@9=*�- y�-7*�0@*=a@M66@9�*�2H;)08AC0�61AC03-79'2C01O808AFE
x619�*
yE<&L:<;%2H;�6G2
N = x2 + 27y2 R»�AC$@y�052H;�682 2H;)0@AC0>0<®@-7E42}-79'2C01O808AFEa6@9�*
b"�-�2H;
N = a2 + 3b2 "%;)0@AC0 b-7Ez9�$D2
*�- y�-IE<-7a�(70Va@M3R
�������������� ����� #(��O�6�9� 8Q��A��#$;5#��0�s�����#�> �5��,)*�6��; �7�O�0���Q1 � #(��O�6�9� 8Q��A��#$;5#$1�7�4��7�C��� > 1 � �m1��o� �m1�>@��+ ���7#+ ��� �C�Q#/#+ ������� R
F@0G2x619�*
ya)0�-I9'240@O80@AFE3EH&L:<;=2H;�682
N = x2 + 27y2 R���0�61AC05O[-�y�089Q2H;�6G2N ≡ 2 (mod 6)
$�AN ≡ 4 (mod 6)
R S#;)08AC08{l$�AC0'KN
-IEQ0by�0@9�6@9�* -7E>9�$D2*�- y�-IE<-7a�(70wa@M
3R�Z8-79�:40
N-IE=0by�089#K
x2 619�* 27y2 J,&�EH25;�6[y�0X2H;)0wE<61J>0�?�6@AC-�2\M]la)$D2H;�0by�089w$�A}a)$D2H;¤$)*�*1`HK�"%;'-+:<;Q-IJ>?�(I-70@E}2H;�6G2x6@9�*
y;�6[y�0�2H;)0,E<61J>05?�6@AC-�2\MsR
Z8-79�:40N-7Ev9�$D2�*�- y�-IE<-7a�(70�a@M
3K@"%0zE�010}2H;�682
x-7Ev9�$D2�*�- y�-IE<-7a�(70za@M
3]+a�0@:468&�Eb03-.{
x"%0@AC0V*�- y�-IE<-7a�(70Va@M3KD2H;)089,2H;)0�E�6@JQ0�"%$'&)(7*>a)0z24AF&L0�{l$�A
x2 + 27y2 = N`<R
�f08��9�0a = 1
2(x+9y)
619�*b = 1
2(x−3y)
R^Z8-79�:40x619�*
y;�6Gy�0 2H;)0�E�6@JQ0
?�6@AC-�2\M�K1a�$D2H;x+9y
6@9�*x−3y
61AC0}0by�0@93r[2H;)0@AC0G{l$�AC0�Ka619�*
b6@AC0 -I9'240@O80@AFE8R � $D240
2H;�682a2 + 3b2 = N
619�*V2H;�682b-7Ev9�$D2N*�- y�-IE<-7a�(70za@M
3]7E<-I9�:H0
x-7Ev9�$D2N*�- y�-IE<-7a�(70za@M
3`<R^S#;'&�E[K
a6@9�*
b61AC0V0<®@6@J>?�(+08E}$'{�2H;)0VAC0@°�&)-IAC01*,-79'2C01O808AFE@R
�Jq RZW�R � RZW�R N�GmGHL q RZW TZGJIBKHL MBGONHT T4K ~sl M_L�L R d ]I \&\ � R �0�O���o����#+ �����\E �mEA��E0� �2���U�;�1��Q� ��; ���,���O#$1����Q+ �6� R
]+6[`�S#;)03¼<08AC$)0@Ef$'{�2H;)0�?�$'(�M�9�$�J=-I6@(P (x) = x2 − 5x + 2
6@AC0�?�AC0@:4-7E�08( M�2H;)0*�-IJQ089)E<-7$�9)E3$'{�6�AC01:C2F619�O[(70%-79X:H089'2C-7J>0G24AC08E@R��f0G2C0@ACJ=-I9�0V2H;)0�?L08AC-IJQ0[2408Af619�*=2H;)061AC086�$'{L2H;)0�AC01:C2F619�O[(70�R
]la1`^S�;)0z¼<08AC$)0@E $'{L2H;)0V?�$'(�M�9�$�J=-I6@(P (x) = x3 − 70x2 + 1629x − 1260061AC03?�AC0@:4-7E�08( M%2H;)0�-I9)9�0@A�*'-7J>0@9)EH-+$�9)EB$'{�6�AC0@:C2C6@9�O[&)(76@ANAC$)$�Ji-795J>0G24AC08E@R��D-I9�*%2H;)0
24$D2F6@(NE<&�AF{+6D:40Q6@AC0@6Q619�*�2H;)05y�$'(I&�J>0X$'{N2H;)0=-79'2C0@AF-+$�Az$'{N2H;)0QAC$)$�J�]�"%;)0@9u*�$)$�AFE619�*�"�-I9�*�$1"VEf6@AC0%:4(7$�Eb0@*D`<R�������������� ��� ��$HO# � �\E � �����s#*1 ���4��+ #$�7�61 �Y��;�#H�,� ��������#T�}�5�,)#$�6�O�����1D�� ���Y��;�#H�,� 1� m1��o� � R
]+6[` F@0[2l619�*
wa)0 2H;)0z*'-7J>0@9)EH-+$�9)E�$'{'2H;)03AC0@:C2C6@9�O[(+0}-I95:40@9'2C-IJQ0[24AC0@E8R^Z8-79�:40
"%0%/19�$1"�2H;�682l619�*
w6@AC0z2H;)03¼<0@AC$)08E $'{
P (x) = x2 − 5x + 2KD"%0V;�6[y�0
P (x) = (x − l)(x − w) = x2 − (l + w)x + lwR
S#;'&�E[Kx2 − 5 + 2 = x2 − (l + w)x + lw
RS#;'-IE%-IJ>?�(I-70@Ez2H;�682
l + w = 56@9�*
lw = 2rT2H;)0@AC0G{l$�AC0�KT2H;)0Q?L08AC-IJQ0[2408AV$'{�2H;)0
AC01:C2F619�O[(70�-7E2(l + w) = 2(5) = 10
:HJ�K�6@9�*52H;)0V6@AC0@6�-IElw = 2
:HJ 2 R
� �
]la1` F@0[2lKwK8619�*
ha)0B2H;)0}*�-IJQ089)E<-7$�9)EN$'{12H;)0 AC0@:C2C6@9�O[&)(76@A�AC$)$�J�-79�JQ0[24AC0@E8R
Z8-79�:40lKwK'619�*
h61AC032H;)0z¼<08AC$)0@Ef$'{
P (x) = x3 − 70x2 + 1629x − 12600KD"%0
;�6Gy�0
P (x) = (x − l)(x − w)(x − h)
= x3 − (l + w + h)x2 + (lh + lw + wh)x − lhwR
S#;'&�E[K
x3 − 70x2 + 1629x − 12600
= x3 − (l + w + h)x2 + (lh + lw + wh)x − lhwR
S#;'-IEf-IJ>?�(I-70@Ev2H;�6G2lh + lw + wh = 1629
619�*lhw = 12600
r12H;'&�EGKD2H;)0�E<&�AF{+6D:4061AC0865$'{�2H;)0V-I9'2408AC-7$�Af$'{�2H;)0%AC$)$�J -7E
2(lw + lh + wh) = 2(1629) = 3258J 2 K619�*52H;)0�y�$'(I&�J>0�-IE
lwh = 12600J 3 R
��I TZGDTZGJI {JR d ~ r7��I V RZW7�HS rJb���� lmS Nhx W M {ZS L R F � q�GHGJI b�� S rJT RZW M b �'K W6�*R r u S N�d � M � q SJW d:j ] k R T�TUb��S N � q�Gmx S I_GOT�� RZW d R TUb � �5b��FZ� ]I \&\ � R �0�s���o���$# + ���>��+��:� �O+/!E �Q�� ��#����1S�7����)#��0�s�,�C�?�������O���7��� 1�����O���7��� 1 ���yE
¹ 9u0@°�&)-.(76G2408AF6@(T24AF-76@9�O[(+0>E<-�2CE�682C$�?x6QEb°'&�61AC0>6@E�-79¤2H;)0*�-I6DOGAF6@JxR ¹ (I(LE<-7*�08E};�6[y�0V(70@9�O�2H;
1R ¹ :4-7AC:4(70V?�6@E�E�0@Ev2H;�AC$'&LO8;
?L$'-I9'2CEPKQK�619�*
RRT��;�6G2�-7Ev2H;)0�AF6D*'-I&�E $'{L2H;)0%:4-IAC:4(+0Co
..
.
..
..
..
..
.
..
..
.
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.....................................................
..............................................................................................................................................................................................................................................
......................................................................................................
.......................................................................................................................................................................................................................
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...............................................................................................................................................
P
Q R
ST
��������������I���y�9�����:�5+5���,% H;�Q� 1��*� � ��� �� � �D85�����1 ���H� ����1�!H�$��� � RF@0G2
Oa)0�2H;)0=:40@9'24AC0=$'{�2H;)0=:4-IAC:4(+0'K#619�*
(+0[2ra)0�-�2CE�AF6D*'-I&�E8R3S�;)05?L08A4?�0@9�*'-+:4&)(I61Af{+AC$�J
?L$'-I9'2P2C$ E<-7*�0
QRa'-7E�01:C2FEQE�01OGJ>0@9'2FE
TS619�*QR
6G23?L$'-I9'2CEM
619�*NKTAC08E�?�01:C2C- y�08( M�R
��05/19�$1"±2H;�682O(I-70@E�$�9=2H;)0�E�01OGJ>0@9'2
MNK
"�-.2H;OM = r − PM = r −
√3
2
619�*ON =
√
r2 − (
12
)2 R^Z8-79�:40OM + ON = 1
K"%0%$)aD2C68-7952H;)0V01°'&�682C-7$�9
r −√
32
+√
r2 − 14
= 1R
..
.
..
..
..
.
..
..
.
..
..
.
..
..
.
..
..
..
..
.
..
..
..
..
.
..
.
..
.
..
.
..
..
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
....................................................................................................................
.........................................................................................................................................................................................................
...
....
...
...
....
...
..
..
...
..
..
..
...
..
..
..
..
.
..
..
.
..
..
.
..
..
..
.
..
..
.
..
..
.
..
..
..
.
..
..
..
..
..
...
..
..
..
...
..
...
...
....
...
...
....
.............................................................................................................................................................................................................
..............................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.................................................................................................................................................................
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
...............................................................................................................................................................
M
N
O
P
Q R
ST
r
��0V9�$1"±E�$'(�y�03{l$�Arq
√
r2 − 14
= 1 +√
32
− rK
r2 − 14
=(
1 +√
32
)2
− 2r(
1 +√
32
)
+ r2 K
r(
2 +√
3)
= 1 +√
3 + 34
+ 14
K{+AC$�J�"%;'-+:<;%"%0V:4$�9�:4(I&L*)0z2H;�682
r = 1R
��I TZGfTZGJI {OR d ~ rc�5I V RZW(�HS r4b��$� lmS N x W M {ZS L R F � q�GHGJI b�� S rOT RZW M b:�sK W6�HR r ] �Jq RZW�R � RZW�R L �7GM N � G W6W�RO� LTZGJIBKHL MBGONHT T4K ~sl M_L�L R d ]
� �
I \&p�^ R �0�O���o����#+ ��� �C�Q# �����9��#$> ������� R¹ (\K�²�0[2F2\M�K�Us0@:4-I(�Kv�f$�AF6�Ks619�*u�D&LO80@9�0=6@AC0>O8$'-79�O%2C$�*�- y�-7*�0
n:4$'-79)Ez61J>$�9�O
2H;)0@J=Eb0G(�y�08E}/19�$1"�-79�O�2H;�6G2 q� RV�1y�08A�M�$�9�0VAC0@:H0G-�y�08E 6G2�(+0861E42v$�9�0%:4$'-79�R� R ¹ ( O80[2CE5{l0["%08A�:H$'-I9)E�2H;�619�²�0G2F2�M#KN"%;)$ O80[2CE5{l0�"%0@A�2H;�619 U#01:4-.(\KN"%;)$ O80[2CE{l0["%08A^2H;�619��f$�AF6)K1"%;)$,O80G2FEf{l0�"%0@A^2H;�619X�'&LO8089�0�R
� RV��61:<;5?L08AFEb$�9=/19�$1"VE $�9�( M�2H;)032C$D2C68(n6@9�*=;)$1"J=619DM=:4$'-79)Ef;)0V$�ANEb;)0�O8$D2[R
��;�682f-IE 2H;)05E<J>68(I(70@E42f?L$�E<E<-7a�(70�y'6@(.&L0,$'{nEH&L:<;=2H;�682 9�$)a�$)*1M¤:C619¤*)01*'&L:H0
2H;)0f9�&�J>a�08As$'{D:4$'-79)E�AC0@:H0G-�y�0@*za@M�086D:<;3$'{82H;)0f$D2H;)0@AFE�"�-�2H;)$'&�2�JQ$�AC0v-I9�{l$�A4J=682C-7$�95o�����)# +>�������&H;��� �O+ �4E7-/#(��s1 �7�� H;�Q� �����,���0#��O+ # �s1\8 �m1��}� � R
F@0G2^2H;)0�9�&�JQa)0@Av$'{#:H$'-I9)EfAC01:408- y�01*=a@M ¹ (\K�²�0[2F2\M�K�Us0@:4-I(�K#�f$�AF6�K'619�*Q�'&LO8089�0a�0
AKBKCKDK�6@9�*
EK�AC0@E<?L0@:C2C-�y�0G(�MsRf��0�;�6Gy�0
1 ≤ A < B < C < D < EK
619�*52H;)0�E<&�J¦$'{L2H;)0%:4$'-79)EB-IEnR
�@®@6@J=-I9�-79�O�y'6@(.&L0@EN$'{n-I9V-I9�:HAC0@6@E<-I9�O}$�AC*)0@A�6@9�*V6@(.(12H;)0 ?L$�E<E<-7a�(70By'68(I&L08E^{l$�A
AKBKCKDK�619�*
E{l$�AB086D:<;
nO[- y�0@Ev2H;)0�{l$'(.(+$1"�-I9�O�2F6Da'(+0$q
n A B C D E��;)$=/19�$1"VE o
15 1 2 3 4 5�@y�0@A�M�$�9�0
16 1 2 3 4 6�@y�0@A�M�$�9�0
17 1 2 3 4 7�f$�AF6��p�'&LO8089�0
17 1 2 3 5 6�f$�AF6��p�'&LO8089�0
18 1 2 3 4 8�f$�AF6��p�'&LO8089�0
18 1 2 3 5 7�D&LO80@9�0
18 1 2 4 5 6U#01:4-.(����'&LO8089�0
19 1 2 3 4 9�f$�AF6��p�'&LO8089�0
19 1 2 3 5 8�D&LO80@9�0
19 1 2 3 6 7�f$�AF6
19 1 2 4 5 7Us0@:4-I(
20 1 2 3 4 10�f$�AF6��p�'&LO8089�0
20 1 2 3 5 9�D&LO80@9�0
20 1 2 3 6 8� $=$�9�0
20 1 2 4 5 8� $=$�9�0
20 1 2 4 6 7�D&LO80@9�0
S#;'&�E[KL{l$�AB2H;)0V2�"%$Q:46@Eb08Ef"%;)0@AC0n = 20
6@9�*E = 8
K�9�$X$�9�05:C619¤*)01*'&L:H02H;)0>9�&�JQa)0@Az$'{v:H$'-I9)E�AC01:408- y�01*xa@Mx086D:<;w$'{^2H;)0Q$D2H;)08AFEGK�619�*
20-7E32H;)0>E<J>68(I(70@E42
E<&L:<;%y'6@(.&L0%$'{nR
�Jq RZW�R � RZW�R N�GmGHL q RZW TZGJIBKHL MBGONHT T4K ~sl M_L�L R d ]
� A
I \&p n R �0�O���o����#+I���>� +��:��s+/!E��Q� � ��#$� ��1R�}�5�,)#$�6�O����� ��� ����O���7��� 1 ����O���7��� 1��� R
Us$�AC*)08(.-76}619�*��s089'2T?�(76[Mz2H;)0 {l$'(.(+$1"�-I9�O OG6@JQ0)R^Us$�AC*)08(.-76�O8$)0@ET��AFEH2�6@9�*z2H;)0bM2C61/10�68(.2C0@AC9)682C0 2C&�AC9)E@R��'6D:<;�Eb0G(+0@:C2CE^639�&�J>a�08A�{+AC$�J
12C$
6-79�:4(.&�E<- y�0f2H;�682^;�6@EN9�$D2
6@(IAC0@61*DM5a)01089�Eb0G(+0@:C240@*3rG2H;)0zOG61J>0z089�*)EN-79�E<- ®zJ>$@y�08E@R ¹ 2�2H;)030@9�*5$'{L086D:<;%J>$@y�0'K2H;)0X?)(I6GM�08A�J=6D/@-79�O,2H;)0XJ>$@y�0=2C61/10@E�2H;)0XEH&�J $'{B68(I(^2H;)0Q9�&�JQa)0@AFE�Eb0G(+0@:C240@*ua@M08-�2H;)0@A�?)(I6GM�08A�&�?�2C$X2H;�682�?L$'-I9'23619�*x:4(I6@-IJ>Ez6@(.(N$'{N-.2FE�?L$�EH-.2C- y�0>*'-�y'-7E�$�AFE@R,��;)0@92H;)0%OG61J>0�-7E $8y�0@A�K'2H;)0VE�:H$�AC0V$'{s0@61:<;=?�(76[M�0@AN-7Ev2H;)0%;'-+O8;)08EH2v9�&�J>a�08A
k{l$�A^"%;'-7:<;
2H;)0�?�(76[M�0@Af;�61E�:4(I6@-IJQ0@*>6@(.(L2H;)0�:H$�9)E�01:4&�2C- y�0�9�&�J>a�08AFE1K2K3K. . .
Kk{+AC$�J
12C$
k-I9�:4(I&�EH-�y�0)RuS#;)0="�-79)9�08Az-7EV2H;)0X?�(76[M�0@A�"�-.2H;�2H;)0¤;'-+O8;)08EH2VE�:H$�AC0 r�-.{ a)$D2H;Y;�6[y�0
2H;)0VE�6@JQ0�Eb:4$�AC0�K�9�08-�2H;)0@A^"�-I9)E 6@9�*52H;)0%OG61J>0�-7Ef6�*�AF6G">RB��$�Av0<®@6@J>?�(+0'K)EH&�?�?�$�Eb02H;)0XE<- ®�J>$@y�08E�6@AC0X61EV{l$'(I(7$1"VE�q%U q
2r � q
4rBU q
1r �4q
3rBU q
5r �4q
6RuS�;)0XAC0@E<?L0@:C2C-�y�0
:4(768-7J=Efa@M,UQ6@AC01K2r1K7r1K3K5K15rD619�*,a@M �¤6@AC0
1K2K3K6r1K2K5K10r1K3K
7K21R^U#$�AC*�0G(I-I6}619�* �s0@9'2T;�6[y�0f2H;)0 E<61J>0 Eb:4$�AC0�K
3K@6@9�*z2H;)0}OG6@JQ0f-7ET6�*)AF6[">R�S�;)0
0<®@6@J>?�(+0f*�$)08E�9�$D2#*�08JQ$�9)E424AF682C0�y�08A�M�O8$)$)*�?)(I6GMsRN!�E#2H;)0@AC0B619DM "V6[M}2H;�682#Us$�AC*)08(.-76:46@9Qa)0V?)AC0by�0@9'2C01*,{+AC$�J�"�-I9)9�-79�Oz61E<E<&�J,-79�OzE�;)0�-IEf?)(I6GM'-79�Oz6@Ef619>0H®D?�0@A�2 o��������������I��� �C�Q#c���s���o���$#��*E
Us$�AC*)08(.-76z:C619%61E<E<&�AC0�6 "�-I9�R^Z1;)03a�0@O[-79)E^2H;)03OG61J>03a@M%Eb0G(+0@:C2C-79�O5KD:4(I6@-IJ=-I9�O
1619�*
5R
Z8&�?�?�$�Eb0f2H;�682 �s089'2TEb0G(+0@:C2CEN6�9�&�J>a�08A^$D2H;)0@A#2H;�6@92R�S�;)0@9�Us$�AC*)08(.-763$�9�2H;)0
2H;'-7AC*3J>$@y�0B:46@9z61:<;'-+0by�0v6fEH&�J$'{12KG619�*368(.2C$)O80G2H;)08A#:4(I6@-IJ³6@(.([2H;)0B9�&�J>a�08AFE�{+AC$�J
124$
6K�-I9�:4(I&�EH-�y�0)R �s0@9'2 :46@9)9�$D2 J=682C:<;=2H;'-IE@R � $D240V2H;�6G2 9�$>EH&�J :46@9¤0H®1:H0@01*
21K
619�*52H;�6G2vE<&�J=E}$'{106@9�*
206@AC0V9�$D2v?�$�E�EH-+a'(+03{l$�A �s089'2[R^S#;)08AC08{l$�AC0'K'2C$=:4(I6@-IJ
5K
�s0@9'2BJ,&�EH2v?)(I6GM3$�9,2H;)0�{l$'&�A�2H;=JQ$8y�0z2C$=61:<;'-+0by�0V6VE<&�J¥$'{
15R}²�&�2N2H;)089 �s0@9'2
:46@9)9�$D2B61:<;'-+0by�0�619DM,E<&�J=E8K12K16K�619�*
20K)6@9�*>;)0@9�:40%:C619)9�$D2B:4(768-7J
4R
� $1"¥EH&�?�?�$�Eb0,2H;�682bKN$�9�2H;)0QEb0@:H$�9�*wJQ$8y�0�K��s089'2�Eb0G(+0@:C2CEV62KN:4(I6@-IJ=-I9�O
1619�*7R¤S�;)0@9Y$�9u;)08AzEb0@:H$�9�*xJ>$@y�0'KNUs$�AC*)08(.-76QEb0G(+0@:C2CE
1RxZ1$�{+6@AbKNU#$�AC*�0G(I-I6¤:C619
:4(768-7J1K2K4K5K)6@9�*
8R^S#;'&�Ef{+6@AbKD2H;)0V9�&�JQa)0@AFE
5K2K1;�6Gy�0%a�0@0@9=Eb0G(+0@:C240@*�RB!�9
$�AC*�08Av24$>:4(768-7J2K �s0@9'2fJ=&�E42f9�$1"iEb0G(+0@:C2B619X0by�0@9Q9�&�J>a�08AB2C$QO80G2f6@9X0by�0@9QE<&�J
]+6G2N2H;)0%0@9�*=$'{ �s0@9'2b� EB2H;'-7AC*=J>$@y�0'KD2H;)0VEH&�J�K21K'-IE $)*�*1`�R
!l{ �s0@9'2�Eb0G(+0@:C2CE4$�9=;'-IEfE�01:4$�9�*5JQ$8y�0�K�;)0V:C619,9�$1"�6@(�24$)O80[2H;)0@A�:4(768-7J
1K2K
3K4K6K7K�619�*
12KLa�&�2v2H;)0@9XU#$�AC*�0G(I-I65E�08(701:C2FE
324$QO80[2f6�E<&�J«$'{
15619�*Q:4(768-7J=E
6@(�24$)O80[2H;)0@A1K2K3K4K5K8K15R��s0@9'2�"�-.(I(19�0by�08AT:4(I6@-IJ
5r�"%;)0@9�:40�K@Us$�AC*)08(.-76B"�-I9)E@R
ªf9Y2H;)0w$D2H;)08A�;�6@9�*#Kv-I{ �s089'2�Eb0G(+0@:C2CE6{l$�A�;'-IE,E�01:4$�9�*_J>$@y�0'Kf;)0�J>61/10@E
2H;)0�EH&�J14619�*Q:46@9X:4(768-7J
1K2K7RfS#;)089XUs$�AC*)08(.-76�� Ef2H;'-7AC*>JQ$8y�0%-IEf24$=Eb0G(+0@:C2
4K
J>61/1-I9�O}6�E<&�Ji$'{18KD68(.2C$)O80G2H;)08A^:4(I6@-IJ=-I9�O
1K2K3K4K5K6K8K9K18R �s089'2N:C619)9�$D2
J>61/10V6VE<&�J¦*�- y�-IE<-7a�(70Va@M4K�6@9�*>Us$�AC*)08(.-76V61OG6@-I9�"�-79)E8R
�Jq RZW�R � RZW�R N�GmGHL q RZW TZGJIBKHL MBGONHT T4K ~sl M_L�L R d ]
� D
)+*�,.-0/�&�( ,�� �R$'&� ,�� �R$
����������������9���� ���� j 8 i : c�h ] ���� � �'&L:4(.-+*XU#$�9'2408EH2�`f!�9X65E�01°'&L0@9�:40,$'{
p¼<08AC$)0@E}619�*
q$�9�08EGK)2H;)0
i���2408A4J¤Kti
K'-IE :468(I(701*,6 H;��� ��;�#c�o����7� -I{ti 6= ti−1
KD{l$�Ai = 2
K3K4K. . .
Kp + q
Rv��$�A0<®@6@J>?�(+0'K�2H;)0�Eb0@°�&L089�:H0
0K1K1K0K0K1K0K1;�61E
p = q = 4K�6@9�*Q�'y�0,:<;�6@9�O80
?L$'-I9'2CEt2Kt4Kt6Kt7Kt8R��'$�A�6@(.('?�$�E�EH-+a'(+0 E�01°'&L0@9�:40@E�$'{
p¼<08AC$)0@E^619�*
q$�9�08E�"�-�2H;
1 ≤ p ≤ qK)*)0G2C0@ACJ=-I9�0
]I-�`�2H;)0VJ,-79�-IJ=&�J 619�*=J=6[®@-7J,&�J¦9�&�J>a�08AB$'{s:<;�619�O80V?�$'-79'2FEGK'619�*]7-.-�`�2H;)0V6[y�0@AF61O80V9�&�J>a�08Af$'{#:<;�619�O80V?�$'-79'2FE@R
S#;)039�$D2F682C-7$�95$'{�2H;'-IE�?�AC$)a'(+08J�J>61/[email protected]�(+$)$)/zE�:46@A�M#KDa�&�2�2H;)03?)AC$)a�(70@Ji-IE�9#� 2AC0@68(I( MVE�$Va)61*�RB²L6@E<-7:468(I( M#K�"%0 61AC0�a)08-I9�O 6@Eb/10@*%;)$1"pJ>6@9DMz2C-7J>0@E�2�"%$V:4$�9)Eb0@:4&�2C-�y�02408A4J=E�-79�6zEb0@°�&L089�:H0z$'{
0Ev6@9�*
1Ev6@AC0z*�-.¶�08AC0@9'2[R���0361AC0zO[- y�0@9V2H;�682�2H;)039�&�JQa)0@A
$'{0E -79,2H;)0�Eb0@°�&L089�:H0V-7E
p6@9�*=2H;)0%9�&�JQa)0@A $'{
1E}-IE
qR3!l{#2H;'-7E}9�$D2C6G2C-+$�9>J>61/10@E
M�$'&%°�&L0861EFM�KG24A�M}"%$�AH/@-79�Of$�9�6 ?)61A�2C-+:4&)(I61As:46@Eb0B(.-+/10p = 5
6@9�*q = 7
R ¹ 9)E4"%08AC-I9�O2H;)0z2�"%$=°'&L0@E42C-+$�9)EB-I952H;'-7EfE�?�01:4-I6@(L:46@Eb0z-7EfE42C-I(.(L6@9,-79'2C0@AC08EH2C-I9�O32F61E�/�R
��0�� (I(DEb$'( y�0�]I-�`L�LAFE42[RB²�0G{l$�AC0B"%0 (I6@&�9�:<;�-I9'24$z-.2FENE�$'(I&�2C-7$�9#KG2H;)0 �LAFE42s2C0@61:<;'-79�O?L$'-I9'2}{+AC$�J«2H;'-7E3?�AC$)a'(+08J·6@AC-IEb08E@R,!�9w$�AC*)0@Af2C$¤E�;)$1"i2H;�682
M-7E}2H;)0,J>6�®1-IJ=&�J¤K
"%0,9�010@*X2C$�*)$Q2�"%$>2H;'-I9�OGE�qv"%0,J=&�E42f�+&�E42C-I{.M>"%;1M>"%0=:46@9)9�$D23;�6[y�0=J>$�AC0�2H;�619M
:<;�6@9�O80¤?L$'-I9'2CE[K�619�*�"%0¤J,&�EH2VE�;)$1"¥2H;�6G23"%0¤:46@9 ;�6[y�0�0H®@6D:C2C( MM
:<;�6@9�O80?L$'-I9'2CE8Ru]I��;1Mw2H;'-IE%E�01:4$�9�*uE42408?-o��08(.(\KT-.{�2H;)08AC0¤:C619#� 2Va�0QJ>$�AC0>2H;�6@9#KNE<6GM�K
10:<;�619�O805?�$'-79'2FE�-I9X65?�6@AF2C-7:4&)(76@A E�01°'&L0@9�:40�K�2H;)089Q2H;)08AC0,:C619#� 2�a)05J>$�AC0%2H;�6@9100008-�2H;)0@A��}²�&�2
M:46@9#� 2fa)0%a)$D2H;
10619�*
1000R `zZ8-7J,-I(I61A^2H;'-79�OGE 9�0@01*52C$=a�0VE�;)$1"V9
{l$�AN2H;)0%J,-79�-IJ=&�J¤K)619�*=J>$�AC0%O80@9�08AF6@(.(�M�K�-79=619DM>$�?'2C-7J,-.¼H6G2C-+$�9=?)AC$)a�(70@J¥].2H;�682v-IEGKJ>6�®1-IJ=&�J¦$�AvJ,-79�-IJ=&�J ?�AC$)a'(+08J5`<R�������������� �,��� � � � F@0G2b� E3�LAFE42 (+$)$)/,{l$�AB2H;)0,J=-I9�-7J,&�J�K
mR�Us08AF2F6@-I9�(�M
m ≥ 0K�a)01P
:468&�Eb0>"%0�:46@9#� 2�;�6[y�0�6�9�0@OG682C- y�0¤9�&�JQa)0@A�$'{ :<;�619�O80�?�$'-79'2FE@R¢U�619u2H;)08AC0wa)00:<;�619�O80V?L$'-I9'2CE o � $sRvZ8-I9�:H0V6@9DM=Eb0@°�&L089�:H0%:H$�9'2F6@-I9)E}a)$D2H;
0E 619�*
1E[KD2H;)0@AC0VJ=&�E42
a�0V609�0H®@2N2C$,6
1E�$�JQ0�"%;)0@AC0)RTS#;)08AC08{l$�AC0'K
m-7Ef682N(+0861E42
1R
Us$'&)(7*>2H;)0�J,-79�-IJ=&�J a�0 � oQ�T08E��_2H;)0�E�01°'&L0@9�:400K0K. . .
K0K1K1K. . .
K1;�61E�$�9�(�M%$�9�0�:<;�6@9�O80�?�$'-79'2[R���0�;�6[y�0}Eb;)$1"V9�2H;�6G2s2H;)0}9�&�JQa)0@A^$'{):<;�619�O80}?L$'-I9'2CE
J=&�E42Ba�0V6G2�(70@6@EH216@9�*>:46@9,-79,{+61:C2Ba�0
1RNZ1$�2H;)0VJ,-79�-IJ=&�J�-7E
1R
¯V$1"�6Da)$'&�2f2H;)0,J=6[®@-7J,&�J�KMouS#;'-IE�-7E}24AF-+:</@-+08A[R5�'6D:<;XE�01°'&L0@9�:40%"�-.2H;
p¼<0@AC$)08Ev619�*q$�9�08Ef;�6@E
p + q2408A4J=E�r82H;'&�EGK
M-IEB:H08AF2F6@-I9�(�M�9�$%(76@ACO80@AT2H;�6@9
p + qR
²�&�2�2H;)0���AFEH2s2408A4J�:46@9)9�$D2�a)036z:<;�6@9�O803?L$'-I9'2N]l:<;)0@:</32H;)03*�0G�L9�-�2C-+$�9D`<RTS�;)0@AC0G{l$�AC0�KM
-IEf9�$5(76@ACO80@A^2H;�619p + q − 1
RUs$'&)(7* 0by�08A�Mu2C0@ACJ¸a�&�2�2H;)0X�LAFE42�a)0¤6w:<;�619�O80�?�$'-79'2 o S�A�Mu��*)*�(.-79�OQ{l$�AV6
J=-I9�&�240X$�A}2�"%$X24$�Eb0@05"%;�682}M�$'&Y:C619u*'-7E�:H$8y�0@A�R>�T$'&u:H$'&)(7*x?�0@A4;�61?)E�24A�Mx6>{l0�"*�-.¶�08AC0@9'2v?L$�E<E<-7a�(70�y'6@(.&L0@EB{l$�A
p619�*
qR
¹ 9DM5(.&L:</ o F@0[2b� Ev(+$)$)/���AFEH2�6G2^2H;)0V:C61E�0p = q
R�!�952H;'-IE :C61E�0�K@M�08EGK12H;)08AC0V:C619a�0
p + q − 1 = 2p − 1:<;�6@9�O80%?�$'-79'2FEGK)a�0@:468&�Eb0z2H;)0VEb0@°�&L089�:H0%:4$'&)(+*=6@(�2408A49)6G240
a�0[2�"%010890619�*
1� {l$�AV0<®@6@J>?�(+0'K
0K1K0K1K
. . .K0K1RYS#;'&�E[K�-.{
p = qKN2H;)0
J>6�®1-IJ=&�J 9�&�JQa)0@AB$'{s:<;�6@9�O80V?L$'-I9'2CEB-7EM = p + q − 1 = 2p − 1
R��;�682f-I{
p < qo � $D2C-+:40%2H;�682}0by�0@A�M¤:<;�619�O805?L$'-I9'2 -I9Dy�$'(�y�08E36
0Ks08-�2H;)0@Af-I9
�
2H;�682}?�$�E<-�2C-+$�9¤$�Af-79>2H;)0,?�$�E<-�2C-+$�9¤a)08{l$�AC0)R5��61:<;¤$'{�2H;)0p¼<08AC$)0@Ez:C619�:4$�9'24AC-7a�&�2C0
24$�682�J>$�EH2T2�"%$5:<;�619�O80z?L$'-I9'2CEf]7619�*�2C$,0H®@6D:C2C( M%2�"%$%-I{�-.2�;�61E1E $�9,a�$D2H;5EH-+*)0@EB$'{
-.2�`<R�Z8-I9�:H032H;)0@AC0V6@AC0p¼<0@AC$)08EGK12H;)0@AC0%:C619>a)0V6G2vJQ$�E42
2p:<;�619�O80V?�$'-79'2FE@R
U#6@9�2H;'-7Ev&�?)?L08Ava�$'&�9�*,a�0z6D:<;'-70by�01*!o,]+ª $�?�E � �_2H;�682b� Ev{+619�:\M,J=682H;)08J>6G2C-+:CEbPE�?�0@61/�{l$�A � U�6193"%0 61:C2C&�6@(.(�Mz��9�*V6}E�01°'&L0@9�:40B"�-�2H;
2p:<;�619�O80 ?L$'-I9'2CE oG�<`v�T0@E � {l$�A
0<®@6@J>?�(+0'K1K0K1K0K1K. . .
K0K1K1K. . .
K1-7EB6�Eb0@°�&L089�:H0�"�-�2H;
p < q"%;'-7:<;,;�61E
2p:<;�6@9�O80V?L$'-I9'2CE ]I2�"%$5{l$�Av086D:<;
0`<R^S�;)0@AC0G{l$�AC0�K
M = 2p-I{
p < qR
� $D2C052H;�6G2 2H;)0,9�&�JQa)0@A3$'{0E�:H$�9'24AC$'(IE�2H;)0=J=6[®@-7J,&�J�9�&�JQa)0@A3$'{�:<;�6@9�O80
?L$'-I9'2CE�;)0@AC0)RfS�;)0�9�&�JQa)0@A $'{1E -IE (+08E�Ef-7J=?L$�A�2C6@9'2bK�a)01:C6@&�E�0V2H;)08AC0�6@AC0%JQ$�AC0
1E
2H;�619=9�010@*�0@*�R��039�010@*%2C$%(7$)$)/�6G2�]I-I- `�9�0H®@2[R���9'2C0@A^EH2F6DO80�(708{I2�2H;)03Eb0@:H$�9�*V24086D:<;'-I9�O�?L$'-I9'2[R
S�$x�LO[&�AC0�$'&�232H;)0X6Gy�08AF6DO80X9�&�J>a�08A%$'{ :<;�6@9�O80¤?�$'-79'2FEGK�"%0X9�010@*w24$w��O[&�AC0¤$'&�22H;)0324$D2F6@(�9�&�JQa)0@Av$'{#:<;�619�O80�?�$'-79'2FE $8y�0@A�68(I(�E�01°'&L0@9�:40@Ef6@9�*=*�- y�-7*�0Va@M�2H;)032C$D2C68(9�&�JQa)0@Av$'{�E�01°'&L0@9�:40@E8RTS#;'-IEfE�0108J>E�2C$5AC0@°�&)-IAC0z(+$)$)/@-79�O�682^-79�*'-�y'-+*'&�6@(�Eb0@°�&L089�:H08E619�*�*�0[2408A4J,-79�-I9�O�2H;)0,9�&�JQa)0@A�$'{�:<;�6@9�O80=?L$'-I9'2CE3-I9w086D:<;XEb0@°�&L089�:H0)R���0,J,-+O8;D22H;)0@9,;�6[y�0}2C$V��O[&�AC0z$'&�2N;)$1"³J>6@9DM�E�01°'&L0@9�:40@EB;�6[y�0 � :<;�619�O803?�$'-79'2bKD;)$1"³J>6@9DM;�6Gy�0 � KT619�*wEb$w$�9�RXS�;'-7E3"%$'&)(7*w6D:C2C&�68(I( Mxa�0>AC0@68(I( Mw?�68-79�{7&)(�Rx!l{^M�$'&s� AC0>{l010G(I-I9�O?�6@AF2C-7:4&)(76@AC( M=6@JQa'-.2C-7$'&�EGK@M�$'&Q:H$'&)(7*>$'{s:4$'&�AFEb0324A�M52H;'-IE �
S#;)08AC0z-IEv6�E<9�0@61/8MV"V6[M%2C$5*�$V2H;'-7E8R�!l{)"%0z:H$'&)(7*,*�0[2408A4J,-79�0�2H;)0�2C$D2C68(�9�&�J>Pa�08Af$'{#E�01°'&L0@9�:40@Ef-I95"%;'-7:<;=?�$�E<-�2C-+$�9 � -7Ef6�:<;�619�O80V?L$'-I9'2bKD2H;)0V9�&�JQa)0@Av-I95"%;'-7:<;?L$�EH-.2C-7$�9 � -IE 6�:<;�6@9�O80%?L$'-I9'2bK)6@9�*>E�$=$�9#KD"%0%:H$'&)(7*=6D*)*,2H;)08Eb0z2C$D2C68(7Ev24$=O80G2�2H;)024$D2F6@('9�&�J>a�08A^$'{):<;�619�O80}?L$'-I9'2CE8R^]+ªB{�:4$'&�AFEb0'KG"%0 EH2C-.(I(';�6Gy�0f2C$%*'-�y'-+*)0�a@Mz2H;)0 2C$D2C68(9�&�JQa)0@AB$'{#Eb0@°�&L089�:H08E@R `�������������� �,� � �A� � � F@0[2b� E���AFEH2���9�*¤2H;)0,24$D2F6@(^9�&�J>a�08A�$'{�Eb0@°�&L089�:H08E@RXZ8-I9�:H0,2H;)08AC061AC0
p+q?L$�EH-.2C-7$�9)E^-79V2C$D2C68(\KG"%03:C6195:<;)$)$�E�0
p?�(761:H08E^24$V?)&�2�2H;)0
0E8R�S#;'-IE�JQ08619)E
2H;�682N2H;)08AC0V61AC0 (p+q
p
) Eb0@°�&L089�:H08Ef-I9524$D2F6@(�R� $1"i:4$�9)E<-7*�08AB?L$�EH-.2C-7$�9
k-I9,2H;)0%Eb0@°�&L089�:H08EGK'"%;)08AC0
k:46@9Xa)0%6@9DM=-79'2C01O808A
{+AC$�J22C$
p + qRf!�9Q$�AC*)0@A�{l$�Av?�$�E<-�2C-+$�9
k2C$>a)0V6�:<;�619�O80V?L$'-I9'2bKD"%0�J=&�E42f;�6[y�0
tk = 06@9�*
tk−1 = 1KT$�A
tk = 1619�*
tk−1 = 0RQ¯�$1" J=619DM¤Eb0@°�&L089�:H08E�61AC0
2H;)0@AC0}"�-�2H;tk = 0
619�*tk−1 = 1
o5Z8&L:<;�6�E�01°'&L0@9�:40�;�61Ep − 1
¼<0@AC$)08Ev619�*q − 1$�9�0@ET2C$V?)&�2�-I9�2H;)0}AC0@J=6@-I9�-79�O
p+q−2?�$�E<-�2C-+$�9)E r[2H;'&�EGK82H;)08AC0�6@AC0 (p+q−2
p−1
) E<&L:<;Eb0@°�&L089�:H08E@RTZ8-7J,-I(I61AF(�M�K[2H;)08AC0 6@AC0 (p+q−2
p−1
) Eb0@°�&L089�:H08E#"�-.2H;tk = 1
6@9�*tk−1 = 0
RS#;)08AC08{l$�AC0'KL2H;)0@AC056@AC0
2(
p+q−2p−1
) Eb0@°�&L089�:H08E "�-.2H;Q6,:<;�619�O805?L$'-I9'2f-79X?L$�EH-.2C-7$�9kR
� $D2C-7:H032H;�682N2H;'-IEv24$D2F6@(�-7EB-I9�*�08?L089�*�089'2f$'{kR
Z8-79�:4032H;)0@AC0�61AC0p + q − 1
?L$�E<E<-7a�(70z?L$�EH-.2C-7$�9)Ev{l$�A�6%:<;�6@9�O80V?�$'-79'2�6@9�*52H;)09�&�JQa)0@A�$'{^:<;�619�O805?L$'-I9'2CE�-I9¤65�'®10@*X?L$�EH-.2C-7$�9Q-7E�6=:H$�9)E42C6@9'2bKL2H;)0%24$D2F6@(s9�&�JQa)0@A$'{s:<;�619�O80V?�$'-79'2FEf-IE
2(p + q − 1)(
p+q−2p−1
) K1"%;'-+:<;,J>0@6@9)E 6@9=6Gy�08AF6DO80%$'{
2(p + q − 1)(
p+q−2p−1
)
(
p+q
p
) =
2(p + q − 1)(p + q − 2)!
(p − 1)!(q − 1)!
(p + q)!
p!q!
=2(p + q − 1)!p!q!
(p + q)!(p − 1)!(q − 1)!=
2pq
p + q
R
� ©
�T)�� 1 ! ��� �N$'&�� !�*�( ,�� ��� &�!��
�H� ���� � � �������
� c�� ��� ��� 8 �Wq ¹ 9 ��� � -7E�6 �2�>?>@#��U�Z�+H �0��C��;5��� ��� �Y� H K^:4$�9)E<-IEH2C-I9�O=$'{^2H;�AC0@0Q01°'&�6@(EH24AF68-+O8;D2z(I-I9�0XE�01OGJ>0@9'2FEV2H;�6G2%;�6[y�0¤01°'&�6@(v6@9�O[(+08E5a)0G2�"%0@0@9u6D*1�l6D:40@9'2�Eb0@OGJQ089'2CE8R��$�Av0<®@6@J>?�(+0$q
..................................................................................................................................................................................................................................................................................................................................................................
.
.
.
.
.
.
..
..
..
.........................
.
.
.
.
.
.
..
..
..
.........................
............
.
.
.
.
.
.
.
.
.
.
.
.
............
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.............................................................................................................................................................................................................................................
.
.
.
.
.
.
.
.
.
.
.
.....................................
......................... . . . . . . . . . . .............
. . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
. . . . . . . . . . . .
..................................................................................................................................................................................................................................................................................................................................................................
.
.
..
.
..
.
..
..
..
.....
.
.
..
.
..
.
..
..
.......
............
.
.
.
.
.
.
.
.
.
.
.
.
............
..................................................................................................................................................................................................................................................................................................................................................................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
¹ 9�Z)» ¹ :H$'&)(7*�a)0_6 ?�$�AF2C-7$�9�$'{�6_AC01O[&)(I61A>?�$'(�M�O8$�9#K�$�A,-�2Q:H$'&)(7*�a)0_6x"%;)$'(+001°'&)-I(I682C0@AF68(�24AF-76@9�O[(+0'K)*�08?L089�*�-I9�OV$�9,-.2FEf619�O[(70�R
!�92H;'-IEY6@AF2C-7:4(+0'K%"%0 "�-I(.(50H®D?�(+$�AC0�Eb$�J>0p?)AC$�?L08AF2C-70@EY$'{=Z)» ¹ Eu619�*�2H;)0G-7A:H$�9)9�0@:C2C-+$�9V"�-�2H;V2H;)0z;�6@A4J>$�9�-+:NJQ08619�RBWwM�AC0861E�$�9�{l$�ANa)08-I9�O}-79'2C0@AC08EH2C01*%-795Z)» ¹ E:46@JQ0}{+AC$�J�6 � ���7- � � ?)AC$)a�(70@J"%;)0@AC0 2�"%$�0@°�&)-.(76G2408AF6@(D24AC-I619�O[(70@ET"%0@AC0�?�(761:H0@*6D*1�l6D:40@9'2�24$w$�9�0>619�$D2H;)08A�$�9¤2H;)0QE<61J>0=(I-I9�0Q]+E�010>?)AC$)a�(70@J¸W � � A � �@�D�;D q A��D© KA��@¨ r �1�1�D q A��@¨ `�R
!+2z-7E�:H$�9Dy�089�-+089'2324$w&�E�0X6@9Y0<®82408A49)68(v6@9�O[(+0X6@E%6¤?)61AF6@JQ0[2408AbKNE�6[MθRxS�;)0
$D2H;)0@Az:H$�9Dy�089�-+089'2�?)61AF6@JQ0[2408A3-IEz2H;)0=(+089�O�2H;u$'{v0@61:<;�(I-I9�0>Eb0@OGJQ089'2bK^E<6GMaRX��0
"�-I(.(LAC08{l08AN2C$�2H;)0VJ,-+*)*�(70�(.-79�0�Eb0@OGJQ089'2v61Ev2H;)0 ���9�$# $'{L2H;)0%Z)» ¹ R
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
......
.
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
........
........................................ ........................................
θ θ
a
a
a
»)(761:H032�"%$,Z)» ¹ E�"�-.2H;,*�-.¶�08AC0@9'2�(70@9�O�2H;,?�6@AF61J>0G2C0@AFEa6@9�*
b6@9�*52H;)0�E�6@JQ0
619�O[(70 ?)61AF6@JQ0[2408Aθ$�932H;)0fE<61J>0 a)6@Eb0B(.-79�0�"�-�2H;�$�9�0f?L$'-I9'2#-I9V:4$�J>J>$�9#K861ETE�;)$1"V9
a�0G(+$1">R
............................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.
.
.
..
.
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
.......
.
.
.
..
.
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
.......
.
.
.
..
.
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
.........
.
.
.
..
.
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
.......
.......................................... ....... ....... ....... ....... ....... .......
A
A1A2
B
B1 B2
O
θ θ θ θ
b
b
a
b
a
a
����������� "!"#c© $&%�%�' ��(&)�(�*�� (&),+-(.#.!�/"01(.#2� 34(&5�6&��32� /2#��
� ¨
µ�$'-79BA1
619�*AB1
R4F@0G2Pa)0f2H;)0}?L$'-I9'2^$'{'-I9'2408AFEb0@:C2C-+$�9%$'{D2H;)0 (I-I9�0@E
BA1619�*OB2
K'619�*,(70G2Qa�0z2H;)0�?L$'-I9'2B$'{�-I9'2408AFEb0@:C2C-+$�9>$'{�2H;)0�(I-I9�0@E
AB1619�*
OA2R
............................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.........
....... ....... ....... ....... ....... .......
A
A1A2
B
B1 B2
O
PQ
θ
S#;)089OP = OQ =
ab
a + b
RTS#;'-IEf-IE $�9�0%;�68(I{s$'{L2H;)0V;�61ACJQ$�9�-7:BJQ08619>$'{a6@9�*
bK
619�*,-IEB-79�*)0@?�0@9�*)0@9'2f$'{L2H;)0�?�6@AF61J>0G2C0@AθR
��0}"�-.(I(�?�AC$8y�0�2H;'-7EvAC0@J=61A4/@6Da'(+03{+61:C2^-79%2�"%$�*�-.¶�08AC0@9'2�"V6GM�EGK'086D:<;5$'{�"%;'-7:<;-7EB-I9)EH24AF&L:C2C-�y�0)R
� ���6��� �0�s����� �m� $D2C0z2H;�682 4A1AB6@9�* 4POB
61AC0VE<-IJ=-.(76@A[RN��0z2H;)0@AC0G{l$�AC0�;�6[y�0OP
OB=
AA1
AB
R Z8-79�:40A1A = AO = a
6@9�*OB = b
K'"%0�$)aD2C68-79 OP
b=
a
a + b
KO[-�y'-79�O
OP =ab
a + b
R Z8-7J,-I(I61AF(�M�K)&�E<-I9�OVE<-IJ=-.(76@A�24AF-76@9�O[(+08EB1BA
619�*QOA
K�"%0Eb0@0z2H;�6G2
OQ =ab
a + b
R��#9HO���Q+ �0�O������� S�;'-7E5?�AC$)$'{f&�E�0@E,:H$)$�AC*'-79)6G2408E@R�F@0[2
O = (0, 0)K
A = (a, 0)K
619�*B = (−b, 0)
R � $D240X2H;�682AA1 = OA = a
R�S�;)0x:4$)$�AC*�-I9)682C0@E=$'{A161AC0 (
a(1 + cos θ), a sin θ) K'619�*%2H;)0V:H$)$�AC*'-79)6G2408Ef$'{
A26@AC0
(−a cos θ, a sin θ)R
S#;)05:H$)$�AC*'-79)6G2408E3$'{B1
61AC0 (−b cos θ, b sin θ) KL6@9�*>2H;)05:H$)$�AC*'-79)6G2408E3$'{
B261AC0
(b cos θ, b sin θ)R
S�$X*�0[2408A4J,-79�0%2H;)0,:4$)$�AC*�-I9)682C0@E3$'{PK)"%0%�L9�*>2H;)0,0@°�&�6G2C-+$�9)Ez$'{s2H;)0�(.-79�08E
BA1619�*
OB2RTS#;)0V?�$'-79'2B$'{L-79'2C0@AFE�01:C2C-7$�9>$'{L2H;)0@E�0z(I-I9�0@EB-IE
P =
(
ab cos θ
a + b,ab sin θ
a + b
) RZ8-7J,-I(I61AF(�M�K'a@M��L9�*'-79�O 2H;)0�0@°�&�6G2C-+$�9)EB$'{)2H;)03(.-79�08E
AB1619�*
OA2K8"%0z$)a'2F6@-I9�2H;)0
:H$)$�AC*'-79)6G2408E}$'{Qq
Q =
(
−ab cos θ
a + b,ab sin θ
a + b
) R
�'AC$�J�2H;'-7E[K'-.2N-7E :4(70@6@AN2H;�6G2OP = OQ =
ab
a + b
R²LAF&L:H0VZ1;�6G"3M�08A
�f0@?)61A�24JQ089'2f$'{^WY6G2H;)0@J=682C-7:4Ef6@9�*>ZG2C6G2C-7E42C-+:CEW_08JQ$�AF-768(#� 9�- y�[email protected]�M>$'{ � 0�"�{l$'&�9�*�(I619�*
ZG2[RDµ�$);�9#� E[K � F'K)U�619)61*)6 ¹ � U D Z ©
��
��� � � �.� � � �T �� � � �����
� ���������� ����� ��� � �)� �
¹ Ev"%0Va�0@O[-79=619�$D2H;)08A^M�0@6@Av619�*=6V9�0["³y�$'(I&�J>0%$'{Nc�d e'g�hBj+k�m>npofq#r3t�niK-.2�-IE�6@?�?)AC$�?�AF-76G240 2C$�(+$)$)/za�6D:</z$@y�08As2H;)0 �@�D�1 9�&�J>a�08AFEB$'{'2H;)0 8Q���J��#$� 619�*�2H;�6@9�/6@(.()2H;)$�Eb0�"%;)$,?)AC$@y'-+*)01*,&�E�"�-.2H;,?)AC$)a�(70@J=EGK):4$�J>J>0@9'2FE 6@9�*=E�$'(I&�2C-7$�9)E�q¹ AH/@61*DM ¹ (�2¯V$'&L*�6 ¹ 9�$'&�9Wu-+O[&L0G( ¹ JQ089�O[&�6@(�Us$8y'61EWu-+:<;)0G(^²L6G2C68-I(.(+0�#$)a�08AF2�²�-I(.-79)E�/1-»)-+08A4AC0,²�$�A49)E4¼C240G-79Us;�AF-7E424$�?�;)0@A^µ8Rs²LAF61*�(70bM²LAC&L:40%U#AC$'{l$)$D2»'6D$'(7$=UL&�EH2C$)*�-
µ�$�E"E0�FG&)-7Ez� E �I68¼<P[²L61ACAC0@AC$ª�y'-+*'-I&¤�1&�AC*�&)-N01$'¶#AC0bM ¹ R �#619�*�6@(.(!\$�6@9)9�-7E �#682FE<-7/1-IET&�EH2F6Gy�$ �sAC-IJQ/108A¹ 9�*1MtFG-I&»'6[y�(7$�E3WY61AF61O8$'&L*)61/1-IE�N0@*�&)(I6 � R#Wu&�AF2�M
S�$�E�;'-+$,Z108-IJ=- M'6¹ :<;'-I(.(+0861EfZ8-79�0G{+6D/1$�?�$'&)(+$�E� -7:</5Z1/1$�J>a)AF-7E²L6Da'-7E ZG2C0@ACO[-7$'&²^R µ8R��N089�/@682F6D:<;�68(76�)*1"V61AC*,S#R7¯=R'�¤619�O�#6@-IJ=-I9�O��8;�61$FG-��8;)$'&
!�68(7E�$3"V619'2s24$z2H;�619�/}µ�$�6@9)9�0zU�619)6@?L03]+99E0@0�F@$�9�Ob"%$�AF2H;1`4K8"%;)$%:H$�9'2C-I9�&L0@E^2C$"%$�AH/%J=-IAF6D:4(70@E�"�-�2H;,J5M,Eb:HAF-+a)a�(70@E[KD2C&�AC9�-79�O�2H;)0@J -I9'24$,;'-+O8;=°�&�68(I-�2\M F �\S � �_��(70@E8R
��0�:H$�9'2C-I9�&L0�9�$1"¡"�-.2H;³2H;)0�AC08J>68-79�-I9�O�?�AC$)a'(+08J>EYEb;)$�A�2C(I-IEH2C01*�{l$�A�2H;)0A;A ��� !CW_ª�-79>µ<61?)619�R=WwMQ2H;�619�/1E�O8$>24$ ¹ 9�*DM1FG-.&sKsU#6@9)6D*'-76@9¤S�0861J F@086D*)0@A�K#{l$�A:H$'(.(+0@:C2C-79�O�2H;)08J {l$�Av$'&�A�&�E�0�R_&_����������! #"��%$&�!�('��%$*)QI+$��!,- I+$&�!��X.$*)/'0)21 I435�6$*7
8:9 �<;>=@?>A�BDCE=EFHGJIK;��2LMANFPOQC' :G� c�i j �
$1_ R F@0G2na�0�6w?L$�EH-.2C- y�0¤-I9'240@O80@A�Kf619�*Y(+0[2
x1 ≤ x2 ≤ · · · ≤ xn
a)0wAC086@(9�&�JQa)0@AFE8R]76G`¤»�AC$@y�032H;�6G2 ( n
∑
i,j=1
|xi − xj |)2
≤ 2(n2 − 1)
3
n∑
i,j=1
(xi − xj)2 R
]la1`>Z1;)$1"2H;�682�2H;)0�0@°�&�68(I-�2\M>;)$'(+*�E -.{s619�*>$�9�(�M,-I{x1K. . .
Kxn
-IE}6@9>61AF-.2H;�J>0G2C-7:Eb0@°�&L089�:H0)R
$ a R F@0[2�R + a)0w2H;)0uEb0[2=$'{36@(.(�?�$�E<-�2C-�y�0xAC086@(�9�&�J>a�08AFE@R¥�1-79�*¢68(I( {7&�9�:C2C-+$�9)Ef :
R + → R + 2H;�682vE<682C-IE<{.M%2H;)0�{l$'(.(+$1"�-I9�O�:4$�9�*�-�2C-+$�9)E�q]76G`
f(xyz) + f(x) + f(y) + f(z) = f(√
xy)f(√
yz)f(√
zx){l$�Aw6@(.(
xKyKz ∈ R + r
]la1`f(x) < f(y)
{l$�Av68(I(1 ≤ x < y
R
�1�
$ s R4F@0G2na�0B6 ?�$�E<-�2C-�y�0�-79'2C01O808AbKG619�*3(70G2
(x1, . . . , xn)619�*
(y1, . . . , yn)a)0v2�"%$
Eb0@°�&L089�:H08Ez$'{�?L$�EH-.2C- y�0�AC0@68(s9�&�JQa)0@AFE8RzZ8&�?�?�$�Eb0(z2, . . . , z2n)
-IE�65Eb0@°�&L089�:H05$'{?L$�EH-.2C- y�0�AC0@68( 9�&�J>a�08AFE=E<&L:<;u2H;�6G2
z2i+j ≥ xiyj
{l$�A�68(I(1 ≤ i
Kj ≤ n
R F@0[2M = max{z2
K. . .
Kz2n} R »�AC$8y�0z2H;�6G2
(
M + z2 + · · · + z2n
2n
)2
≥(
x1 + · · · + xn
n
) (
y1 + · · · + yn
n
) R
/[8 h i � ���� 8 j � d��X _ ROF@0G2
x1K. . .
Kxn
6@9�*y1K. . .
Kyn
a�0VAC086@(L9�&�J>a�08AFE@R�F@0[2A = (aij)1≤i,j≤na�0z2H;)0�J>6G24AC- ®z"�-.2H;=089'24AC-70@E
aij =
{
1-.{
xi + yj ≥ 0r
0-.{
xi + yj < 0R
Z8&�?�?�$�Eb0�2H;�682B-IEB6@9
n × nJ>6G24AC- ®3"�-�2H;50@9'24AF-+08E
0K1EH&L:<;%2H;�6G2T2H;)0zE<&�J $'{)2H;)0
08(70@J>0@9'2FEV-79u086D:<;wAC$1"¥619�*u086D:<;x:H$'(.&�J>9u$'{B-7E%01°'&�6@(^2C$¤2H;)0X:H$�ACAC0@E<?L$�9�*'-79�O
E<&�J�{l$�AN2H;)0�J>6G24AC- ®ARf»�AC$8y�0z2H;�6G2
A = BR
X a Rv�@y�0@A�M,?L$'-I9'2^"�-�2H;�-79'2C01O808AB:4$)$�AC*�-I9)682C0@EB-I952H;)0�?�(76@9�0z-7E�2H;)0V:H089'24AC0%$'{�6%*'-7E�:"�-.2H;,AF61*�-.&�E
1/1000R
]76G`¤»�AC$@y�0Y2H;�6G2>2H;)0@AC0 0<®@-7E42CE�6@9�01°'&)-I(I682C0@AF68(324AC-I619�O[(70Y"%;)$�E�0uy�0@A�2C-+:40@E¤(.-+0_-I9*�-.¶�08AC0@9'2B*'-7E�:4E8R
]la1`�»�AC$@y�0�2H;�6G2B0by�08A�M=0@°�&)-.(76G2408AF6@(�24AC-I619�O[(70�"�-�2H;Vy�0@A�2C-+:40@Ev-I9=*�-.¶�08AC0@9'2v*�-IEb:CEf;�61EE<-7*�0@PH(70@9�O�2H;=OGAC08682C0@A^2H;�619
96R
X s R F@0[2f(k)
a)0p2H;)0�9�&�JQa)0@AY$'{,-79'2C01O808AFEn2H;�6G2uE�6G2C-7EH{.M2H;)0p{l$'(I(7$1"�-79�O
:H$�9�*'-.2C-7$�9)E�q]I-�`
0 ≤ n < 10kK)E�$�2H;�682
n;�6@E}0<®@61:C2C(�M
k*'-+O[-�2CE}]I-79Q*)01:4-IJ>68(�9�$D2C6G2C-+$�9D`4KD"�-�2H;
(+086D*'-79�O�¼<08AC$)0@Ef68(I(7$1"%01*[r]7-.-�`�2H;)0z*'-+O[-�2CEB$'{
n:46@9,a)03?L08A4J,&�240@*�-I95EH&L:<;�6}"V6[MV2H;�6G2T2H;)0bM�M'-+0G(+*%619%-79'2C01O808A
*�- y�-IE<-7a�(70Va@M11R
»�AC$@y�032H;�682f(2m) = 10f(2m − 1)
{l$�Av0by�0@A�M,?L$�EH-.2C- y�0z-79'2C01O808AmR
�9c 8 hHc � jt�� a RF@0[2
ABCa�03619%-7E�$�Eb:408(70@E^24AF-76@9�O[(+0 "�-.2H;
AC = BCK8"%;)$�E�0�-79�:40@9'24AC0�-7E
IRF@0G2
Pa)0¤6�?�$'-79'2%$�9u2H;)0�:4-IAC:4&�JQ:4-IAC:4(+0�$'{v2H;)0Q24AF-76@9�O[(+0
AIB( M�-I9�OQ-I9)E<-7*�0Q2H;)0
24AC-I619�O[(70ABC
RBS#;)0V(.-79�08Ef2H;�AC$'&LO8;P?�6@AF6@(.(+0G(L24$
CA619�*
CBJQ0@0G2
AB6G2
D619�*EK�AC0@E<?L0@:C2C-�y�0G(�MsRBS#;)0V(.-79�0�2H;�AC$'&LO8;
P?�6@AF6@(.(+0G(L2C$
ABJQ0@0G2FE
CA6@9�*
CB682F619�*
GK�AC08E�?�01:C2C- y�08( M�R�»�AC$8y�0�2H;�682�2H;)0�(I-I9�0@E
DF619�*
EG-I9'2408AFEb0@:C2f$�9=2H;)0
:4-7AC:4&�J>:4-7AC:4(70%$'{L2H;)0324AC-I619�O[(70ABC
R�Gs R%�'6D:<;Q?�68-7A}$'{^$�?�?�$�E<-�240�E<-7*�08E3$'{^6,:H$�9Dy�0H®Q;)0H®@6DO8$�9¤;�61E}2H;)05?)AC$�?L08AF2�MQ2H;�6G22H;)0�*'-7E42C6@9�:H0¤a�0[2�"%01089x2H;)08-IA�J=-7*�PH?L$'-I9'2CE%-7E�01°'&�6@(N24$ √
3/22C-IJQ08EV2H;)0¤EH&�J $'{
2H;)08-IA�(70@9�O�2H;�E8R »�AC$8y�0z2H;�6G2v6@(.()2H;)0V6@9�O[(+08E}$'{�2H;)0%;)0<®@61O8$�9=61AC0V01°'&�6@(�R
� �
� � RF@0[2ABC
a)036�24AC-I619�O[(70 "�-.2H;%Eb08J=-I?L08AC-IJQ0[2408As6@9�*%-79)AF61*�-.&�E
rR�S#;)03Eb08J=-7P
:4-7AC:4(70@E�"�-.2H;%*�-I61J>0G2C0@AFEBC
KCA
KAB
6@AC03*)AF6["V95$�9�2H;)0z$'&�2FE<-7*�0�$'{�2H;)0 24AF-76@9�O[(+0ABC
RTS#;)0%:4-IAC:4(+032F619�O8089'2�2C$,6@(.()2H;�AC010�Eb08J=-7:4-7AC:4(70@E ;�6@EfAF6D*'-I&�EtRf»�AC$@y�032H;�682
s
2< t ≤ s
2+
(
1 −√
3
2
)
rR
f�� h�i$c�j )� c 8 jt���a R ¹ 9¤-79'2C01O808A
n-7EzE<6@-7*X2C$�a)0 ;5����+ -I{ |n| -IEz9�$D2 2H;)0=E�°�&�6@AC0>$'{N6@9¤-79'2C01O808A[R�f0G2C0@ACJ=-I9�0}68(I(1-I9'240@O80@AFE
m"�-�2H;32H;)0f{l$'(I(7$1"�-79�OB?)AC$�?L08AF2�MWq
m:46@9%a�0 AC0@?)AC0@E�0@9'2C01*�K
-79p-79���9�-.2C08( M�J=619DM "V6GM�EGK�6@E¤6 EH&�Jº$'{z2H;�AC0@0 *�-IEH2C-I9�:C2XO8$)$)*p-I9'240@O80@AFEQ"%;)$�E�0?�AC$)*'&L:C2v-IEB2H;)0�Eb°'&�61AC0V$'{#619>$)*)*,-I9'240@O80@A�R� s RtF@0G2
pa)0=6>?�AF-7J>0=9�&�JQa)0@A�Rw»�AC$@y�052H;�6G2}2H;)08AC0Q0H®1-IEH2FE�6>?)AC-IJQ0=9�&�J>a�08A
qE<&L:<;�2H;�682bK'{l$�Av0by�0@A�M5-79'2C01O808AnKD2H;)0V9�&�J>a�08A
np − p-7Ef9�$D2B*'-�y'-7EH-+a'(+0Va@M
qR
� � R S�;)0YE�01°'&L0@9�:40a0K
a1K
a2K
. . .-7EX*)08��9�01*�6@E>{l$'(I(7$1"VE�q
a0 = 2K36@9�*
ak+1 = 2a2k − 1
{l$�Ak ≥ 0
R«»�AC$@y�0w2H;�6G25-I{36@9p$)*)*¢?�AF-7J>0p*�- y�-7*�08E
an
K2H;)0@9
2n+3 *'-�y'-+*)0@E p2 − 1R
� � R�F@0[2pa)0,65?�AF-7J>0,9�&�J>a�08AbK#6@9�*Q(+0[2
Aa�056,E�0G2}$'{T?L$�EH-.2C- y�0%-79'2C01O808AFE�2H;�6G2
E�6G2C-7EH��08Ev2H;)0z{l$'(I(7$1"�-79�Oz:H$�9�*'-.2C-7$�9)E�q]I-�`�2H;)0 Eb0[2�$'{�?)AC-IJQ0}*�- y�-IEb$�AFE^$'{12H;)0�0G(+08JQ089'2CET-I9
A:H$�9)EH-7E42CE^$'{
p−108(70@J>0@9'2FE�r
]7-.-�`5{l$�AV619DMY9�$�9�P<0@J=?�2�M_E<&La�Eb0[2�$'{AKN2H;)0�?)AC$)*�&L:C2�$'{B-.2FE,0G(+08JQ089'2CE�-IE59�$D2%6
?L08AC{l0@:C2p��� ?�$1"%0@A�R
��;�682�-IEv2H;)0�(I61ACO808EH2v?�$�E�EH-+a'(+0�9�&�J>a�08AB$'{s08(70@J>0@9'2FEB-79Ao
²�08{l$�AC0%2C&�A49�-I9�O%24$QEb$'(.&�2C-+$�9)E "%0,O[-�y�056=AC086D*)0@A�� E�:4$�J>J>0@9'2�$�9¤6,Eb$'(.&�2C-+$�9?)&La'(I-IEb;)0@*�-79X2H;)0Qª :C2C$)a�08A �1�1�D 8Q���J��#$� R=S�;)0=?�AC$)a'(+08J·-IEz{+AC$�J 2H;)0QZ8-I9�OG61?�$�AC0WY682H;)08J>6G2C-+:C6@(�ªB(�M�J>?�-761* � �1�1� D q � � D BD� � r �1�1�D q �@¨ �� R� Rv�1y'68(I&�6G240
2002∑
k=1
k · k!
2k−
2002∑
k=1
k!
2k− 2003!
22002
R
8Q��>?>@#���� ��� &(��)*�,+>�Q�Z��+ ��# �1��}�5�,)#$�6�O����� ��� � ���6�Q#$1 �D�s���Q� 1 ����1��}� � ES#;)0V?�&La�(.-7E�;)01*=E�$'(I&�2C-7$�9=0@E42C61a�(.-7E�;)0@Ev2H;)0z{l$�A4J,&)(76
n∑
i=1
(k − 1)k!
2k=
(n + 1)!
2n − 1
a@M¤&�E<-I9�O�-79�*'&L:C2C-+$�9w$�9nRX²�&�2}-�2}-IE�0@6@E<-70@A}6@9�*�J>$�AC0=0@9�(.-+O8;D2C0@9�-I9�O�24$¤$)a)E�0@A�y�0
�D�
2H;�682bK�EH-79�:40(k − 1)k! = (k + 1)k! − 2(k!)
KD"%0V;�6Gy�0n∑
i=1
(k − 1)k!
2k=
n∑
k=1
(k + 1)!
2k−
n∑
k=1
k!
2k−1
K
"%;'-+:<;�240G(+08Eb:4$�?L08Ev2C$�2H;)0VE42C6G240@*=AF-+O8;D2HP<;�619�*=EH-+*)0�R
��0�9�$1"iE�;'-I{I2v2C$>AC086D*)0@AFE[�)Eb$'(.&�2C-+$�9)Ef2C$>?�AC$)a'(+08J>E -79=2H;)0 � $8y�0@J>a�08A �@�D�;D8Q���J��#$� 619�*X2H;)0=Z108(701:C2C-7$�9¤S)0@E42}{l$�A 2H;)0 © ��� � 682C-7$�9)6@(^ªB( M)J=?)-I6D*�$'{ ²�$�E<9�-76=6@9�*¯V08AF¼<0@O8$@y'-79)6 �1�1�'� K�O[- y�0@9=682f� �@�D�;D q A5�@ Rn ROF@0G2
xKyK�619�*
za)0VAC0@68(L9�&�JQa)0@AFE E<&L:<;�2H;�6G2
x + y + z = 36@9�*
xy + yz + xz = a]a-7E^6}AC086@(D?)61AF6@JQ0[2408A�`<Rf�f0[2408A4J,-79�0B2H;)0vy'6@(.&L0�$'{12H;)0}?)61AF6@JQ0[2408A
a{l$�AL"%;'-+:<;32H;)0
*�-.¶�08AC0@9�:40 a�0[2�"%01089�2H;)0BJ>6�®1-IJ=&�J�6@9�*zJ,-79�-IJ=&�J�?L$�E<E<-7a�(70Ny'6@(.&L0@ET$'{x01°'&�6@(IE
8R
�����)# + ��� ����#��J�O# �}���J��� �$��#*����1 � ��� �$���-��%��2G �Z�,#*1 � �Z� �JHO#�� ���+ ���5)$������ ��4��;��0�5+5� � � �'1 ���6�O#��2�'1 �7�s##�HJ#5E � # ;���)#��I�}���J��� �$��#*����� ���$��Z�����,���7E
��0z;�6Gy�0y + z = 3 − x
6@9�*yz = a − x(3 − x)
R�!+2^-7Ev0@6@EFMV24$zy�0@AF-I{.MV2H;�6822H;)0VEFM)E42408J
y + z = sK
yz = pK
"�-.2H;z&�9�/D9�$1"V9)Ey619�*
zK1;�61EN6�AC086@('E�$'(I&�2C-7$�9�-I{�619�*%$�9�(�M�-.{
s2 −4p ≥ 0R�¯�0@9�:40�K
2H;)0=&�9�-7°�&L0>:H$�9�*'-.2C-7$�9X"%0>;�6[y�0,2C$�E<682C-IE<{.MX-IE(3 − x)2 ≥ 4(a − 3x + x2)
KT$�A3(x − 1)2 ≤ 4(3 − a)
R^S�;�682�-IEGKa ≤ 3
6@9�*
1 − 2√
1 − 13a ≤ x ≤ 1 + 2
√
1 − 13aR
S#;)0Y*'-I¶s0@AC089�:H0ua�0[2�"%01089 2H;)0uJ=6[®@-7J,&�J 6@9�*¢J,-79�-IJ=&�J£?L$�E<E<-7a�(70¤y'6@(.&L0@EQ$'{x01°'&�6@(IE
8-I{#6@9�*>$�9�(�M5-.{
4√
1 − 13a = 8
K1"%;'-+:<;5-7J=?)(.-+08Ev2H;�6G2a = −9
R\ R�S�AF-76@9�O[(+0
ABC-IEzO[- y�0@9X-I9¤6,?)(I619�0)R=�BAF6G" 2H;)0=a'-7E�01:C2C$�AFEz$'{^68(I(�2H;�AC010,$'{T-.2FE
619�O[(70@E8R�S#;)089w*)AF6["�2H;)0,(.-79�0�2H;�6G23:H$�9)9�0@:C2CE�2H;)0=?�$'-79'2FE}"%;)0@AC0�2H;)0>a'-7E�01:C2C$�AFE�$'{619�O[(70@E
ABC619�*
ACBJQ0@0G2B2H;)0�E<-7*�08E
AC619�*
ABKLAC0@E<?L0@:C2C-�y�0G(�MsR3S�;�AC$'&LO8;
2H;)0X?L$'-I9'2V$'{v-79'2C0@AFE�01:C2C-7$�9Y$'{�2H;)0¤a'-7E�01:C2C$�AV$'{B6@9�O[(+0BAC
6@9�*w2H;)0X?�AC0by'-+$'&�EH(�M*)AF6["V9%(.-79�0'KD*�AF6G"�619�$D2H;)08A�(.-79�0'K1?�6@AF6@(.(+0G(124$32H;)0�E<-7*�0
BCR4F@0G2s2H;'-IE^(I-I9�0}-I9'2408AFEb0@:C2
2H;)0zEH-+*)0@EAB
6@9�*AC
-795?�$'-79'2FEM
619�*NRB»�AC$@y�0}2H;�682
2MN = BM + CNR
�����)# +������ ��;���#H� �Y>@#���; ����@8Q�5)$�9�s1�8������ � �A; �5#��Z�1 �����B�,��� HZ��1 �$��������� � �Q+��# � ���s# ��� E �Q� �Q+5���B��1m- �>@+ #���1Y8 �H1p�o� � E�� #f;���)# �C��##�:�Z�,�,# %H�$� �����Y>@#���; ����8Q��)$�2�'1�>@��+ � ��# +>��� �C��# # + �,�,���0�E0 c�h h �LRz!�9>24AC-I619�O[(70
ABCK)2H;)05a�-IEb0@:C24$�Af$'{
∠ACBJ>010[2CE 2H;)0�EH-+*)0
AB6G2
DK
619�*=2H;)0�a'-7E�01:C2C$�Af$'{∠ABC
JQ0@0G2FE 2H;)0%EH-+*)0AC
682ERMF@0G2
Pa)0%619DM>?L$'-I9'2 $�9
2H;)0VE�01OGJ>0@9'2DE
ROF@0G2XKYK�6@9�*
Za)0z2H;)0V$�AF2H;)$)O8$�9)68(L?�AC$1��0@:C2C-+$�9)E $'{
P$�9'2C$
2H;)0VE<-7*�08EBC
KAC
K�619�*AB
K)AC08E�?�01:C2C- y�08( M�RTS�;)0@9PX = PY + PZ
R
�"�
�0�s����� � F@0G2R6@9�*
Sa�0u2H;)0 $�A�2H;)$)O8$�9)6@(
?�AC$1��0@:C2C-+$�9)EV$'{D$�9'24$Q2H;)0=E<-7*�08E
BC6@9�*
ACK AC0@E<?L0@:C2C-�y�0G(�MsR F@0G2
T619�*
Ua)0�2H;)0
$�AF2H;)$)O8$�9)68(Y?�AC$1��0@:C2C-+$�9)E $'{E
$�9'24$«2H;)0E<-7*�08E
BC6@9�*
ABKzAC08E�?�01:C2C- y�08( M�R � $D2C0
2H;�682DR = DS
619�*ET = EU
RF@0G2r = DP/DE
RBS�;)0@90 < r < 1619�*
PE/DE = 1 − rR�Z8-I9�:H0
PY ‖ DSK
"%03;�6Gy�0PY = (1−r)DS = (1−r)DR
rE<-IJ=-.(76@AC( M#K#EH-79�:40
PZ ‖ EUK�"%0,6@(IEb$¤;�6[y�0
PZ = rEU = rETR�S#;)089
..
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
.....................................................................................................................................
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
......................................................................................................................................................
......................................................................................................................................................................................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................
.
.
.
.
.
.
.
.
.
.
..................................
............................................
.
.
.
.
.
.
.
.
.
.
..................................
.
.
.
.
.
.
.
.
.
.
..................................
.
.
.
.
.
.
.
.
.
.
..................................
............................................
.
.
.
.
.
.
.
.
.
.
..................................
A
B C
D
E
P
R
S
T
U
X
Y
Z
PY + PZ = (1 − r)DR + rET = PXK
E<-I9�:H0DR ‖ PX ‖ ET
619�*DP/PE = r/(1 − r)
R� $1"³"%032C&�AC9,2C$�2H;)0%O[- y�0@9=?�AC$)a'(+08JxR
¹ Ex-79³2H;)0¢(+08J>J=6)KV"%0¢(+0[2D
619�*Ea)0
2H;)0�?�$'-79'2FEB"%;)0@AC0�2H;)05a'-7E�01:C2C$�AFE�$'{∠ACB619�*
∠ABCJQ0@0G2v2H;)0%E<-7*�08E
AB6@9�*
ACK
AC0@E<?L0@:C2C-�y�0G(�MsR F@0[2Pa)0>6@9DMw?L$'-I9'2z$�9¤2H;)0
Eb0@OGJQ089'2DE
RT] F86G2408AbKb"%0�"�-I(.(@AC01°'&)-7AC0P2C$
a�0V$�952H;)0Va'-7E�01:C2C$�Av$'{∠BAC
K'61EfAC0@°�&)-IAC01*-7952H;)0V?)AC$)a�(70@JwR `F@0G2
[UV W ]*�089�$D240 2H;)0¢61AC086p$'{�6
24AC-I619�O[(70UV W
R�Z8-I9�:H0MN ‖ BC
K�"%0
..
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
..
..
...
..
..
..
..
...
..
..
..
...
..
.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................................................
..........................................................................................................................................................................................................................................................................................................................
.....................................................................................................................................
.
.
.
.
.
.
.
.
.
.
..................................
............................................
.
.
.
.
.
.
.
.
.
.
..................................
A
B C
D
E
MN
P
X
Y
Z
Eb0@0z2H;�6G2[MBP ] = [MXP ]
6@9�*[PCN ] = [PXN ]
Rv¯�0@9�:40�K
[MBP ] + [PCN ] = [MXP ] + [PXN ] = [MXN ]R
Z8-79�:40�2H;)0x61AC086Y$'{ 4MBPJ=6GM a�0u0H®D?)AC0@E<Eb0@*¢a@M 1
2BM · PZ
KB2H;)0x6@AC0@6u$'{4PCN
a@M 12CN · PY
K'619�*�2H;�6G2v$'{ 4MXNa@M 1
2MN · PX
K@"%0�J=6GMV"VAC-�2402H;)0V6Da)$@y�0V01°'&�682C-7$�9=61E
BM · PZ + CN · PY = MN · PXR ] � `
� $1"�61E<E<&�J>0�2H;�682P(.-+08E�$�9X2H;)0=a�-IEb0@:C24$�A�$'{
∠BACR%S#;)089
PZ = PY619�*�KLa@M,2H;)0�(+08J>J=6)KPX = 2PY
RvZ8&La)E42C-.2C&�2C-I9�O�-I9'24$>] � `N6@9�*Q*'-�y'-+*'-79�OVa@MPY
K"%0%$)aD2C68-7952H;)0V*�08E<-IAC01*,AC0@EH&)(.2[Rp RN!l{
n-IET6}9)6G2C&�AF6@(D9�&�J>a�08AbK@?)AC$@y�0B2H;�6G2�2H;)0 9�&�J>a�08A
(n+1)(n+2) · · · (n+10)-7Ef9�$D2v6V?L08AC{l0@:C2vEb°'&�61AC0)R��������������I��� ����#��J�O# �}���J�-� �$�,#H�6��1 � ��� �$���-��%��2G �Z�,#*1 � �Z� �JHO#�E
¹ E�EH&�JQ0'Ks{l$�A 2H;)0=?�&�A4?�$�Eb0>$'{�:H$�9'24AF61*�-7:C2C-+$�9#K#2H;�6G2n-7Ez6>9)682C&�AF68(^9�&�JQa)0@A
E<&L:<;�2H;�682(n + 1)(n + 2) . . . (n + 10)
-IEf6V?L08AC{l0@:C2BE�°�&�6@AC0�R¹ :H$�J=JQ$�9,*'-�y'-7E�$�A�$'{L6@9DM%2�"%$5$'{)2H;)0z9�&�J>a�08AFE
n + 1Kn + 2
K. . .
Kn + 10-7E�682 J>$�EH2
9R ¹ (7E�$�KL61J>$�9�OV2H;)08Eb0
109�&�JQa)0@AFE[K#6G2}J>$�EH2v2�"%$X:46@9�a)05J,&)(.2C-I?)(70@E
$'{76@9�*5682NJ>$�EH2T2�"%$5:C619,a�03J=&)(�2C-7?�(+08Ef$'{
5R�!+2N{l$'(.(+$1"VE^2H;�682N682^(70@6@EH2
6$'{)2H;)08Eb0
��A
9�&�JQa)0@AFE�;�6[y�0,$�9�0,$'{�2H;)0�{l$�ACJ>Ex2 K 2x2 K 3x2 K#6@9�* 6x2 R5²�M>2H;)05?)-7O801$�9�P<;)$'(+0?�AF-79�:4-I?)(70�KD2H;)08AC0V0<®@-7E42
iKj ∈ {1
K2K. . .
K10} "�-�2H; i < j
EH&L:<;�2H;�682n + i = kx2619�*
n + j = ky2 {l$�A�E�$�JQ0�?L$�EH-.2C- y�0z-I9'240@O80@AFE x619�*
y6@9�*
k ∈ {1K2K3K6} R� $1"
k(y2 − x2) = j − iKD619�*V2H;'&�EGK
1 ≤ k(y2 − x2) ≤ 9R ¹ (IEb$�KG"%0z;�6[y�0
kx2 = n + i ≥ 2RzS�;)05?L$�E<E<-7a�-.(I-�2C-+08E}{l$�A
(x, y, k)E�6G2C-7EH{.M�-I9�O�2H;)0@E�0,:4$�9�*�-�2C-+$�9)E
61AC0>61Ez{l$'(I(7$1"VE�q(1, 2, 2)
K(1, 2, 3)
K(2, 3, 1)
K(3, 4, 1)
K^6@9�*(4, 5, 1)
Rw!�9u0by�0@A�M:46@Eb0
kx2 ≤ 16K�6@9�*x2H;)08AC08{l$�AC0>2H;)0XEb0@°�&L089�:H0
n + 1K
n + 2K
. . .K
n + 10-IE
:H$�9'2F6@-I9�01*x-I9x2H;)0QEb0[2 {2K3K
. . .K25} R¢²�&�2z2H;)089x2H;)0QEb0@°�&L089�:H0XJ,&�EH2V:H$�9'2F6@-I908-�2H;)0@A
11$�A
17R�Z8-79�:40V9�$=J>$�AC0z2H;�6@9Q$�9�0�{+61:C24$�Av$'{
11$�A
17:46@9Qa)0V?�AC08Eb089'2v-I9
2H;)0z?)AC$)*�&L:C2(n +1)(n +2) · · · (n +10)
K8"%0z:H$�9�:4(.&L*�0}2H;�682�2H;'-7Ev?)AC$)*�&L:C2^-IEv9�$D26%?�0@AF{l01:C2vEb°'&�61AC0'K1"%;'-+:<;=:4$�9'24AF6D*'-+:C2FE}$'&�AN-79�-�2C-768(L61E<E<&�J=?�2C-7$�9�R_ R�F@0G2
aKbK#6@9�*
ca)05AC0@68(�9�&�JQa)0@AFEzEH&L:<;>2H;�682
a2 + b2 + c2 = 1R,»�AC$8y�0�2H;)0
-79�0@°�&�68(I-�2\Ma2
1 + 2bc+
b2
1 + 2ca+
c2
1 + 2ab≥ 3
5
R�����)# +���� � �A; �5#*� �Y>@#���; ���� 8Q�5)$�9�s1 8������ � ��;���#$�4��1 �����B�,��� HZ��1 �������6� � ����#��J�O#�}���J�-� �$�,#H�6��1 � ��� �$���-��%��2G �Z�,#*1 � �4�� HJ# � �Y)*�,+ ��� � ��s+Q����1����4�5+ #����61 � # ���,#$�J�� �+H;���A; �� �7����)#��0�s�,�C��1 �����6�>?� �*��� 1 ���U1j�o� � ������)*�,��� � ��4��;��0�5+5� � � �'1 ���6�O#��2�'1�7�s##�HJ# ��� �Q+ 0# +Q���6���yE � � �H���1�& �5)#��41 � �m1��o� � E�� # ;��,)# � ��s+Q��� � ���$��Z�����,���7E
!�9_y'-+0�"�$'{ 2H;)0�-79�0@°�&�68(I-�2\M2xy ≤ x2 + y2 619�*_2H;)0x$)a)E�0@A�y'6G2C-+$�9_2H;�6G2
1 + 2bc = a2 + (b + c)2 > 0K)0G2C:GR+K1"%0�Eb0@0z2H;�6G2
a2
1 + 2bc+
b2
1 + 2ca+
c2
1 + 2ab≥ a2
1 + b2 + c2+
b2
1 + c2 + a2+
c2
1 + a2 + b2
=a2
2 − a2+
b2
2 − b2+
c2
2 − c2
RF@0G2
f : (0, 1) → Ra)0%*�0G�L9�0@*=61Ef(x) = x/(2 − x)
R^S�;)0@9
f ′(x) =2
(2 − x)26@9�*
f ′′(x) =4
(2 − x)3R
S#;'&�E[Kf-IE :H$�9Dy�0H®�K�E<-I9�:H0
f ′′(x) > 0R�� EH-79�O}µ�089)Eb089#� E�!�9�01°'&�6@(.-.2�M#K1"%0%O80[2
f(
x + y + z
3
)
≤ f(x) + f(y) + f(z)
3
RS)61/1-I9�O
x = a2 K y = b2 KL619�* z = c2 O[- y�0@E f(a2) + f(b2) + f(c2) ≥ 3f(
13
) r2H;�682�-IEGK
a2
2 − a2+
b2
2 − b2+
c2
2 − c2≥ 3 ·
1
3
2 − 1
3
=3
5
R� $D2C0!q�0@°�&�68(I-�2\M5-7E $�9�( M=?�$�E�EH-+a'(+0z-.{
a2 = b2 = c2 = 13
Ra R F@0[2
p6@9�*
qa�0>*'-I¶s0@AC089'23?)AC-IJQ0,9�&�JQa)0@AFE8R>Z1$'( y�052H;)0,{l$'(.(+$1"�-I9�O5EFM�EH2C0@J·$'{
01°'&�682C-7$�9)Ef-I952H;)0VE�0G2B$'{L-79'2C01O808AFE�qz + p
x+
z − p
y= q
Kz + p
y− z − p
x= q
R
�"D
��������������I��� ����#��J�O# �}���J�-� �$�,#H�6��1 � ��� �$���-��%��2G �Z�,#*1 � �Z� �JHO#�E� $D2C0z2H;�6G2
xy 6= 0RNU�(70@6@AC-I9�O�2H;)0%*)0@9�$�J,-79)6G24$�AFE[K1"%0%O80[2
(z + p)y + (z − p)x = qxyK
(z + p)x − (z − p)y = qxyR
²�M,��AFEH2vEH&La'24AF61:C2C-79�O�2H;)0%0@°�&�6G2C-+$�9)E[KD2H;)089>61*�*'-79�O�2H;)0@J¤K1"%0%O80[2zy = px
K ] � `xz + py = qxy
R ] � `Wu&)(.2C-I?)( M�-I9�Ox] � `�a@M
z6@9�* &�EH-79�Ow] � `HKB"%0uO80[2
xz2 + xp2 = qpx2 KB"%;'-7:<; -IE01°'&)-�y'68(+089'2^24$z2 + p2 = qpx
RTS�;'-7Ef0@°�&�6G2C-+$�9�-7J=?)(.-+08E�2H;�6G2p*'-�y'-+*)0@E
z]+EH-79�:40
p-7E ?�AF-7J>0[`<RvS�;'&�EGKp2 *�- y�-7*�08E qpx
RfZ8-I9�:H0q 6= p
619�*q-IE}?)AC-IJQ0'K)-�2�2H;)089>{l$'(.(+$1"VE
2H;�682p*'-�y'-+*)0@E
xR F@0G2
x = pa619�*
z = pcKL"%;)0@AC0
a619�*
c61AC05-79'2C01O808AFE�6@9�*
a 6= 0RTS#;)0�EFM)E42408J -IEf9�$1"
cy = paK ] � `
pac + y = qayR ] A `
�'AC$�J¦] A `HK1"%0%*)01*'&L:H0z2H;�6G2a*'-�y'-+*)0@E
yK�E�6[M
y = abRTS#;'&�E[K
cb = pK ] D `
pc + b = qabR ] `
Z8-79�:40p-7Ef?)AC-IJQ0'K'-.2N{l$'(I(7$1"VEB{+AC$�J¦] D `�2H;�682
(b, c) ∈ {±(p, 1)K ±(1, p)
} R/l� ��c >�R
(b, c) = (p, 1)R
�'AC$�J«] `HKD"%0�;�6Gy�0qa = 2
K'"%;'-+:<;,-7J=?)(.-+08EB2H;�682q = 2
6@9�*a = 1
RBS�;)0@9(x, y, z) = (p, p, p)
K1"%;'-7:<;,-IEf6VEb$'(.&�2C-+$�95{l$�Ap619DM>$)*)*=?)AC-IJQ0)R
/l� ��c�k R(b, c) = (−p, −1)
R¹ Es-I9zU#6@Eb0 � Kb"%0f$)aD2C68-79
q = 2619�*
a = 1RsS#;'&�E[K
(x, y, z) = (p, −p, −p)R
/l� ��c�� R(b, c) = (1, p)
R�)°'&�682C-7$�9,] `�E<-IJ>?�(I-.��08EN2C$
p2 + 1 = qaR�S#;)089
qJ,&�EH2�a)0z6z?�AF-7J>0�*'-�y'-7E�$�A
$'{p2 + 1
K'-79�"%;'-7:<;=:46@Eb0
(x, y, z) =
(
p(p2 + 1)
q,
p2 + 1
q, p2
) R/l� ��c&m R
(b, c) = (−1, −p)R
¹ EB-79>U�61E�0 � K1"%0z�L9�*52H;�6G2qJ=&�E42B*�- y�-7*�0
p2 + 1RTS#;)089
(x, y, z) =
(
p(p2 + 1)
q, −p2 + 1
q, −p2
) RS#;)08AC08{l$�AC0'Kz2H;)0 Eb$'(.&�2C-+$�9)E¤6@AC0
(x, y, z) ∈ {
(p, p, p)K
(p, −p, −p)} K�-.{
q = 2K'{l$�Av6@9DM=?)AC-IJQ0
p > 2K�619�*
(x, y, z) ∈{(
p(p2 + 1)
q,
p2 + 1
q, p2
) K (p(p2 + 1)
q, −p2 + 1
q, −p2
)} K{l$�A^619DM�?)AC-IJQ08E
p6@9�*
qEH&L:<;V2H;�682
q*'-�y'-+*)0@E
p2 +1]."%;'-7:<;%-IJ>?�(I-70@E^2H;�682
p 6= q`<R
�1
s R�F@0[2B2H;)0�y�08AF2C-7:H08E3$'{s2H;)05:4$�9Dy�0<®=°'&�6D*�AC-.(76G2408AF6@(ABCD
6@9�*>2H;)0%-I9'2408AFEb0@:C2C-79�O?L$'-I9'2
S$'{�-.2FE�*'-761O8$�9)6@(IE�a)0>-79'2C01O808Az?L$'-I9'2CEV-I9�2H;)0Q?)(I619�0)R F@0[2
Pa�0=2H;)0Q6@AC0@6
$'{B2H;)0�°�&�61*)AF-I(I682C0@AF68(ABCD
619�*P1
2H;)0�6@AC0@6w$'{v24AF-76@9�O[(+0ABS
R³»�AC$8y�0X2H;)0{l$'(I(7$1"�-79�O�-79�0@°�&�68(I-�2\M3q
√P ≥
√
P1 +
√2
2
R��������������I��� ���5)$����� � ��4��;��0�5+5� � � �'1 ���6�O#��2�'1 �7�s##�HJ#$1�>@��+ � ��# +>��� �C��# # + �,�,���*E0 c�h h �LR^!l{
X1KX2
K8619�*X3
6@AC0 9�$�9�P<:H$'(.(I-I9�0@6@A#?L$'-I9'2CE#"�-�2H;3-79'2C01O808A�:4$)$�AC*�-I9)682C0@E[K2H;)0@9
[X1X2X3] ≥ 12
R�0�s����� � F@0G2
Xi
;�6[y�0�:4$)$�AC*�-I9)682C0@E(ai, bi)
KG"�-�2H;ai
Kbi ∈ �
KG{l$�Ai = 1
K2K3R�S�;)0@9
2[X1X2X3] =
∣
∣
∣
∣
∣
∣
a1 b1 1a2 b2 1a3 b3 1
∣
∣
∣
∣
∣
∣
∈ � ∗+
R
¯V089�:H0'K2[X1X2X3] ≥ 1
r@2H;�682�-IEGK[X1X2X3] ≥ 1
2
RF@0G2
P2KP3KD619�*
P4a)0�2H;)0z61AC0861Ef$'{)24AC-I619�O[(70@E
BCSK
CDSKADS
KLAC08E�?�01:C2C- y�08( M�Rf��0�;�6[y�0P1/P2 = AS/CS
Ka�0@:468&�Eb0 4BAS
;�61Ex2H;)0pE�6@JQ0�;)0G-+O8;D2w{+AC$�J 2H;)0�a)6@Eb0AS
6@E *�$)08E 4BCS{+AC$�J´2H;)0�a)6@Eb0
CSR�Z8-7J,-I(I61AF(�M�K
P4/P3 = AS/CSR S�;'&�EGK
P1/P2 = P4/P3R§S#;)089
P1 = P2P4/P3 ≤ 2P2P4K�EH-79�:40
P3 ≥ 12
a@M=2H;)0�(70@J=J>6�R¯V089�:H0'K
√
2P1 ≤ 2√
P2P4 ≤ P2 + P4K
.................................................................................................................................................................................................................................................................................................................................................................................
...............
...............
...............
......................
...............
...............
...............
...............
...............
...............
...............
...............
......................................................................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................
A B
CD
S
P1
P2
P3
P4
&�E<-I9�O32H;)0 ¹ W B BW¡!�9�01°'&�6@(.-.2�M�R^S�;)0@9
P = P1 + P2 + P3 + P4 ≥ P1 + P3 +√
2P1
≥ P1 + 12
+√
2P1 =(√
P1 +√
22
)2 RS)61/1-I9�O�E�°�&�6@AC0VAC$)$D2CE O[- y�0@Ev2H;)0%*)0@EH-7AC0@*=AC08E<&)(�2[R
� 0H®@2#"%0 (+$)$)/36G2TEb$'(.&�2C-+$�9)ET{+AC$�J�$'&�A�AC0@61*�08AFE^24$V?)AC$)a�(70@J=Ev$'{D2H;)0 A ��� ¯V$�9�O�s$�9�O5WY6G2H;)0@J=682C-7:468(#ªB( M)J=?)-I6D*=O[-�y�089=682 � �1�1� D q A5�D©� R\ RN�D-I9�*z68(I(@?L$�EH-.2C- y�0v-79'2C01O808AFE
nEH&L:<;�2H;�6G2L2H;)0 01°'&�682C-7$�9
x3+y3+z3 = nx2y2z2;�61Ef?L$�EH-.2C- y�0z-79'2C01O808AvEb$'(.&�2C-+$�9)E8RB²�0VEH&�AC0z2C$=O[- y�0V6V?�AC$)$'{CR��������������I��� ����#��J�O# �}���J�-� �$�,#H�6��1 � ��� �$���-��%��2G �Z�,#*1 � �Z� �JHO#�E
F@0G2n ≥ 1
a)036@9%-I9'240@O80@ATEH&L:<;�2H;�682s2H;)0@AC030<®@-7E42T?L$�EH-.2C- y�0}-I9'240@O80@AFExKyK16@9�*
zE<682C-IE<{.M'-79�O
x3 + y3 + z3 = nx2y2z2 R ] � `
�D©
�X-.2H;)$'&�2f(7$�E�E3$'{TO80@9�08AF6@(.-.2�M#K)"%0�J>6[MX6@E�EH&�JQ0V2H;�682x ≥ y ≥ z
R3S�;)0@9>"%05;�6[y�0nx2y2z2 ≤ 3x3 K1"%;'-7:<;5(+086D*�Ev24$
ny2z2 ≤ 3xR ] � `
�'AC$�J¦] � `HK1"%0�6@(IEb$,;�6Gy�0x2(ny2z2 − x) = y3 + z3 > 0
K'-7J=?)( M�-I9�O�2H;�682
ny2z2 ≥ x + 1R ] � `
� $D2C0V2H;�682(y3 − 1)(z3 − 1) ≥ 0
K�"%;'-+:<;=(+086D*�E 24$1 + y3z3 ≥ y3 + z3 R� E<-I9�O�] � `^619�*52H;)089Q] � `HK1"%0VO80[2
1 + y3z3 ≥ x2(ny2z2 − x) ≥ x2 R ] A `S#;'&�E[K�&�EH-79�O�] � `HK1"%0V*�0@*�&L:40z2H;�682
9(1 + y3z3) ≥ n2y4z4 R ] D `/l� ��c >�R
y = z = 1R
�'AC$�J¦] A `HK1"%0�Eb0@0z2H;�6G2x = 1
K�6@9�*52H;)0%O[- y�0@9>0@°�&�6G2C-+$�9�M�-708(7*)En = 3
R/l� ��c�k R
yz > 1R
S#;)089y4z4 ≥ 2y3z3 > 1 + y3z3 R�S#;'-IE32C$)O80G2H;)08Af"�-�2H;�] D `f-IJ>?�(I-70@E�2H;�6G2
n2 < 9RB¯�0@9�:40�K
n = 1$�A
n = 2R
��$�An = 1
K'-.2N-7E 0861EFM�24$Vy�0@AF-I{.M�2H;�682(x, y, z) = (3, 2, 1)
-IEf6VEb$'(.&�2C-+$�9�R��$�A
n = 2K@&�E<-I9�O�] D `HK8"%03;�6[y�0
y3z3(4yz −9) ≤ 9KG"%;'-+:<;�{l$�AC:H08E
yz ≤ 2R
S#;)089yz = 2
K�"%;'-+:<;Q-IJ>?�(I-70@E 2H;�682y = 2
6@9�*z = 1
R5!+2 {l$'(I(7$1"VE 2H;�682x ≥ 2
K619�*¤2H;)0>O[-�y�089u0@°�&�6G2C-+$�9wAC0@*�&L:40@Ez2C$
x3 + 9 = 8x2 R>S�;)0@9 x2 J=&�E42z*�- y�-7*�0 9K
"%;'-+:<;,-7J=?)(.-+08EB2H;�682x = 3
R�²�&�2v-.2v-IE}0@6@EFM,2C$�y�08AC-.{.M,2H;�682�2H;'-IEf-7E 9�$D2B6%Eb$'(.&�2C-+$�9�RS#;)08AC08{l$�AC0'K
n = 2-7EB-IJ>?�$�E�EH-+a'(+0)R
S#;)08AC08{l$�AC0'K'2H;)0V*�08E<-IAC01*�y'68(I&L08EB{l$�An6@AC0
n = 16@9�*
n = 3R
p R���$�A 0@61:<;Q-I9'240@O80@Ak ≥ 4
K#?)AC$@y�0%2H;�682f-I{F (x)
-IE36,?�$'(�M�9�$�J=-I6@(L"�-.2H;>-79'2C01O808A:H$)0��X:4-70@9'2FE�"%;'-7:<; E<682C-IE<�L0@Ep2H;)0 :H$�9�*'-.2C-7$�9
0 ≤ F (c) ≤ k{l$�A�0by�08A�M
c = 0K1K. . .
Kk + 1
KD2H;)089F (0) = F (1) = · · · = F (k + 1)
R��������������I��� ����#��J�O# �}���J�-� �$�,#H�6��1 � ��� �$���-��%��2G �Z�,#*1 � �Z� �JHO#�E
S#;'-IE�?)AC$)a�(70@J¦-7E {+AC$�J 2H;)0=!CW_ª � � � © E�;)$�AF2C(.-7E42[R ¹ Eb$'(.&�2C-+$�9Q:C619Xa�0V{l$'&�9�*-79X»�RL²�$�A49)E4¼C240G-79#K �92#*; �>?���C� K �N&)-7a�08AF2bKD?LasR�� ¹ � R �'© RN$�Af�zRL�N�+&L/1-=E:�K �3R8µ<619�/1$8y�-=E:�K!HR#WY682C-=E:�K � Rs»�0[24AC$@y'- E:�K �2��#����� 8Q��> �o#��Q+ �Z�> K)Z@?�AF-79�O808A[R_ RTS#;)08AC0V61AC0
212?L$'-I9'2CEB-79)EH-+*)0V$�AB$�9=6%:4-IAC:4(+0�"�-.2H;5AF6D*'-I&�E
1Rf»�AC$@y�032H;�682N2H;)08AC0
61AC0�682�(70@6@EH22001
?�68-7AFE $'{L2H;)08Eb0�?L$'-I9'2CE ;�6[y�-I9�OV*'-7E42C6@9�:H08E 6G2vJ>$�EH21R
��������������I��� ����#��J�O# �}���J�-� �$�,#H�6��1 � ��� �$���-��%��2G �Z�,#*1 � �Z� �JHO#�E��0zJ=6GM�?�6@AF2C-�2C-+$�9%2H;)0�*'-7E�/z-79'2C$
6:H$�9�OGAF&L0@9'2�E�01:C2C$�AFEGK'086D:<;�"�-�2H;�6V:40@9'24AF68(
619�O[(70�$'{60◦ Kva@MuAF6GM�E�{+AC$�J|2H;)0¤:H089'24AC0
O$'{�2H;)0�*'-7E�/�R_Z8-I9�:H0>2H;)0¤9�&�J>a�08A%$'{
O[-�y�089X?L$'-I9'2CE -7E �L9�-�240'K�"%0�J=6GMX:<;)$)$�E�0V2H;)0�AF6[M)E}Eb$,2H;�6G2 9�$�9�05$'{#2H;)0�?�$'-79'2FE}(I-70
�1¨
$�9X65AF6[M#K�0<®1:40@?'2B2H;�682v2H;)05:H089'24AC0OJ>6[MXa�05$�9�05$'{s2H;)0�?�$'-79'2FE@R�!�9X0@61:<;X$'{s2H;)0
6Eb0@:C24$�AFE[KD2H;)0%*'-7E42C6@9�:H0Va�0[2�"%01089>6@9DM52�"%$,?�$'-79'2FEB-7Ef682vJ>$�EH2
1ROF@0G2
n1K. . .
Kn6a�0z2H;)0�AC0@E<?L0@:C2C-�y�0V9�&�J>a�08AB$'{L2H;)0%O[-�y�089=?L$'-I9'2CEB-79>0@61:<;=$'{L2H;)0
6E�01:C2C$�AFE ]7-.{
O-IE
$�9�0%$'{L2H;)0�?L$'-I9'2CE[K'-.2N-7E :4$'&�9'240@*>$�9�(�M=$�9�:H0'K�E�6[M5-79n1`<R^S�;'&�EGK 6
∑
i=1
ni = 212R
!�9¤E�01:C2C$�AiK�2H;)059�&�J>a�08A}$'{T?�68-7AFE3$'{T?L$'-I9'2CE}-IE 1
2ni(ni − 1)
K�6@9�*¤0@61:<;X$'{2H;)0@E�0=?�68-7AFEV;�6@EV*'-7E42C6@9�:H0=6G23J>$�EH2
1RX¯�0@9�:40�K#2H;)052C$D2C68(^9�&�J>a�08A
N$'{N?�68-7AFE�$'{
?L$'-I9'2CE ;�6Gy'-79�O�*�-IEH2F619�:40@Ef6G2vJQ$�E421E�6G2C-7EH��08E
N ≥6∑
i=1
1
2ni(ni − 1) =
1
2
( 6∑
i=1
n2i − 212
) R
S#;)089#K�&�EH-79�O}µ�0@9)E�0@9#� E3!�9�0@°�&�68(I-�2\M,{l$�A^2H;)0�{7&�9�:C2C-7$�9f(x) = x2 K1"%0V*)01*'&L:H0z2H;�6G2
N ≥ 1
2
(
1
6
( 6∑
i=1
ni
)2
− 212
)
=1
2
(
1
6(212)2 − 212
)
> 3639R
S#;'&�E[KN ≥ 3640
KD"%;'-+:<;5-IE a�0[2F2408AN2H;�619=AC0@°�&)-IAC01*sR
� 0H®@2u"%0�(+$)$)/ 6G2 AC086D*)0@AFE[�,E�$'(I&�2C-7$�9)E 2C$�?�AC$)a'(+08J>E�$'{Q2H;)0 � D ��� !�AF-7E�;WY682H;)08J>6G2C-+:C6@(�ªB(�M�J>?�-761*#KL�1-7AFE42�»'6@?L08AbK)O[-�y�089,-79 � �1�1� D q A5�D© B�A5� � Rn RV!�9¤6%24AC-I619�O[(70
ABCKAB = 20
KAC = 21
K�619�*BC = 29
R3S�;)05?�$'-79'2FED619�*
E(I-70=$�9X2H;)05(I-I9�0,Eb0@OGJQ089'2
BCK�"�-.2H;
BD = 8619�*
EC = 9R,U�6@(7:4&)(76G240
2H;)0V619�O[(70∠DAE
R�����)# + ��� � ����� >?>@#+ �Y�2��O�A���1 ���U�Z�9�"���0� ��;1 � �4�� HJ# � � �A; �5#*� �Y>@#���; ����8Q��)$�2�'1 8����6� � ��;���#$�4��1 � ���A����� H��1 �$��������� �Z���� �5� � �Q��� �s� � � �s1 �h�C�Q#����'1 �7�s##�HJ# �����)*�,��� � ��4��;��0�5+5� � � �'1 ���6�O#��2�'1 �7�O# #9HO#��0���+ ��# � ���s# ��� E �Q� �Q+5���B��1�- � >@+ #$��178 �H1�o� � E�� # ;��,)# �Q� �Q+5���B� � ����������������}E
....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
......................................................................................................................................................................................................................................................................................
........................................................................
A
B CD E
20 21
8 12 9
α β γ
β+γ α+β
F@0G2α = ∠BAD
Kβ = ∠DAE
K)6@9�*γ = ∠EAC
RBZ8-79�:40BA = BE
KD"%0;�6Gy�0
∠AEB = α+βr@E<-IJ=-.(76@AC( M#K1EH-79�:40
CA = CDKG"%03;�6Gy�0
∠ADC = β +γR
S#;)08AC08{l$�AC0'K180◦ = (α + β) + (β + γ) + β = (α + β + γ) + 2β
R²�&�2
α + β + γ = 90◦ K�EH-79�:40 202 + 212 = 292 RNUs$�9)E�01°'&L0@9'2C( M#K β = 45◦ R
� �
\ RY]+6[` ¹ OGAC$'&�?_$'{B?L0@$�?)(70X6G2F24089�*)E�6¤?)61A�2\MsR �'6D:<;x?L08AFEb$�9Y;�61E%682�J>$�EH232H;�AC0@06D:4°�&�68-79'2F619�:40@Ef-7952H;)0VOGAC$'&�?sK)6@9�*,-I{L2�"%$5?L0@$�?)(70%*)$,9�$D2B/D9�$1" 086D:<;=$D2H;)0@A�KD2H;)0@92H;)0bMV;�6Gy�0 6}J,&�2C&�6@(16D:4°�&�68-79'2F619�:40 -7932H;)0}OGAC$'&�?�R���;�6G2s-IEs2H;)0 J>6�®1-IJ=&�J9�&�JQa)0@A$'{#?L0@$�?)(70V?)AC0@E�0@9'2 o
]+aD`z!l{�KT-I9x6D*)*�-�2C-+$�9#K�2H;)0QOGAC$'&�?Y:H$�9'2F6@-I9)Ez2H;�AC010QJ,&�2C&�6@(N6D:4°�&�68-79'2F619�:40@E�].2H;�682-7E[K)2H;�AC0@0�?L0@$�?)(705086D:<;Q$'{�"%;)$�J«/D9�$1"VEf2H;)05$D2H;)0@A�2�"%$1`HK'"%;�682B-IEf2H;)0�J>6�®1-IJ=&�J9�&�JQa)0@AB$'{#?L0@$�?)(70"o�������������� ��� ���,#$�J�s# �}���J��� �$��#*����1 ����� ��������% ��9G �Z��#$1 � �Z� �JHO#*1 >@��+ ���7#+���� �C�Q## + �,�,���*E
]+6[`TS#;)0�J>6�®1-IJ=&�J�-7E10RF@0G2
Aa�0¤619DMY?L08AFEb$�9Y-79x2H;)0wOGAC$'&�?�R³!l{
A;�6@E,9�$u6D:4°�&�68-79'2F619�:40@E[K�2H;)0@9
AJ=&�E42}a�0%2H;)0,$�9�(�MQJQ08JQa)0@A}$'{s2H;)0=OGAC$'&�?�R�!l{
A;�61Ez$�9�( M¤$�9�0561:H°'&�6@-I9'2C6@9�:H0'K
E�6[MBK'2H;)089>619DM>$D2H;)0@AvJ>0@J>a�08A
CJ=&�E42fa)0V619>61:H°'&�6@-I9'2C6@9�:H0�$'{
B]la)01:C6@&�E�0
A619�*CJ,&�EH2 ;�6[y�0�6�J=&�2C&�68(s6D:4°�&�68-79'2F619�:40[` r)2H;)0@9#KLEH-79�:40
B:C619X;�6Gy�0%682fJ>$�EH2
36D:4°�&�68-79'2F619�:40@E[K'2H;)0%OGAC$'&�?Q;�6@Ef682vJ>$�EH24J>0@J>a�08AFE@R
Z8&�?�?�$�Eb0%2H;�682A;�6@Ez0<®@61:C2C(�M>2�"%$Q61:H°'&�6@-I9'2C6@9�:H08EGK#E�6[M
B6@9�*
CR��'6D:<;X$'{
B6@9�*
C:C619Q;�6Gy�0%6G2fJ>$�EH2�2�"%$,{7&�A�2H;)0@AB6D:4°�&�68-79'2F619�:40@E8R Z8-79�:40�6@9DM>?L08AFEb$�9>9�$D2
6D:4°�&�68-79'2C01*V"�-.2H;AJ=&�E42�a�036D:4°�&�68-79'2C01*V"�-�2H;508-�2H;)0@A
B$�A
CK8"%0z:H$�9�:4(.&L*�0�2H;�6G2
2H;)0@AC0V6@AC0V6G2vJQ$�E421 + 2 + 2 · 2 = 7
?�01$�?�(+0z-7952H;)0VOGAC$'&�?TR� $1"pE<&�?)?L$�E�0 2H;�6G2
A;�6@E�0<®@61:C2C(�Mz2H;�AC0@0�61:H°'&�6@-I9'2C6@9�:H08EGK1E�6[M
BKCK@6@9�*
DR
¹ 9DM>$D2H;)08A�?L08AFEb$�9,-I952H;)0VOGAC$'&�?>J,&�EH2Ba)0�6D:4°�&�68-79'2C01*�"�-�2H;BKCK)$�A
DK�6@9�*,{l$�A
0@61:<;w$'{BKCKs6@9�*
D2H;)0@AC0=6@AC0=682�J>$�EH2 2�"%$X{7&�A�2H;)0@A�6D:4°�&�68-79'2F619�:40@E8RX¯�0@9�:40�K
2H;)0@AC0V6@AC0V6G2vJQ$�E421 + 3 + 3 · 2 = 10
?�01$�?�(+0z-I952H;)0%OGAC$'&�?TRS#;'&�E[K�2H;)0wOGAC$'&�?¢:C619 ;�6Gy�0�9�$YJ>$�AC0X2H;�6@9
10J>0@J>a�08AFE@Rpªf9u2H;)0w$D2H;)0@A
;�619�*�KL6�?)61A�2\M5"�-.2H;10?L0@$�?)(70V-7E ?L$�E<E<-7a�(70�K)61E}?)AC$@y�0@*Xa@M,2H;)0�OGAF61?�;X$�9,2H;)0V(+0G{I2
a�0G(+$1" ]�"%;)0@AC0�?L0@$�?)(70V6@AC03y�08AF2C-7:H08E 6@9�*>01*)O80@Ef61AC0�6D:4°�&�68-79'2F619�:40@EC`�R..........................................................................
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
...............................................................................................................................................................................................................................................................................................................................................................................................................................................
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
....
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
.....
..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.........................................................................................................................................................................................................................................................................................................
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
....
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
...
..
...
...
..
...
...
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
..
..
.
..
..
..
.
..
..
..
.
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
..
.
..
..
.
..
..
..
.
..
..
.
q
q
q
q
q
q
q
q A
B
C
D
EF
G
H
I
J............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
.
.
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
............................................................................................................................
.................................................................................................................................................................................
.................................................................................................................................................................................
q
q q
q
q q
A
B C
D
E F
GH
]la1`TS#;)0VJ=6[®@-7J,&�J -IE8RF@0G2
AKBKs6@9�*
Ca�052H;�AC0@0=J=&�2C&�68(^6D:4°�&�68-79'2F619�:40@E8RX!l{
A;�6@E�9�$X{7&�A�2H;)0@A
6D:4°�&�68-79'2F619�:40@E[K#2H;)089w68(I(T$D2H;)0@A}?L08AFEb$�9)EzJ,&�EH23a)0=61:H°'&�6@-I9'240@*Q"�-.2H;¤08-�2H;)0@AB$�A
CKL6@9�*¤086D:<;Q$'{#2H;)0@J :46@9¤;�6[y�0�682fJ>$�EH2 $�9�0%{7&�A�2H;)0@Af6D:4°�&�68-79'2F619�:40�R}S#;'-IE3O[- y�0@E
6�J>6�®1-IJ=&�J«$'{3 + 1 + 1 = 5
?L0@$�?)(70V-79,2H;)0�OGAC$'&�?TR�!l{A*)$)0@E};�6Gy�0%6%{7&�A�2H;)0@A
6D:4°�&�68-79'2F619�:40�K�E<6GMDK�2H;)089>619DM>$D2H;)0@Av?L0@$�?)(70%J,&�EH2fa)0%61:H°'&�6@-I9'240@*5"�-.2H;
BKCK
$�ADRz�'$�A 086D:<;Q$'{
B6@9�*
CK�2H;)08AC0�-IE�6G2fJQ$�E42 $�9�0�EH&L:<;Q61:H°'&�6@-I9'2C6@9�:H0'K�619�*Q6G2
JQ$�E42N2�"%$�{l$�ADR^S�;'-7EB(70@61*)Ev24$5682vJ>$�EH2
4 + 1 + 1 + 2 = 8?L0@$�?)(70�R
¹ ?�6@AF2�M="�-�2H;8?L0@$�?)(70%-IE�?�$�E�EH-+a'(+0'KL6@E�?)AC$@y�0@*¤a@M=2H;)05OGAF6@?L;X$�9=2H;)0�AF-+O8;D2
6Da)$@y�0%]�"%;)0@AC0�?L0@$�?)(70V6@AC03y�08AF2C-7:H08E 6@9�*>01*)O80@Ef61AC0�6D:4°�&�68-79'2F619�:40@EC`�R
�1�
p R��1-79�*V6@(.(124AF-7?�(+08E�$'{�?L$�EH-.2C- y�0 -I9'240@O80@AFE(p, q, n)
KG"�-�2H;p6@9�*
q?�AF-7J>0�K@E<&L:<;z2H;�6G2
p(p + 3) + q(q + 3) = n(n + 3)R
�����)# + ����� ����� >?>@#+ �Y�2��O�A���1 ���U�4�2�2���0��Z;�1 � �Z� �JHO#������,#$�J�s# �}���J��� �$��#*����1� ��� �$���-��%��2G �Z�,#*1 � �4�� HJ# � ���+ ���5)$����� ��� �Z�;5�0��+5� � � �s1 ���6�O#��2�'1E�7�s##�HJ#5E � #m;���)#�Y�9� �s�B�6� � �8�$��Z�����,���7E
�D-IAFEH2¤$'{�68(I(�Kz"%0¢$)a)E�0@A�y�0_2H;�6G2bK%{l$�AX6@9DM�?�$�E<-�2C-�y�0_-79'2C01O808AmK�"%0¢;�6[y�0
m(m+3) ≡ 1 (mod 3)-.{
3 - mK@619�*
m(m+3) ≡ 0 (mod 3)-I{
3 | mRTZ8-79�:40
"%0zAC0@°�&)-IAC0p(p + 3) + q(q + 3) ≡ n(n + 3) (mod 3)
K'6G2^(70@6@EH2N$�9�0�$'{p619�*
qJ=&�E42Na�03R�S#;'&�E[K8"%0�J>6[M%61E<E<&�J>0�KG"�-.2H;)$'&�2�(+$�E<Ev$'{�O80@9�08AF6@(.-.2�M#K@2H;�6G2
p = 3R���0
;�6Gy�0n ≥ q +1
K8"%;'-+:<;�J>0@6@9)EN2H;�682n(n +3) − q(q +3) ≥ 2q +4 > 3(3 +3)&�9�(+08E�E
q ≤ 7RBU#;)01:</@-79�O32H;)0%?)AC-IJQ08E
q ≤ 7K'"%0���9�*,2H;�6G2
q = 2619�*
q = 761AC0
2H;)0%$�9�(�M,E�$'(I&�2C-7$�9)E@RS#;'&�E[KL2H;)0=$�9�( MXEb$'(.&�2C-+$�9)E�61AC0
(p, q, n) = (2, 3, 4)K(p, q, n) = (3, 2, 4)
K(p, q, n) = (3, 7, 8)
K�619�*(p, q, n) = (7, 3, 8)
R_ ROF@0G2N2H;)0�Eb0@°�&L089�:H0
a1Ka2Ka3Ka4K. . .
a)0%*)08��9�01*>a@Ma1 = 1
Ka2 = 1
Ka3 = 1
K|6@9�*an+1an−2 − anan−1 = 2
K{l$�Av68(I(
n ≥ 3Rf»�AC$@y�032H;�682
an
-7Ef6V?�$�E<-�2C-�y�0 ���7��#$;5#�� {l$�A�6@(.(n ≥ 1
R�����)# + ����� ����� >?>@#+ �Y�2��O�A���1 ���U�4�2�2���0��Z;�1 � �Z� �JHO#������,#$�J�s# �}���J��� �$��#*����1� ��� �$���-��%��2G �Z�,#*1 � �4�� HJ# � ���+ ���5)$����� ��� �Z�;5�0��+5� � � �s1 ���6�O#��2�'1E�7�s##�HJ#5E � #m;���)#�Y�9� �s�B�6� � �R�:�Z�,�,# %H�$�DE
¹ 9u0861EFM�-I9�*�&L:C2C-7$�9x?)AC$@y�08Ez2H;�682an = 4an−1 − an−2
-I{n-IE%0by�089#K^6@9�*
an = 2an−1 − an−2-.{
n-7E $)*)*�RTS�;'&�EGK
an
-IEf6V?L$�EH-.2C- y�0z-I9'240@O80@AN{l$�A�6@(.(n ≥ 1
Ra ROF@0G2
0 < aKbKc < 1
Rf»�AC$8y�0z2H;�6G2a
1 − a+
b
1 − b+
c
1 − c≥ 3
3√
abc
1 − 3√
abc
R�f0G2C0@ACJ=-I9�0z2H;)0%:C61E�0V$'{s01°'&�6@(.-.2�M�R�����)# + ��� � ��� �>?>@# + �Y�2��O�A���10���U�4�2�2���0��Z;�1 � �4�� HJ# � � ��;���#H� �Y>@#���; ����78Q��)$�2�'18������ � �A; �5#��Z�1 � ���A����� H��1Y�������6� � ����#��J�O#>�}���J��� �$��#*����1 � ��� �$���-��%��2G �Z�,#*1 � �4�� HJ# �!�����2# ����� � &h2� � �$% �Q� �J�O#��O� 1q�7����)#��0�s�,�6��� �0��A�,� �#�H4�5�&H�� + # 8���������� �����1 �Q�� HJ#*����� ��1�������6� � �Y)*�,+ ��� � � �O+Q���A1 ���4��+ #$�7�61 � # ���,#$�J� � �+H;���A; ����7����)#��0�s�,�C��1������6�>?� �*��� 1��61p�o� � � 0#+Q����� �yE � � �H���1\& �5)#��41�� �m1p�}� � �����5)$����� � ��4��;��0�5+5� � � �'1 ���6�O#��2�'1�7�s##�HJ# �m� �Q+ -/#$� �;� ���+HZ��s+ � 1 � # +�;� � � )#$�6� 8Q��A��#$;5#
(8m��� �
)1��Q�s����� ��� ��1 � ��1
�o� � E�� # ;��,)#y�C�Q#U�:�Z�,�,# %H�$� ��� ���+HZ��s+ �0ES#;)0}{7&�9�:C2C-7$�9
f(x) = x/(1−x)-IEv:4$�9Dy�0<®z$�9V2H;)0 -79'2C0@A�y'68(
(0, 1)Rv!�9z"%;�682
{l$'(I(7$1"VEGK@"%0z&�Eb0�µ�089)Eb089#� E3!�9�0@°�&�68(I-�2\M=6@9�*52H;)089,2H;)0 ¹ W B BW£!�9�0@°�&�68(I-�2\M3qa
1 − a+
b
1 − b+
c
1 − c≥ 3 ·
a+b+c
3
1 − a+b+c
3
≥ 33√
abc
1 − 3√
abc
R�)°'&�6@(.-.2�M>;)$'(7*)EB-I{#6@9�*>$�9�( M,-.{
a = b = cR
� �
s R ¹3 × n
OGAF-+*�-IEz��(.(+0@*w6@E�{l$'(.(+$1"VE8R=S#;)0=��AFEH23AC$1"¥:4$�9)E<-IEH2FEV$'{N2H;)0=9�&�JQa)0@AFE{+AC$�J
124$
n6@A4AF6@9�O801*Q{+AC$�J«(+0G{I2B2C$QAC-7O8;D2 -I9¤6@Eb:40@9�*'-79�O�$�AC*�08A[R3S�;)05Eb0@:H$�9�*QAC$1"
-7Ef6%:�M�:4(.-+:BE�;'-I{I2v$'{L2H;)0z2C$�?>AC$1">RTS�;'&�EGKD2H;)0%$�AC*)0@AvO8$)0@EiKi + 1
K. . .
Kn − 1
KnK1K2K. . .
Ki − 1
{l$�ABEb$�J>0iRBS�;)0V2H;'-IAC*>AC$1"�;�6@Ef2H;)0%9�&�JQa)0@AFE
124$
n-I9QE�$�JQ0%$�AC*�08AbK�EH&LaD��0@:C2v2C$
2H;)0�AC&)(70V2H;�6G2B-79X086D:<;Q$'{#2H;)0n:4$'(I&�J=9)EGK�2H;)0�E<&�J«$'{#2H;)0V2H;�AC0@0�9�&�JQa)0@AFE}-IE 2H;)0
E�6@JQ0)R��$�AL"%;'-7:<;�y'6@(.&L0@EN$'{
n-IET-�2�?L$�E<E<-7a�(70B2C$z�)(I(@2H;)0}OGAC-7*V61:H:4$�AC*�-I9�Ov24$32H;)0 6Da)$@y�0
AC&)(70@E o=��$�A^619n{l$�A�"%;'-7:<;%2H;'-IEv-IEB?�$�E�EH-+a'(+0'K'*�0[2408A4J,-79�0�2H;)0z9�&�J>a�08Av$'{�*�-.¶�08AC0@9'2
"V6GM�E $'{���(.(I-I9�O�2H;)0%OGAF-+*sR��������������I��� ����#��J�O# �}���J�-� �$�,#H�6��1 � ��� �$���-��%��2G �Z�,#*1 � �Z� �JHO#�E
¹ E�EH&�JQ032H;�6G2^"%0VJ=6GM5��(.()2H;)0%OGAF-+*,6D:4:H$�AC*'-79�O�24$�2H;)0�AC&)(70@E8RS#;)089>2H;)0�EH&�J $'{#2H;)0�9�&�JQa)0@AFE}-79Q68(I(L2H;)0V2H;�AC010�AC$1"VE -7E
3n(n + 1)/2R�!+2
{l$'(I(7$1"VEB2H;�682�2H;)0�:H$�J=JQ$�9>E<&�J¦-I9X086D:<;>:H$'(.&�J>9>-IE3(n + 1)/2
K�"%;'-+:<;,{l$�AC:H08En24$�a)0=$)*�*sR F@0G2
n = 2p + 1R�S�;)0@9#Ks61J>$�9�O%2H;)0=9�&�J>a�08AFE
1K2K. . .
KnK�2H;)08AC0
61AC0p + 1
$)*�*59�&�J>a�08AFEf6@9�*p0by�089,9�&�JQa)0@AFE8R�F@0[2
k + 1a�0�2H;)0z9�&�J>a�08Av$'{�2H;)0
:H$'(.&�J>9>:H$�9'2F6@-I9�-79�On-7952H;)0�Eb0@:H$�9�*,AC$1">RTS#;)089,2H;)0VOGAC-7*,-IEf61EB{l$'(I(7$1"VE�q
1 2 . . . k k + 1 k + 2 . . . nn − k n − k + 1 . . . n − 1 n 1 . . . n − k − 1
a1 a2 . . . ak ak+1 ak+2 . . . an!�9V0@61:<;�$'{12H;)0 :H$'(.&�J>9)E12H;�AC$'&LO8;
k+1K�2H;)0 EH&�J $'{@2H;)0f��AFEH2L2�"%$39�&�JQa)0@AFE
;�61E>2H;)0uE�6@JQ0u?�6@AC-�2\M�6@Ek]7{l$�A>0H®@61J=?)(70�K}-79p:H$'(.&�J>9�9�&�JQa)0@A
kK}2H;)0uE<&�J -IE
n − 1 + k = 2p + k`4Ks"%;'-.(+0=-I9u086D:<;x$'{^2H;)0X:4$'(I&�J=9)E
k + 22H;�AC$'&LO8;
nK�2H;)0
E<&�J«$'{L2H;)0V�LAFE42�2�"%$=9�&�J>a�08AFE3;�6@Ef2H;)0V?�6@AC-�2\M>$�?�?�$�E<-�240z24$kRfZ8-I9�:H0z2H;)0%E<&�J¥$'{
2H;)0 9�&�J>a�08AFE^-I9V0@61:<;�:H$'(.&�J>9z-IE^:H$�9)E42C6@9'2bKG-.2�{l$'(I(7$1"VEs2H;�6G2a1Ka2K. . .
Kak+1
;�6[y�02H;)0�E�6@JQ0}?�6@AC-�2\M%6@9�*
ak+2K. . .
Kan
;�6Gy�0 2H;)0�E<61J>0�?)61AF-.2�M�R�S�;'&�EGKk+1 = p+1$�A
k + 1 = pRYW_$�AC01$8y�0@A�KNEH-79�:40=2H;)0QEH&�J|-79u0@61:<;u:4$'(I&�J=9x-IE
3(n + 1)/2K^-.{
2H;)0�y'6@(.&L0V$'{k-7Ef/D9�$1"V9#K12H;)0@9�2H;)0�y'6@(.&L0V$'{
ai
-7Ef*�0[2408A4J,-79�0@*5{l$�Av0@61:<;iK1"%;'-7:<;
JQ08619)Ev2H;�682N2H;)08AC0�-7Ef6G2vJQ$�E42B$�9�0�"V6[M52C$5�)(I()2H;)0%OGAF-+*5{l$�Av6%O[- y�0@9kR
/l� ��c >�Rn = 2p + 1
6@9�*k + 1 = p + 1
R��0VJ=6GM5�)(I()2H;)0%OGAF-+*,61EB{l$'(.(+$1"VE�q
1 2 . . . k k + 1 k + 2 k + 3 . . . 2k + 1k + 1 k + 2 . . . 2k 2k + 1 1 2 . . . k2k + 1 2k − 1 . . . 3 1 2k 2k − 2 . . . 2/l� ��c�k R
n = 2p + 16@9�*
k + 1 = pR
��0VJ=6GM5�)(I()2H;)0%OGAF-+*,61EB{l$'(.(+$1"VE�q1 2 . . . k k + 1 k + 2 k + 3 . . . 2k + 1
k + 3 k + 4 . . . 2k + 2 2k + 3 1 2 . . . k + 22k + 2 2k . . . 4 2 2k + 3 2k + 1 . . . 1S#;)08AC08{l$�AC0'K)2H;)05OGAC-7*QJ=6GMXa)0%�)(I(701*=-I{s619�*X$�9�( M>-.{
n-7E�$)*�*sR ¹ 9�*�K�{l$�ABE<&L:<;
619nKD2H;)0@AC0�61AC032�"%$%"V6[M)Ev24$���(.()2H;)0%OGAF-+*sR
�D�
� RNZ8&�?�?�$�Eb0n-IEf6%?)AC$)*�&L:C2B$'{�{l$'&�AB*'-7E42C-79�:C2�?�AF-7J>0@E
aKbKcKdEH&L:<;�2H;�682
]76G`a + c = d
r]la1`
a(a + b + c + d) = c(d − b)r
]l:F`1 + bc + d = bd
R�f0G2C0@ACJ=-I9�0
nR
�����)# + ����� ����� >?>@#+ �Y�2��O�A���1 ���U�4�2�2���0��Z;�1 � �Z� �JHO#������,#$�J�s# �}���J��� �$��#*����1� ��� �$���-��%��2G �Z�,#*1 � �4�� HJ# �E��# � ���s# ��� E �����+5���A�A1�- �>@+ #���178 �H1��o� � �5� �Q+ ���5)$������ ��4��;��0�5+5� � � �'1 ���6�O#��2�'1 �7�s##�HJ#5E � # ;���)# �����+5���A�;� �:;5#��Q#��Z���B� �s�����,���7E
W_$�AC0%O8089�0@AF68(I( M#K1"%0�"�-I(.(�*)0G2C0@ACJ=-I9�0V6@(.(�-79'2C01O808AvEb$'(.&�2C-+$�9)Ef$'{L2H;)0VEFM�EH2C0@J
a + c = dK ] � `
a(a + b + c + d) = c(d − b)K ] � `
1 + bc + d = bdR ] � `
!l{d = 0
K�-.2�-7E50@6@EFMx24$uEb0@0Q2H;�6G2�2H;)0�$�9�(�MYE�$'(I&�2C-7$�9)E�{l$�A(a, b, c, d)
61AC0(1, 1, −1, 0)
619�*(−1, −1, 1, 0)
RF@0G2N&�EB61E<E<&�J>0d 6= 0
Rv�'AC$�J ] � `T619�*=] � `HK@"%0�;�6Gy�0a(b + 2d) = c(d − b)
K"%;'-+:<;=:46@9>a�0�?)&�2�-7952H;)0z{l$�A4J
b(a + c) = (c − 2a)dR�� EH-79�O�] � `^61OG6@-I9#K1"%0%O80[2
c = 2a + bR ] A `
S#;)089#K�-I9�y�-70[" $'{#] � `HKd = 3a + b
R ] D `� E<-I9�O3] A `�6@9�*,] D `#-795] � `HK8"%0z$)a'2F6@-I9
1+ b(2a + b)+3a + b = b(3a + b)K8"%;'-+:<;
:46@9Qa)0VAC0�"VAC-�2F24089=61Eb + 1 = a(b − 3)
R^S�;)0@AC0G{l$�AC0�Kb 6= 3
K�6@9�*
a =b + 1
b − 3= 1 +
4
b − 3
R
S#;'&�E[K 4
b − 3
-7E�6@9%-79'2C01O808A rG2H;�682T-IEGKb−3 ∈ {±1
K ±2K ±4} R�S#;)03:H$�J=?)(70G2C0}2F6Da'(+0$'{#Eb$'(.&�2C-+$�9)Ef:46@9>9�$1"±a�0V0@6@E<-.(�M=:4$�9)EH24AF&L:C240@*Wq
a b c d1 1 −1 0
−1 −1 1 05 4 14 19
−3 2 −4 −73 5 11 14
−1 1 −1 −22 7 11 130 −1 −1 −1
!l{L"%0%9�$1"±AC01°'&)-7AC0�2H;�6G2aKbKcKda�0V?�AF-7J>0@E[K'2H;)0@9
(a, b, c, d) = (2, 7, 11, 13)K
"%;'-+:<;5-7J=?)(.-+08Ev2H;�6G2n = 2 · 7 · 11 · 13 = 2002
R
�"�
� R �f0@9�$D2C0¢a@M��·2H;)0 Eb0[2�$'{%AF682C-7$�9)6@(V9�&�J>a�08AFE@Rº�f0[2408A4J,-79�0 6@(.(�{7&�9�:C2C-+$�9)Ef :� → �pEH&L:<;52H;�6G2
f(
x + f(y))
= y + f(x)K�{l$�A�6@(.(
xKy ∈ �VR
�����)# + ����� ����� >?>@#+ �Y�2��O�A���1 ���U�4�2�2���0��Z;�1 � �Z� �JHO#������,#$�J�s# �}���J��� �$��#*����1� ��� �$���-��%��2G �Z�,#*1 � �4�� HJ# � ���+ ���5)$����� ��� �Z�;5�0��+5� � � �s1 ���6�O#��2�'1E�7�s##�HJ#5E � #m;���)#�Y�9� �s�B�6� � �R�:�Z�,�,# %H�$�DE
Z10G2F2C-I9�Ox = 0
-I9,2H;)0VO[-�y�089Q:4$�9�*�-�2C-+$�9#K1"%0z�L9�*�2H;�682f(
f(y))
= f(0) + y{l$�Af68(I(y ∈ �VR�!l{ f(x) = f(y)
{l$�ABE�$�JQ0xKy ∈ ��K�2H;)089 f
(
f(x))
= f(
f(y)) r
;)0@9�:40�Kf(0) + x = f(0) + y
K'{+AC$�Ji"%;'-+:<;%"%0VO80[2x = y
R^S�;'&�E[Kf-7EB-I9D��01:C2C- y�0�R
Z10G2F2C-I9�Oy = 0
-I9X2H;)0=O[- y�0@9w:4$�9�*�-�2C-+$�9#K�"%0=$)a'2F6@-I9f(
x + f(0))
= f(x){l$�A�6@(.(x ∈ �VRNZ8-79�:40 f
-7Ev-79D��0@:C2C-�y�0'K@"%0VO80[2x = x + f(0)
K�619�*�2H;'&�E[Kf(0) = 0
RS#;)089
f(
f(y))
= y{l$�Av68(I(
y ∈ �%R� $1",K'{l$�A�68(I(xKy ∈ ��K
f(x + y) = f(
x + f(
f(y)))
= f(x) + f(y)R
¯V089�:H0'K�a@Mu6@9Y0861EFMw-I9�*�&L:C2C-7$�9#Kf(nx) = nf(x)
{l$�Az6@(.(n ∈ � ∗ R � $1"¦(+0[2
xa�0�619DMQ?L$�EH-.2C- y�0�AF6G2C-+$�9)68(s9�&�JQa)0@A�R�Z10[2F2C-79�Ox = r/s
"�-�2H;rKs ∈ � ∗ K�"%05;�6[y�0
sf(x) = f(sx) = f(r) = rf(1)r�;)0@9�:40�K
f(x) = (r/s)f(1) = xf(1)R
� $1"³"%0�9�$D2C0z2H;�6G2bK'{l$�Av68(I(x ∈ ��K
0 = f(0) = f(x − x) = f(x) + f(−x)K
619�*x2H;)08AC08{l$�AC0'Kf(−x) = −f(x) = −xf(1)
R_S�;'&�EGKf(x) = xf(1)
{l$�AV68(I(x ∈ �%R�W_$�AC01$8y�0@A�K#a@M¤E�0G2F2C-I9�O x = f(1)
K�"%0���9�*>2H;�682f(
f(1))
= f(1)f(1)R
�'AC$�J|6Da)$@y�0'Ks"%0Q;�6[y�0f(
f(1))
= 1RXS�;'&�EGK
f(1) = ±1RX��0Q:4$�9�:4(I&L*)0=2H;�6G2
08-�2H;)0@Af(x) = x
{l$�A�6@(.(x ∈ ��$�A f(x) = −x
{l$�Av68(I(x ∈ �VR!+2�-7Ef9�$1" 0861EFM�2C$=:<;)01:</�2H;�682
f(x) = x619�*
f(x) = −x6@AC0VE�$'(I&�2C-7$�9)E@R
� Rv�'$�AN0@61:<;�AC0@68()9�&�JQa)0@AxK'*)08��9�0 bxc 24$5a)0�2H;)0zOGAC0@6G2408EH2T-79'2C01O808A^(+08E�E^2H;�619,$�A01°'&�6@(�2C$
xR�F@0[2
α = 2 +√
3Rf»�AC$@y�032H;�682
αn − bαnc = 1 − α−n K {l$�An = 0
K1K2K. . .
R�����)# + ����� ����� >?>@#+ �Y�2��O�A���1 ���U�4�2�2���0��Z;�1 � �Z� �JHO#������,#$�J�s# �}���J��� �$��#*����1� ��� �$���-��%��2G �Z�,#*1 � �4�� HJ# � ���+ ���5)$����� ��� �Z�;5�0��+5� � � �s1 ���6�O#��2�'1E�7�s##�HJ#5E � #m;���)#�Y�9� �s�B�6� � �8�$��Z�����,���7E
��0>;�6Gy�0α−1 = 2 − √
3 < 1r#2H;)0@9
0 < α−n < 1R�²�MX2H;)0�²�-79�$�J,-768(
S#;)0@$�AC0@J¤K12H;)0�$)*�*�?L$1"%08AFEf$'{ √3-I9�2H;)0z0<®1?�6@9)E<-7$�9,$'{ (
2 +√
3)n
+(
2 − √3)n
:46@9�:H0G(#$'&�2[Rv¯V089�:H0'Kαn + α−n
-IEf619,-79'2C01O808A[RTS#;'&�E[K
αn + α−n = bαnc + 1R
� ��+ �������"� �D�Q� �,# � WY61AF61O8$'&L*)61/1-IE^?L$'-I9'2CE^$'&�2�2H;�682s6}E�$'(I&�2C-7$�9#K86@(7a�0G-.2L"�-�2H;�*�-.¶�08AC0@9'2-79'2C01O808AFEf-79Dy�$'( y�01*=]
α = 3 +√
5`HK'-IEf?)&La'(I-IEb;)0@*,-79 � �1�1� D q �1¨�A5 R
��A
n ^ R F@0[2ABC
a�0=6524AC-I619�O[(70�"%;)$�E�0=EH-+*)0,(70@9�O�2H;�Ez61AC0=68(I(�-79'2C01O808AFEGKs619�*¤(70G2D619�*
Ea)0¤2H;)0w?L$'-I9'2CE,682�"%;'-7:<;Y2H;)0¤-79�:4-IAC:4(+0w$'{
ABC24$'&L:<;)08E
BC6@9�*
ACK
AC0@E<?L0@:C2C-�y�0G(�MsRv!l{ |AD2 − BE2| ≤ 2K�Eb;)$1"³2H;�682
AC = BCR
�����)# + ��� � ��� �>?>@# +��Y�2��O�A���15���U�Z�9�"���0� ��;1 � �Z� �JHO#������+����5)$����� ��� �Z�;5�0��+5� � � �s1���6�O#��2�'1 �7�s##�HJ#5E � # ;��,)# �Y�9� �s�B�6� � �8�$��Z�����,����1�>@��+ � ��# +>��� �C�Q#/# + �,�,���*E
��0%;�6[y�0CE = CD = (a + b − c)/2
R ²�M�2H;)0VUs$�EH-79�0�F86G",KAD2 = b2 +
(a + b − c)2
4− b(a + b − c) cos C
619�*BE2 = a2 +
(a + b − c)2
4− a(a + b − c) cos C
RS#;'&�E[K
BE2 − AD2 = a2 − b2 − (a + b − c)(a − b) cos CR
²�M52H;)0VUs$�EH-79�0�F86[",K1"%0V;�6Gy�0cos C =
a2 + b2 − c2
2ab
K�6@9�*>;)089�:H0'K
BE2 − AD2 = a2 − b2 − (a + b − c)(a − b)a2 + b2 − c2
2ab
=a − b
2ab
(
a2(−a + b + c) + b2(a − b + c) + c2(a + b − c)) R
��$�AN2H;)0%?�&�A4?�$�Eb0%$'{�:H$�9'24AF61*�-7:C2C-+$�9#KD"%0V61E<E<&�J>0z2H;�682a 6= b
R��X-.2H;)$'&�2v(7$�E�E$'{�O80@9�08AF6@(.-.2�M#KG"%0}J>6[M%6@E�EH&�JQ0
a > bR�!l{
c = 1K82H;)089#K1E<-I9�:H0
a < b+cKG"%0�;�6Gy�0
b < a < b + 1K1"%;'-+:<;5-IEf-IJ>?�$�E�EH-+a'(+03{l$�A�-I9'240@O80@AFE
a6@9�*
bRTS#;)08AC08{l$�AC0'K
c ≥ 2RF@0G2
a − b = kRv!l{
k = 1KD2H;)089
BE2 − AD2 =(b + 1)2(c − 1) + b2(c + 1) + c2(2b + 1 − c)
2(b + 1)b
R��0>J=&�E42�;�6Gy�0
2 ≤ c ≤ a + b − 1 = 2bRx!+23-IEV9�$D2�;�6@AC*�24$�Eb;)$1"�2H;�682�2H;)0
J=-I9�-7J,&�J±y'68(I&L03$'{f(c) = c2(2b+1− c)
{l$�A2 ≤ c ≤ 2b
-7Ef(2) = 4(2b−1)
RS#;)08AC08{l$�AC0'K
BE2 − AD2 ≥ (b + 1)2(1) + b2(3) + 4(2b − 1)
2(b + 1)b=
4b2 + 10b − 3
2b2 + 2b
= 2 +3(2b − 1)
2b2 + 2b> 2
R!l{
k ≥ 2KD2H;)0@9
BE2 − AD2 =k
2(b + k)b
[
(b + k)2(c − k) + b2(c + k) + c2(2b + k − c)]
≥ 2
2(b + k)b
[
(b + k)2 + b2]
=2b2 + 2kb + k2
b2 + kb> 2
RS#;'&�E[K)-79Qa)$D2H;>:46@Eb08EGK
BE2 − AD2 > 2K�6�:H$�9'24AF61*�-7:C2C-+$�9�R}¯�0@9�:40�K
a = br
2H;�682�-IEGKAC = BC
R
S#;�6G2 :4$�J>?�(+0[2408Ef2H;)0 8Q���J��#$� {l$�A�2H;'-IE -IE�EH&L0�RfZ10@9�*>JQ0zM�$'&�AB9�-7:H0%E�$'(I&�2C-7$�9)E]+6@9�*=Eb$)$�9D`�{l$�A�?�AC$)a'(+08J>Ev2H;�682B;�6[y�0V6@?�?�0@6@AC01*,-I9 �1�1�D 9�&�J>a�08AFE}$'{L2H;)0 8Q���J��#$� R
�"D
" � � � �����3��� ���� �%��� � ���z��� ��}� ���S�<���
99�0����7� � ���������,#$�6�$#�H'�������
²�M_¯�6@9)EV�¤6@(IEb08AbK�24AF6@9)E<(I682C01*w{+AC$�J·2H;)0X$�AC-7O[-79)68(v�08A4J=619ua@M¢»�0[2408A%¯z-.(.2C$�9u6@9�*µ�0@6@9,»�0@*�08AFEb089#K8?)&La'(I-IEb;)0@*�a@M3S�;)03WY682H;)08J>6G2C-+:C6@( ¹ E�E�$):4-76G2C-+$�9z$'{ ¹ JQ08AC-7:46�K �@�D�1 R!\Z)² �³� P ¨D¨;�1¨;D P D;D"� P A K�;�6@AC*�:4$@y�08AbK � D¨ ?�61O80@E[K)�}Z� A�¨ R D@� R��#�)*�,#�� # + ��� �����������������������Q1 �� ��; ��4�?8Q��B�,#*;�#*1 ��� HJ�0��)#$�41 ��8 R
!l{T2H;�AC010=E424AF6@-7O8;D2}(I-I9�0@E3?�6@E�E�2H;�AC$'&LO8;�6>:4$�J>J>$�9�?L$'-I9'2bKL"%0>:C6@(.(s2H;)0,(.-79�08E:H$�9�:4&�ACAC0@9'2[R���0 6@(.(DAC0@J>0@J>a�08As-79)E42C6@9�:H08E^{+AC$�J�2H;)0�O80@$�JQ0[24A�M%$'{124AC-I619�O[(70@Es"%;)08AC02H;�AC010z(.-79�08Ef6@AC0V:4$�9�:4&�A4AC089'2 q#2H;)0�2H;�AC010�68(.2C-�2C&L*�08E ;�6[y�0�6%:4$�J>J>$�9,?L$'-I9'2bK'61EN"%08(.(61E�2H;)0�2H;�AC0@0zJQ0@*�-I619)E[K12H;)0�2H;�AC0@0�?�0@AC?L089�*�-7:4&)(76@Ava'-7E�01:C2C$�AFEf$'{)2H;)0zE<-7*�08EGKD6@9�*�2H;)02H;�AC010V6@9�O[(+0Va�-IEb0@:C24$�AFE8R
!�9,2H;'-IE}a)$)$)/V2H;)08AC0V61AC099E<?L0@:C2C61:4&)(76@AB0<®@6@J>?�(+08E}$'{#:H$�9�:4&�ACAC0@9�:40�O[-�y�089>6@E
?)-7:C2C&�AC0@E�PC"�-�2H;)$'&�2HPC"%$�AC*�E@RS#;)0}a)$)$)/3$�?�0@9)E#"�-.2H;z6�a�AC$�6D*z-I9'24AC$)*�&L:C2C-7$�9V*�-IEb:4&�E<E<-I9�O�2H;)0f-79'2C0@AFE�01:C2C-7$�9V$'{
2H;�AC010�$�A JQ$�AC0 EH24AF68-+O8;D2 (.-79�08E�6@9�*«:4&�A�y�0@E8R S�;)0�$)a@y�-7$'&�E�-79'2C0@AFE�01:C2C-7$�9«$'{*�-I6DO8$�9)68(7EV$'{�6X*)$)*�0@:461O8$�9w-7Ez{l$'(.(+$1"%0@*wa@M¤2H;)0=-79'2C0@AFE�01:C2C-7$�9x$'{v:4-7AC:4(70@E[K��'$'&�AC-70@A�D(7$1"%0@AFE,]I{7&�9�:C2C-+$�9)E5{+AC$�J£��$'&�AF-+08A�0H®D?)619)EH-+$�9)EC`HKv619�*_Us;)0@a@M)E�;)0by�»�$'( M)9�$�J,-768(7E8RS#;'-IEV-79'24AC$)*'&L:C2C-+$�9xE�0108J>E�(I-7/10Q6XOG682C0X$�?L089�-79�O524$w6XOGAC0@6G2VOG6@AC*�089#K^{+AC$�J�"%;'-7:<;2H;)0V6@&�2H;)$�Av:<;)$)$�Eb08E $�9�0V:H$�AC9�0@AN2C$=0H®D?�(+$�AC0z-7952H;)0�J>68-79=?)61A�2f$'{�2H;)0%a)$)$)/�R
!�9w2H;)0>Eb0@:H$�9�*x?)61A�2�$'{N2H;)0Xa)$)$)/�K99
?�-+:C2C&�AC08EbPF"�-.2H;)$'&�2HPF"%$�AC*)EVAC0@?)AC0@E�0@9'299
0H®@61J=?)(70@Ez$'{^:H$�9�:4&�ACAC0@9�:40�R%Z1$�J>0,$'{�2H;)08Eb05?�-+:C2C&�AC08E �1OGAF61?�;�Ez61AC059)6@JQ0@*�]7{l$�A0<®@6@J>?�(+0'K � ¯V$�J=6DO80Q2C$¢»'M'2H;�61O8$�AF61E[�@K � ²�&�2F2C0@AF¬�-70@E[�@K � �#-7E<E<-I9�OXU�-IAC:4(+08EG�<`4KvEb$�J>0:H$�J>0�"�-.2H;56�(I-�2408AF682C&�AC0�AC08{l08AC0@9�:40�K�6@9�*=68(I(L:H$�J>0�"�-.2H;%2H;�AC010�?)-7:C2C&�AC0@Ev2H;�6G2vEb;)$1"2H;)0zEH2C0@?)Ev(70@61*�-I9�O 24$V2H;)0G-7AN:H$�9)E424AC&L:C2C-7$�9�Rv�'6D:<;�?�61O803-IEf*)0by�$D240@*%24$5$�9�0z0<®@6@J>?�(+0$�9�(�M�KD6@9�*%9�$%?�AC$)$'{+E�$�A^0<®1?)(I619)6G2C-+$�9)E�6@AC03O[-�y�089%-79�2H;'-7E�?)61A�2[R ¹ 95$)a@y'-+$'&�E�AC0861E�$�9{l$�A32H;�6G2z-IE�08J>?�;�61EH-.¼<0@*Ya@M�2H;)0Q6@&�2H;)$�A�;'-7J=Eb0G(I{�-I9w2H;)0>{l$�AC0["%$�AC*�24$¤2H;)0Xa�$)$)/)K"%;)0@AC0z;)030<®1?)(I6@-I9)EB;'-IEv;)$�?L0}2H;�6G2�2H;)0za)$)$)/�"%$'&)(7*�0@9�:4$'&�AF6DO80}2H;)03AC086D*)0@A�2C$V��9�*2H;)0B?�$'-79'2FE�$'{@-I9'2408AFEb0@:C2C-+$�93$�9z;'-IE �1;)0@A�$1"V9#K[?L$�E<E<-7a�( Mf"�-�2H; 2H;)0B;)08(I?�$'{8-79'2C0@AF61:C2C-�y�0O801$�J>0G24A�M3E�$'{I2�"V61AC0)R�]lZ1$'{I2�"V6@AC0B?L$�?�&)(76@AL-79V�D&�AC$�?L0v-IEsAC0@?�(761:H0@*3-I9}2H;)0�24AF619)EH(76G2C-+$�9a@M=?�61:</@6DO808E J>$�AC0%:4$�J>J>$�9�(�M5&�Eb0@*,-I952H;)0%� 9�-�240@*>ZG2C6G2408E@R `
¹ {I2408AL��9�*�-I9�O�6fEH&�A4?)AC-IE<-I9�Ov:H$�9�:4&�ACAC0@9�:40�K�2H;)0v9�0<®82�EH2C0@?�{l$�AL6BJ>6G2H;)0@J=682C-7:4-76@9$�Av619DM>$D2H;)08AB:4&�AC-7$'&�E J=-I9�*,-7Ev24$5��O[&�AC0%$'&�2^"%;1M52H;)0%:H$�9�:4&�ACAC0@9�:40�;�61?)?L089)E}6@9�*(+$)$)/5{l$�A}O80@9�08AF6@(.-.¼H6G2C-+$�9)E8R�!�9Q2H;)0%2H;'-IAC*¤?)61A�2�$'{s2H;)0,a�$)$)/)KL2H;)0568&�2H;)$�A}?)AC$@y'-+*)0@EEb0by�08AF6@( ?�AC$)$'{}EH24AF6G240@O[-+08E52H;�68250<®1?)(I6@-I9 J>6@9DM¢$'{f2H;)0x?�-+:C2C&�AC08E,-79_2H;)0w?�AC0by'-+$'&�EEb0@:C2C-+$�9�RsS�;)0 E424AF682C01O[-70@ETAF619�O80f{+AC$�JE<-IJ>?�(+0 O80@$�JQ0[24AC-7:468(DE42408?�E[K[2H;�AC$'&LO8;�y�0@:C24$�AFE[K24$uUs0by'6�� E%S�;)01$�AC08J§619�*�µ<61:H$)a'-\� E%S#;)0@$�AC0@JwR_Z1$�J>0¤$'{�2H;)0@E�0="%$'&)(7*Y*�0G�L9�-�240G(�M0@9�OG61O80>;'-7O8;¤Eb:<;)$)$'(T$�A�0by�089w0G(+08JQ089'2C6@A�M�E�:<;)$)$'(�EH2C&L*)0@9'2FE@R,��E<?L0@:4-768(I( M�0H®1:4-.2C-I9�O619�*�?L$1"%08AC{7&)(G-7EL2H;)0 � »'MD2H;�6DO8$�AF6@EbP4{+AC010B*)0@AF-�y'6G2C-+$�93$'{82H;)0,F86G" $'{DU#$�E<-I9�0@E[�@K�"%;'-7:<;-7Ef6%a@M)?)AC$)*�&L:C2f$'{�619,-79Dy�08EH2C-7OG682C-7$�9>$'{�2H;)0=»�$'-79'2v$'{T!�9'2408AFEb0@:C2C-+$�9
84R
Z1$�JQ0�$'{D2H;)0}68(+O80@a)AF68-+:T?�AC$)$'{+E^61AC0�°'&)-.2C0�:H$�J=?)(.-+:C682C01*V6@9�*VAC01°'&)-7AC0f2H;)0�;)08(I?$'{s6�Us$�J=?)&�2C0@A ¹ (+O80@a)AF6�Z�M)E42408JxR�S�;'-7E[K�$'{�:4$'&�AFEb0'K�(+086Gy�08EB2H;)0V6@&�2H;)$�Av619�*=J>6@9DM$D2H;)0@ABO80@$�JQ0[24A�M,(7$@y�08AFE};)$�?�-79�O3{l$�Av6@9Q0G(+08JQ089'2C6@A�M>O801$�J>0G24AF-+:C6@(L?�AC$)$'{CRf�#0@61*�08AFE
�1
61AC03$�?L089�(�MV-79Dy'-.2C01*�24$%;)08(I?���9�*�6@9508(70@J>0@9'2F61A�M�O801$�J>0G24AF-+:C6@(�?)AC$)$'{){l$�As2H;)0�»�$'-I9'2$'{L!�9'2408AFEb0@:C2C-+$�9
79]+68(7E�$V:C6@(.(+0@*32H;)0 � »�AC$�?�08(.(+08Ab�<`4K["%;'-+:<;32H;)0 6@&�2H;)$�As?�AC$8y�01*3"�-�2H;
&�Eb0V$'{^WY61?�(+0)R�X-.2H;wO80@$�JQ0[24A�MwJQ$�AC0>6@9�*wJQ$�AC0>9�0@O[(+0@:C240@*xa@M�2H;)0>E�:<;)$)$'(N:4&�A4AF-+:4&)(.&�J|-I9
JQ$�E42f$'{ � $�AF2H; ¹ JQ08AC-7:46�K�2H;'-IE�a)$)$)/,a�AC-I9�OGE�6%AC0@9�0�"%01*>?�6@E�EH-+$�9={l$�AN2H;)0%E<&La1��01:C2[R!+2^$�?L089)E�9�0�"�?)682H;�ET24$V0<®1?)(7$�AC0�6@9�*%E�;)$1"VE�*�-IAC01:C2C-7$�9)E^{l$�AT0<®1?)(7$�AC-I9�O�R�S#;)$�E�0B"%;)$(I-7/10%O80@$�JQ0[24A�M=6@9�*=?)&�¼C¼4(70@E�"�-.(I(L0@9D��$8M=?L$�9�*)0@AF-79�O�0@61:<;,?)-7:C2C&�AC0�R
��#$�����U� � ���,�,#f�5#��Z��# �²�Mx�B6@9�-+0G(N�zRT²�$�9)6@Af6@9�*wWu-7:<;�6D0G(Lµ8R �T;)$'&�A�M�KL?)&La'(I-IEb;)0@*Xa@M=2H;)0=WY682H;)08J>6G2C-+:C6@(¹ E�E�$):4-76G2C-+$�9=$'{ ¹ JQ08AC-7:46�K �1�1�D R!\Z)² �³� P ¨D¨;�1¨;D P © A!D P K�;�6@AC*�:4$@y�08AbK1®@-I-�� �@ �A ?)6DO808EGK)�}Z� A � R � D R��#�)*�,#�� # +���������� ��� ������� ��� ����������� ��1.�7����)#��0�s�,�C� ��� � #�� �Q�*� ���f�Y�+H���1� �O# + #$�Z�&H'������1 ��� R
¹ 9=089'2C-7AC0za�$)$)/V*�0@*�-7:46G240@*�2C$�AC0@68()-I9��L9�-�240zE�0@AF-+08E � ¹ 2^��AFEH2B!�"V61EvEH&�A4?)AC-IEb0@*24$uEb0@0¤E<&L:<;Y6wa)$)$)/�R S�;)0�EH- ®w:<;�61?'2408AFE,6@9�*u2H;�AC0@0�6@?�?�0@9�*'-+:40@E,:H$�9'2F6@-I9_6�AC-7:<;:H$'(.(+0@:C2C-+$�9¤$'{TJ>6G2H;)0@J=682C-7:468(s-7*�0861E8RVUs;�6@?�2C0@AFE � 2H;�AC$'&LO8; � :4$�JQa'-79�0%24$Q?)AC$@y'-+*)06>*)0G2F6@-.(+0@*�$8y�0@A�y'-+0�" $'{T(I619�O[&�61O80�Ks:H$�9Dy�08ACO80@9�:4052C0@E42CE[KL2C01:<;�9�-7°�&L08EGK#619�*¤E<?L0@:4-768(Eb08AC-70@E[KG"�-.2H;%?)61A�2C-+:4&)(I61A^6G2F24089'2C-+$�9V2C$�2H;)0z;�6@A4J>$�9�-+:NE�0@AF-+08E@RsS#;)03AC08J>68-79�-I9�O 2H;�AC0@0:<;�61?'2408AFE¤]+?�(I&�E¤$�9�0u6@?�?�0@9�*'- ®G`5J=6D*)0Y6YE<-7OG9�-I�L:46@9'2,-IJ>?)AC0@E<E<-7$�9#Kz;'-7O8;'(I-7O8;D2CEX$'{"%;'-+:<;,61AC0�Eb;�6@AC01*=;)0@AC0)R
Us;�6@?�2C0@A A :H$�9)EH-7E42CE,$'{ � �'© � �08J>E[�DR¢S#;)0�6@&�2H;)$�AFEV"VAC-�240$q � S#;)08AC0�61AC0�69�&�JQa)0@A $'{�AC0@6@Eb$�9)EB"%0�J=6GMQ;�6Gy�0%(I6Da)08(701*>65AC08E<&)(�2f6,O808JxR}�08J>E}J>6[MX:4$�9��LACJ6�:4$�J>J>$�9�-I9'2C&)-.2C- y�039�$D2C-7$�9�"�-.2H;%6z:4(76@AC-.{.M�-I9�O}?�AC$)$'{�KD$�A^J=6GM%?�AC$8y�-7*�036z:H$'&�9'2C0@A4P0<®@6@J>?�(+0X2C$u-79'2C&)-�2C-+$�9�R S#;)0bM_J=6GM_0<®D;'-+a'-.2%6�?�6@AF2C-7:4&)(76@AC( M_EH(I-7:</x$�AV&�9�0H®D?�01:C2C01*?�AC$)$'{�240@:<;�9�-+°'&L0�R�Z1$�J>0VO808J>E�"%08AC0V:<;)$�E�0@9=a�0@:468&�Eb032H;)0bM,E�0108JQ0@*�24$�&�Ef6G2N�LAFE4261E424$�9�-IEb;'-I9�Oza�&�2T"�-�2H;56�JQ$�J>0@9'2b� EfE42C&L*DM,$'{�2H;)0z?�AC$)$'{�a�0@:46@JQ0�68(7J>$�EH2�:4$�J>J>$�9Eb089)Eb0)R+�
Us;�6@?�2C0@A D :4$�9)E<-IEH2FE}0@9'2C-IAC08( MX$'{ ;� ?)AC$)a�(70@J=E {+AC$�J|»)&�249)61J«Us$�J=?L0[2C-.2C-7$�9)E{+AC$�J � � A)� 2H;�AC$'&LO8; �1�1�'� -I9�:4(I&�EH-�y�0)Rf�f0G2F6@-.(+0@*=E�$'(I&�2C-7$�9)Ef6@AC0V?)AC$@y'-+*)01*sR
�T0G2bKsJQ$�E42}-IJ>?)AC0@E<E<- y�0,-7E�Us;�6@?�2C0@A K^�D-I9)6@(B�v-�y�08AFE<-7$�9)EGK#-79Q"%;'-+:<;Q2H;)0=6@&LP2H;)$�AFE�Eb;�6@AC0�K � 61EN63?�6@AF2C-I9�O�O[-I{I2fR8RGRN63E�$�AF2^$'{�*�08E�E�0@A�2bK7�f?�&�¼C¼4(+08EN-79Dy�$'( y�-I9�O -I9��L9�-�240Eb08AC-70@EN619�*%?�-+:C2C$�AC-I6@('?�AC$)$'{+E�$'{La�61EH-+:T{+61:C2CEN6Da)$'&�2T-I9��L9�-�240�E�0@AF-+08E@RTZ108(701:C2C-7$�9)EN{+AC$�J� �'68(I(I6D:4-70@E[K1�1(76["VEGKG6@9�*%�D(.-7J,¬L6@J��N] 8Q��B�,#*;�# �����C�7EH!��0��J� ��� 619�* �D$�A��)*�²L61A4a�086@&s� EW ¹^¹ a�$)$)/�a@M52H;)0�E�6@JQ0�9)6@JQ0�`s"�-79�*5&�?,2H;)0%:<;�6@?�2C0@A qTªf9�0�E<&L:<;=0<®@6@J>?�(+0z-7Ef6@96D*�61?'2C6G2C-+$�9>$'{#6 � � � cD~[�[�,n��Dk\mL�[�,�Dklj����@~��8��?)AC$)a�(70@JwR
S#;)0x6@?�?�0@9�*'-+:40@EQ$'¶�08A�y'6@(.&�6Da'(+0xAC0G{l0@AC089�:H08E@R ¹ ?�?�0@9�*'- ® ¹ -IE,&�9�&�E<&�68(}-I92H;�6823-�23{l0@6G2C&�AC0@E � � � S�AF&L0 �'�'68(7E�0X°'&L0@E42C-+$�9)E8RY�f0[2C68-I(701*x0<®1?)(I619)6G2C-+$�9)EV6@AC0Q61OG6@-I9?�AC$8y�-7*�0@*�R³S#;'-IE52H;)$�AC$'&LO8;�9�0@E<E=619�* Eb089)Eb0w$'{3:4$�J>?�(+0[2C-+$�9 ?�0@A�y'61*�0¤2H;)0u089'2C-7AC0a�$)$)/LR���;)0[2H;)0@A�M�$'&�2C0@61:<;%:C6@(7:4&)(I&�E[K["%$�A4/ "�-�2H;,»)&�249)61J ?)61A�2C-+:4-I?�6@9'2CE[K1$�AsE<-IJ>?�(�M"�-7E�;Q2C$¤:H$�9)EH-+*)0@A}65{+61J,-I(.-76@A}EH&LaD��0@:C2}{+AC$�J�6=AC0G{+AC0@E�;'-79�O5?�0@AFE<?L0@:C2C-�y�0'K�2H;'-7Eza�$)$)/"�-I(.(�9�$D2v*�-IE�6@?�?�$'-79'2�M�$'&TRBWwM,61?)?�AC0@:4-76G2C-+$�95{l$�AT2H;)032C$�?)-7:^"V6@Ef0@9�;�6@9�:H0@*>a@M%2H;'-7Ea�$)$)/)K�?�6@AF2C-7:4&)(76@AC( M52H;)0Va�(70@9�*>$'{#O80@J=E 6@9�*>*�- y�0@AFEH-+$�9)E8R
�D©
� ).!�* � ��" � �%- �R*�! � � ��,���� !�( &
�X� � � �,� ��� �
�U�?>@#$>@��� � �����o����#��H� �Q��J�Z�6��;������I�5#H��;9�!�9w2H;)0 �o��� > �0����+�8Q���J��#$�y� $sR �1�D� $'{3c�d e'g hBj+k�mYnpoBqsr3t)niK k�� KN9�$sR A
]�WY6[M#K �1�1�'� `HK�65E�08(701:C2C-7$�9X$'{�?�AC$)a'(+08J>E}{+AC$�J¦2H;)05ZG2[RT»�0G2C0@AFE�a�&�ACO=WY682H;)08J>6G2C-+:C6@(ªB(�M�J>?�-761*)Ev-IEfO[- y�0@9,a@M=ªB(701O�!7y�AC-.-�619�*X�#$)a)0@A�2}²L61ACAC-I9�O�24$�9 F@08-7O8;sRTS#;)0�2H;'-IAC*=$�9�0� � K�?TR �@¨ � Ks»�AC$)a'(+08J � ] �� ;D ` -IE
¹ OG6@JQ0>E42C6@AF2FE�"�-.2H;�6Q;)0@6@?u$'{25
a)0@6@9)E@R=S'"%$�?)(I6GM�08AFEV68(.2C0@AC9)682C08( MAC0@J>$@y�0
1K2K8$�A
3$'{82H;)08JxRs��;)08936@(.([2H;)0fa�08619)E�;�6Gy�0fa)01089}2C61/10@9#K�2H;)0
"�-79)9�08A^-IEN2H;)0z?�(76[M�0@As"%;)$5;�61Ev619,0by�089,9�&�J>a�08A�$'{�a�08619)E8R ¹ E<E<&�J,-79�O?L08AC{l0@:C2v?)(I6GM�K)*�$)08Ev2H;)0���AFEH2v?�(76[M�0@Av$�AN2H;)0�Eb0@:H$�9�*=;�6Gy�0�6%EH&�AC03"�-I95oS#;)0 �o��� > �0����+ 8Q���J��#$� 01*'-.2C$�A�AC01:40@9'2C( M,AC01:408- y�01*,6�AC01°'&L0@E42�{l$�A�6�Eb$'(.&�2C-+$�9�R
S#;)0z?�AC$)a'(+08J�-7E�{+AC$�J�6z(.-7E42�$'{LE<&�?)?)(70@J>0@9'2F61A�M5?�AC$)a'(+08J>E r1-�2�J=6GM�9�$D2v;�6[y�0�a�0@0@9&�Eb0@*#K�619�*,9�$,Eb$'(.&�2C-+$�95-7E O[- y�0@9,-I952H;)0%a�$)$)/LR
S#;'-IEVOG61J>0Q*'-I¶s0@AFEz{+AC$�J 2H;)0,&�E<&�68(N/1-I9�*x$'{T2C61/101PH6G"V6[MwOG61J>0,-79¤2H;�6G2�-�2�-IE9�$D2B-7J=?�6@AF2C-I6@(�R ¹ {I2408AB65J>$@y�0�;�6@E�a�0@0@9QJ>61*�0'KL0by�0@9>-I{LM�$'&="%08AC0�9�$D2N"V682C:<;'-79�ODKM�$'&>:46@9�2C08(.('"%;'-+:<;�?)(I6GM�08A�;�61EBJ>$@y�0@*=a@M59�$D2C-7:4-79�Of"%;)$,;�6@EB6D*)*�0@*�2C$,;'-IEf$�A�;)08A:H$'(.(+0@:C2C-+$�9�$'{Na�08619)E8R%S#;'&�E[KL"%0=:46@9)9�$D2}&�Eb0�2H;)0=Z@?)AF6DO[&L0@P�^AC&�9�*1M�S#;)0@$�A�M3r�2H;�6G2-7E[K@"%0V:C619)9�$D2v:468(+:4&)(I682C0�9�-IJQPFy'6@(.&L0@E/� � K)U#;�61?�R �� RNªf9�2H;)0V$D2H;)08Av;�619�*�K'-.2^-7EB9�$D26@( "V6GM�E 6V(I61E42HP<?�(76[M�0@A4PC"�-I9)9�-79�O�OG6@JQ0 rD2H;)0@AC0�-IE 6%J,- ®82C&�AC0%$'{s9�$�ACJ>68(�619�*=J,-7E"?0@AC0]7(I61E42HP<?�(76[M�0@A4PH(7$�E<-I9�Ob`^?�(76[M � � K�Us;�6@?TR � �� KD"%;'-7:<;=J>0@6@9)EB2H;�682^"%0%:46@9)9�$D2v&�Eb0z2H;)0Us$�9D"V6[M,S�;)01$�A�M � � K)U#;�61?�R � 08-�2H;)0@A�R
��0�&�E�0V6VAF6G2H;)0@ABa�AC&�2C01P4{l$�AC:H0VJ>0G2H;)$)*�K�-I9>08¶s01:C2v*)AF6["�-79�O�2H;)03"%;)$'(70%OG6@JQ024AC010'K�2H;)$'&LO8;}"%0vE�6[y�0B6 O8$)$)*z*)0@68(1$'{1E�?)6D:40fa@M3-7*�089'2C-I{.M'-79�O�9�$)*�08Es-79z$'&�A��D-7O[&�AC0 � R
¯V08AC03-IEv6�E�$'(I&�2C-7$�9�{l$�A�2H;)0�OG6@JQ0'KD?)(I6GM�0@*V"�-.2H;%619DM,$)*)*59�&�J>a�08A�$'{�a�08619)E8R��$�Av;)0861?)E $'{
8k + 3K8k + 5
K)$�A8k + 7
a)0@6@9)EGK12H;)0z�LAFE42v?�(76[M�0@AT"�-I9)E�r1{l$�A;)0@6@?�E�$'{
8k + 1a�08619)E[K�2H;)0,E�01:4$�9�*¤?)(I6GM�08AB"�-79)E8R,¯V089�:H0'KL"�-�2H; �;D a�08619)E[K�2H;)0
Eb0@:H$�9�*,?)(I6GM�08AB:46@9�"�-79�RS�$ E�010�2H;�6G2�2H;'-7E=-7E,2H;)0u:46@Eb0'KB"%0xAC0@?)AC0@E�0@9'2,?L$�EH-.2C-7$�9)E=-I9 2H;)0YOG6@JQ0ua@M
b(f, s)Ks"%;)0@AC0
b-7Ez2H;)0>9�&�JQa)0@A�$'{Ba)0@6@9)E%AC08J>68-79�-I9�O=-I9�2H;)0X;)0861?#K^619�*
f6@9�*
s6@AC0¤2H;)0¤2C$D2C68( 9�&�J>a�08AFEQ$'{�a)0@6@9)E=6@(IAC0@61*DM :4$'(I(701:C2C01* a@MY2H;)0���AFEH25]+9�0H®@2�`�6@9�*
Eb0@:H$�9�*=]+?)AC0by�-7$'&�E4`^?)(I6GM�08AFEGK'AC0@E<?L0@:C2C-�y�0G(�MsR��0z"�-I(.(L&�Eb0
d6@9�*
e{l$�AB61A4a�-�24AF61A�MQ$)*)*Q6@9�*Q0by�0@9Q9�&�J>a�08AFEGK�AC0@E<?L0@:C2C-�y�0G(�MsR
S#;)0z$�?�0@9�-I9�O}?L$�EH-.2C-7$�9%-IEv$'{�E�;�61?�0d(0, 0)
KD619�*%E<&La�Eb0@°�&L089'2N?�$�E<-�2C-+$�9)EN6@(.(�E�6G2C-7EH{.Mb + f + s = d
R � $D2C-7:H0%2H;�6G2bK�"%;)089�6,J>$@y�0�-7E�J>61*�0'KL2H;)0,AC$'(70@E3$'{T�LAFE42�]79�0<®82�`619�*XEb0@:H$�9�*¤]+?)AC0by�-7$'&�E4`v?)(I6GM�08A}6@AC0�-79'2C0@AC:<;�6@9�O801*sRV��;)089Q2H;)059�0<®82}?�(76[M�0@AB2C61/10@Eta�08619)E {+AC$�J
b(f, s)K)2H;)0�?L$�EH-.2C-7$�9Xa)01:4$�JQ08E
b−t (s, f +t)R S#;)08AC0%-7E}9�010@*>{l$�A
Eb$�J>01$�9�0324$,*�0by'-7E�0V6VJ>$�AC0V?�0@AFE<?)-7:4&L$'&�E 9�$D2C6G2C-+$�9 ������������ "!"#
c© $&%�%�' ��(&)�(�*�� (&),+-(.#.!�/"01(.#2� 34(&5�6&��32� /2#��
�1¨
� c ���?� : c � b 8": ��� � 8 �¹ ?L$�EH-.2C-7$�9X-I9¤E<&L:<;X6>OG6@JQ05-IEGK)"�-.2H;¤a)0@E42}?)(I6GM�K�0G-.2H;)08A}6%"�-79X{l$�Af2H;)0��LAFE42
]+9�0H®@2�`B?�(76[M�0@A�Ks$�AB2H;)0,E�01:4$�9�*�]7?�AC0by'-+$'&�EC`B?�(76[M�0@A�R�S#;)08Eb056@AC0=$'{I2C0@9X(I6Da)08(.(+0@* N B?L$�EH-.2C-7$�9)Ef619�* P B ?�$�E<-�2C-+$�9)E[K�AC08E�?�01:C2C- y�08( M�Rb 8": ��� � 8 ��Rv!�9,2H;)0VOG61J>0%*)0@E�:HAC-7a�0@*=61a�$8y�0�K� R}2H;)0 P B ?L$�EH-.2C-7$�9)Ef6@AC0 8k+1 (e, e)
619�*8k+5 (d, d)
r� R}2H;)0 N B ?L$�EH-.2C-7$�9)EN61AC0
8k+7 (d, d)K8k+7 (e, e)
K8k+6 (d, e)
K8k+6 (e, d)
K8k+5 (e, e)
K8k+3 (e, e)
K8k+3 (d, d)
K8k+2 (d, e)
K8k+2 (e, d)
6@9�*8k+1 (d, d)
R� R�S#;�6G2T(+086Gy�08E
8k+4 (d, e)619�*
8k+4 (e, d)K8"%;'-+:<;%6@AC0 "�-I9)EN{l$�A
dK8"%;)$)0by�08A
EH2F61A�2CE[K1619�*8k (d, e)
619�*8k (e, d)
K8"%;'-7:<;�6@AC0 "�-I9)EN{l$�AeKG"%;)$)0by�0@A^E42C6@AF2FE@R
S#;�6G2f-7E[K8k+4 (d, e)
6@9�*8k (e, d)
61AC0 N B ?L$�EH-.2C-7$�9)EGK�"%;'-I(708k+4 (e, d)619�*
8k (d, e)6@AC0 P B ?L$�EH-.2C-7$�9)E@R
�0�s����� � ��0�&�E�0�-I9�*�&L:C2C-7$�9�R^S)$,EH2F61A�2N2H;)0z-79�*'&L:C2C-+$�9#K1"%0V:H$�9)EH-+*)0@Ak = 0
RP
0 (d, e)-IEV6,"�-79�{l$�A
ea@Mu*)08��9�-.2C-7$�9#KT619�*u;)089�:H0'K
1 (e, e)"�-.(I(�a)0>6Q(7$�E�E[K
619�*1 (d, d)
63"�-79#KD{l$�AN2H;)0�9�0<®82B?�(76[M�0@A�RP
2 (d, e)-7Ev6�"�-79�{l$�AT2H;)0z9�0<®82�?�(76[M�0@A�KD-I{�;)0�O8$)0@EN24$
1 (e, e)K'61E�-7E
2 (e, d)-.{
Eb;)032F6D/108E}a)$D2H;=a�08619)E[K)O8$'-79�O�24$0 (d, e)
RPx!l{�2H;)08AC0%6@AC03�+&�EH2
3a�08619)E AC08J>68-79�-I9�ODK�2H;)0���AFEH2v?�(76[M�0@AN2C61/10@Ev2H;)0@J¦6@(.(L-.{s;)0
-7EdK)a�&�2B$�9�( M
2$'{L2H;)0@J�-I{#E�;)0z-7E
eR
PY²�&�24 (d, e)
6@9�*4 (e, d)
6@AC0xa�$D2H;x"�-I9)E5{l$�AdR!l{
dEH2F61A�2CE[K ;)0wO8$)0@E�2C$
1 (e, e)K�"%;'-.(+0Q2H;)0x$�9�( M_?L$�E<E<-7a�(70�J>$@y�08E5{l$�A
e6@AC0
3 (d, d)K2 (d, e)
Kf$�A1 (d, d)
K1"%;'-+:<;,61AC0�6@(.(L9�0<®82HP<?�(76[M�0@AT"�-79)E8RPXS#;'&�E[K@{+AC$�J
5 (e, e)K82H;)0}9�0<®82T?)(I6GM�08A#"�-.(I('O8$z24$
4 (e, d)RB²�&�2�{+AC$�J
5 (d, d)K
2H;)0�9�0<®82v?�(76[M�0@A�J,&�EH2vO8$�2C$4 (d, e)
K3 (d, d)
K)$�A2 (d, e)
K1"%;'-7:<;,6@AC0�9�0<®82HP?)(I6GM�08A^"�-I9)E@R
P6 (d, e)
6@9�*6 (e, d)
6@AC0 N B ?�$�E<-�2C-+$�9)E rdO8$)08E524$
4 (e, d)$�A
eO8$)08E524$
5 (d, d)R
Px�'AC$�J7 (e, e)
K12H;)0z9�0H®@2N?)(I6GM�08AvO8$)08E�2C$4 (e, d)
K@"%;)0@AC0}"%0�;�6[y�0�E�01089�2H;�6G2d"�-79)E[K^"%;'-I(70X{+AC$�J
7 (d, d)2H;)0¤9�0H®@2%?�(76[M�0@A%:46@9 O8$x2C$x2H;)0 P B ?�$�E<-�2C-+$�9
5 (d, d)R
Px�'AC$�J8 (e, d)
Ke"�-I9)E^a@M�O8$'-I9�ON24$
5 (d, d)"%;'-.(+0v{+AC$�J
8 (d, e)KdJ=&�E42s?�(76[M
24$7 (e, e)
K6 (e, d)
K'$�A5 (e, e)
6@9�*V2H;)039�0<®82N?�(76[M�0@A�2H;)089V"�-I9)EBa@M5O8$'-I9�O 2C$4 (e, d)
K5 (d, d)
K)$�A4 (e, d)
K�AC08E�?�01:C2C- y�08( M�R!�9�$�AC*)0@A�24$�O80G2#2H;)0f-79�*'&L:C2C-+$�9%$'¶�2H;)0�OGAC$'&�9�*�KG"%0 9�010@*z2C$VO8$�6�a�-�2s{7&�A�2H;)0@A�K
"�-.2H;,2H;�AC0@05:C61E�0@E�$'{k = 1
KLEH-79�:40V2H;)0%?)(I6GM�08AFE}J>6[MQAC08JQ$8y�0%&�?>2C$,2H;�AC0@05a�08619)E8R�'AC$�J
9 (d, d)KN2H;)0X9�0<®82%?�(76[M�0@A3"�-I9)E,a@M_O8$'-I9�O=24$
8 (d, e)KT"%;'-I(70
9 (e, e)-IE�6
P B ?�$�E<-�2C-+$�9#K12H;)0zEb0@:H$�9�*5?)(I6GM�08A�61a�(7032C$,O8$%2C$ 5 (d, d)-I{�2H;)03�LAFE42�?)(I6GM�08A^2F6D/108E
1$�A3K)$�AN2C$
4 (e, d)-.{L2H;)0���AFEH2�?)(I6GM�08AN2C61/10@E
2R
� �
N –positions P–positions N –positions.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
8k+8 (e, d) 8k+8 (d, e)
8k+7 (d, d) 8k+7 (e, e)
8k+6 (e, d) 8k+6 (d, e)
8k+5 (d, d) 8k+5 (e, e)
8k+4 (d, e) 8k+4 (e, d)
8k+3 (e, e) 8k+3 (d, d)
8k+2 (d, e) 8k+2 (e, d)
8k+1 (e, e) 8k+1 (d, d)
8k (e, d) 8k (d, e)
8k−1 (d, d) 8k−1 (e, e)
8k−2 (e, d) 8k−2 (d, e)
8k−3 (d, d) 8k−3 (e, e)
8k−4 (d, e) 8k−4 (e, d)
@@
@@@R
QQQs
PPPPPq
��
���
��
���+
�����)@
@@
@@R
QQQs
PPPPPq
��
���
��
���+
�����)@
@@
@@R
QQQs
PPPPPq
��
���
��
���+
�����)
�D-7O[&�AC0 � qTU#$�9�*�089)Eb0@*>OG61J>0z24AC0@0�'AC$�J
10 (d, e)K
d"�-79)E5"�-.2H;
9 (e, e)6@9�*¢{+AC$�J
10 (e, d)K
e"�-79)E5"�-.2H;
8 (d, e)Rx�'AC$�J�;)08AC0X$�9#KN:4$�?1M�2H;)0>E424AF682C01ObM�{+AC$�J
8a�08619)E%a)61:</5rT{l$�Az0<®@6@J>?�(+0'K
{+AC$�J11 (d, d)
JQ$8y�0z2C$8 (d, e)
6@9�*,{+AC$�J11 (e, e)
24$9 (e, e)
RUs;)0@:</ 2H;)0 *�0[2C68-I(701*�E�$'(I&�2C-7$�9p(I-IEH2C01*p6Da)$@y�0 6DOG68-79)E42w�1-+O[&�AC0 � K3"%;'-+:<;p-IE
?L08AC-7$)*�-7:v-7952H;)0�Eb089)Eb032H;�682�-�2vAC08?L08682FEf-�2CE�08(.{s0by�08A�M8AC$1"VE8Rv��$�Av0H®@61J=?)(70�K
2H;)0VE424AF682C01ObM5{l$�A8k + 3 8k + 4 8k + 5 8k + 6 8k + 7-7Ev2H;)0�E�6@JQ0�61EB{l$�A8k − 5 8k − 4 8k − 3 8k − 2 8k − 1AC0@E<?L0@:C2C-�y�0G(�MsR5S�;)0=61ACAC$1"VEz-I9u�D-7O[&�AC0 � 6@AC052H;)0>$�9�( MQ"�-I9)9�-79�O5JQ$8y�0@E8R5S�;)0@AC0,-IE
619 61ACAC$1"�{+AC$�J£0by�08A�M N B ?�$�E<-�2C-+$�9u24$_6 P B ?L$�EH-.2C-7$�9�R ¹ (I( $D2H;)08A%(701OG68(fJQ$8y�0@E[K$'{�"%;'-+:<;,2H;)08AC0�61AC0�2H;�AC010%{+AC$�J«0@61:<; P B ?�$�E<-�2C-+$�9>619�*=2�"%$=9�$�9�PC"�-I9)9�-79�OVJQ$8y�0@E{+AC$�Ji086D:<; N B ?L$�EH-.2C-7$�9#KG(70@61*324$ N B ?L$�EH-.2C-7$�9)E�r�2H;)0bM�6@AC0 9�$D2�E�;)$1"V9�-I9z2H;)0f�LO[&�AC0�RU#6@9>6@9DM�$�9�0VEH&�?�?�(�M=6%O8089�0@AF68(�2H;)01$�A�M5{l$�AvEH&L:<; � {l$'&�AB$'&�2C:H$�J>0��3OG6@JQ08E o
� c � c�j�c � d c?�� � �D(�"3M�9 ²�0@AF(+0@/@61J=?sK#µ�$);�9xU#$�9D"V6GM�KT619�* ��-+:<;�6@AC*x^&'M�K � �6��������; � ���2�9�����
���0�� �����C�Q#�>?�����&H��������6���9� K ����� 0@*�-�2C-+$�9#K ¹ R �vR�»�0[2408AFEGK � 682C-7:</�K�W ¹ K �1�1� � B�@�D��A R
� �� �zR �3Rw�'$�J=-I9#K ���4E �0#���#��0�2�0��Z; � ���C��#$>?�����+HZ��� �o��� > �0����+��s1 �������� ���� � K»�$'(.-.2C01:<;�9�-7/@6)K)ZG2[Rs»�0[2408AFEba'&�ACODK � � � A R
�#-7:<;�61AC*>T&'M�f08?�6@AF24J>0@9'2B$'{^WY682H;)08J>6G2C-+:CE ��ZG2C6G2C-7E42C-+:CE
� 9�- y�[email protected]�M>$'{#U#68(+OG6@A�MU#68(+OG6@A�M#K ¹ ²�KDS �5� � �OA
U#6@9)6D*�[email protected]
A)�
� £� " ��� � �` >C4D@�EGF6>H:?8�E6>�M�AO>�P465 QR8sFG:IELK?F�8sF�8|8�@�5�8|K�>�@�4L;S7HAbADF6c5�:�>�4�7�E65�A�ELK?7H:RV��lY�� Y�����\C]H]?^�_
� :=78�E65�ADF�8t�(?)
7e6ED5 A�7�:�@HQUP�5�A�FG:i;HF6a67�ED5�8IELK?7�Em7IM�AO>�P465 Q gs78pM�AO>�Ml>8 5;0g�F6ELK�>�@?Em78�>C4D@?EGF6>H:�_
nf7�aoKpM�AO>�P465�QqF�8srCF6cC5�:SFG:Un?:frC4 F�8tKR7H:i;UuHAO5�:�avKfwxELK�5B>yIaDF�7C4i4�7H:fr�@7Cr�5�8�>ekz�7H:?7?;f7i_{L:WF�8|8�@�5�89}�w�~�wm��w�7H:�;0�|w�nf:?rC4 F�8|KdghF 4 4kM�AO5ab5;i5UufAO5 :�aoK?w�7H:i;#FG:WF�8O8C@?5 8S��w��w���w�7H:i;=� wufAO5 :�aoKWghF 4 4�M�AO5Hab5C;i5Snf:?rC4 F�8|K�_B{L:WELK�5�8 >C4D@�EGF6>H:?8m8�5a|EGF6>H:fwsELK�5�M�AO>�P465 Q g�F 4 4�P�5�8�E�7�ED5C;RFG:ELK�5�4�7H:?r�@C7Cr�5B>elELK�5�M�ADFGQR7HAo<�eD5�7�Ev@HAO5C;B8�>C4D@?EGF6>H:�_
�K�5�5C;HF6ED>HA�ELK�7H:��H8��C5 7H:��|��7HAOas�5�AvAoF65 A�7H:i;R��7HA|EGFG:N�k>C4L;f8�E65�FG:R>e�ELK�5p��:fF6c5 AD8OF6EG<p>e�0>H:�ELAO5�7C4�e6>HA�ELAD7H:�8b4�7�EGF6>H:�8�>elELK�5�M�AO>�P465 QR8C_
��05;�6[y�05*'-7E�:H$8y�0@AC0@*=2H;�682fE�$�JQ0%AC01:40@9'2C( MQ?�$�Eb0@*Q?)AC$)a�(70@J=E�61AC0%AC0@?�0@6G2CE�$'{0@6@AC(.-+08AL?�AC$)a'(+08J>E�q�?�AC$)a'(+08J � � ¨'� � �@�D�1 q A� � K A) �A5 -7E�2H;)0vE<61J>0B6@Es?)AC$)a�(70@J �@� � � �@�D�;D q D A;A K D�A�©� rG?)AC$)a�(70@J � � ¨ D � �1�1�D q A) ;� K A� D� -IE�2H;)0BE<61J>0v61Es?)AC$)a�(70@J � � �"D� �@�D��A q � © A K � ©8 KL619�*>?�AC$)a'(+08J � �� ¨ � �1�1�D q D � K D � ¨ -IEf2H;)0�E<61J>0%?�AC$)a'(+08J61E � � ¨'© � �1�1�D q A� � K A� D� R=Z8-79�:40>68(I(s2H;�AC010>*'&�?)(.-+:C682C-7$�9)EV;�6Gy�0>$�9�( M�61?)?L0861AC0@*"�-.2H;'-I9%2H;)0}(76@EH2s2�"%$V-7E<E<&L08EGKG"%0�61AC0�AC0@?�(761:4-79�O 2H;)08Ji-I9%2H;'-IE�-IE�EH&L0�R ¹ 9DM�E�$'(I&�2C-7$�9)E{l$�A 2H;)0>$�AC-7O[-79)68(^?)AC$)a�(70@J=E � � ¨D� 6@9�* � � ¨ D "�-I(.(^a�0,-7OG9�$�AC01*�K�E<-I9�:H0,Eb$'(.&�2C-+$�9)E�2C$2H;)$�Eb0B?�AC$)a'(+08J>ET;�6Gy�0f68(7AC086D*1Mz6@?�?�0@6@AC01*[r86@9DMzEb$'(.&�2C-+$�9)E�24$}2H;)0 $�AF-+O[-I9)6@( � � � ¨ "�-I(.(a�0�24AC08682C01*¤6@EzE�$'(I&�2C-7$�9)E}24$ � � ¨'© R�ªB&�Af2H;�6@9�/DE�2C$xWu-+:<;)0G(B²L6G2C68-I(.(+0�{l$�A}a)AF-79�O[-I9�O2H;'-7Ev24$,$'&�Av6G2F24089'2C-+$�9�R
p n �S\ Rf�#08?)(I6D:40@J>0@9'2[R �0�O���o����#+>��� �Y� ���5+�� �\�,�617� ��/!����$#$1D8 �m1��o� � RF@0G2
aKbK16@9�*
ca)0�619DMV?L$�EH-.2C- y�0}AC0@68(�9�&�J>a�08AFEGK1619�*V(70G2
pa�0}63AC086@('9�&�JQa)0@A
E<&L:<;�2H;�6820 < p < 1
R]76G`¤»�AC$@y�032H;�6G2
a
(b + c)p+
b
(c + a)p+
c
(a + b)p≥ 1
2p
(
a1−p + b1−p + c1−p) R
]la1`�»�AC$@y�032H;�6G2bK'-I{p = 1/3
K'2H;)0@9a
(a + b)p+
b
(b + c)p+
c
(c + a)p≥ 1
2p
(
a1−p + b1−p + c1−p) R
]l:F` ? »�AC$@y�0V$�Av*�-IE�?)AC$@y�0a√
a + b+
b√b + c
+c√
c + a≥ 1√
2
(√a +
√b +
√c) R
p n �Sa R �#08?)(I6D:40@J>0@9'2[R �0�O���o����#+ ���y����� �� � ���,�,#*1 ���4�5+ #����61 ���� HJ#������ �2��$# ���#9HO���Q+5� �;� �$H;������A1�� �6�Q+������41 ��� R
F@0G2an
*)0@9�$D2C0}2H;)0}&�9�-.2FEB*'-+O[-�2N$'{(4n)(3n)(2n)n R��D-I9�*�68(I(�?�$�E<-�2C-�y�0 -79'2C01O808AFE
nEH&L:<;�2H;�682 n
∑
i=1
ai ≥ 4nR
A �
p n � � Rf�#08?)(I6D:40@J>0@9'2[R �0�O���o����#+>����� �+H;��#H���Q�������B�A��#$1 ���0�5#���1 � �4�� HJ# RF@0G2
p = 2n+1a)0f6 ?)AC-IJQ0'K86@9�*z(70G2
sa)0f619DM3-I9'240@O80@A#EH&L:<;�2H;�6G2
1 ≤ s ≤ nR
»�AC$@y�032H;�682 q]76G`
4sn−s∑
k=0
(
2s + 2k − 12s − 1
)
≡ 1 (mod p)K
]la1`4s
n−s∑
k=0
(
2s + 2k2s − 1
)
≡ −1 (mod p)R
p&\�^ n R �0�O���o����#+I��� �\E �QEo-/#��Q+ #��0������1��7��s+ #$� -h�B�A�A1Y8�� > ����#H�A�+H4�s�����61����h1��6�>@#�>@���;� ��� � ��J�4���y� E �Q��� > � ��� R
^- y�0@9Q?L$�EH-.2C- y�0%-I9'240@O80@AFEm619�*
nK�:H$�9)EH-+*)0@Av2H;)0�AC086@(sJ>$�9�-+: ?�$'(�M�9�$�J=-I6@(IE
P (x) =m∑
i=0
aixi619�*
Q(x) =n∑
j=0
bjxj"�-.2H;�9�$�9�PH9�01OG6G2C-�y�0>:H$)0��X:4-70@9'2FE@R=��0
61AC0z-79'2C0@AC08EH2C01*,-I9�"%;)0G2H;)08AP619�*
QE<682C-IE<{.M%2H;)0%:4$�9�*�-�2C-+$�9
P (x)Q(x) =
m+n∑
k=0
xk R
]76G`¤»�AC$@y�032H;�6G2�-I{m619�*
n6@AC0%a)$D2H;=$)*�*�K'2H;)08AC0V6@AC0V9�$,EH&L:<;=?�$'(�M�9�$�J=-I6@(IE@R
]la1`�»�AC$@y�032H;�6G2�-I{m = n
KD2H;)08AC0V61AC0�9�$,E<&L:<;,?�$'(�M�9�$�J=-I6@(IE@R]l:F`=Z1;)$1"�2H;�6G2�{l$�Az086D:<;
m2H;)0@AC0,-7E�6@9�-79���9�-.2C0>E�0G23$'{Ty'68(I&L08E%$'{
n{l$�A "%;'-7:<;
2H;)0@AC0V*�$=0H®1-IEH2�E<&L:<;,?�$'(�M�9�$�J=-I6@(IE@R]l*1`�»�AC$@y�0�2H;�6G2v2H;)0�:4$)0��Q:4-+089'2CE -I9X0by�08A�MQEH&L:<;>?)6@-IAf$'{�?�$'(�M�9�$�J=-I6@(IE}6@AC0�08-�2H;)0@A
0$�A
1R
]lU#$�J>?)61AC0V?�AC$)a'(+08J �1 1 -79X��*D"V6@AC*5µ8Rs²L6@AHa)0@68&sK#Wu&�ACAF6GM>Z�R �s(I61J>/1-I9#K)6@9�*�X-I(.(I-I61J ª�R µ8R±W_$�E�0@A�K � ��)#�- � �Q+5�s#+ �����C�Q#�>?�����&H����98-� ���A��#���;�#(� Ki"%;)08AC0m = n = 5
R `p&\�^S\ R �0�O���o����#+>��� & E !E0� >@##�� ��1�������6����>?>@#H��1��C�Q# � #��C�Q#��Z��� �Q+�� R
F@0G2Γa�036�:4-IAC:4(+0 "�-�2H;�AF61*�-.&�E
rK1(70G2
Aa)03619DM�?L$'-I9'2�$�9
ΓKD6@9�*�(70G2
ta)0�2H;)0
2C6@9�O80@9'2s(I-I9�0B2C$Γ6G2
AR4F@0G2
B6@9�*
Ca)0 ?L$'-I9'2CEN$'{
t$�9%$�?)?L$�EH-.2C0 EH-+*)0@E^$'{
AE<&L:<;
2H;�682AB = mr
6@9�*AC = nr
{l$�A�Eb$�J>0�?L$�EH-.2C- y�0�AC0@68(�9�&�J>a�08AFEm6@9�*
nROF@0G2
Pa)0�619DM,?�$'-79'2v$'{
Γ*�-.¶�08AC0@9'2N{+AC$�J
ARNZ1;)$1"³2H;�682
cot ∠APB + cot ∠APC-7E
65:4$�9)EH2F619'2v{l$�Av6@(.(#EH&L:<;=?L$'-I9'2CEPK)6@9�*Q*�0[2408A4J,-79�0�2H;'-IE}:H$�9)E42C6@9'2�y'68(I&L0�-79,2C0@ACJ>E
$'{m6@9�*
nR
p&\�^Sp R �0�O���o����#+>��� & E !E0� >@##�� ��1�������6����>?>@#H��1��C�Q# � #��C�Q#��Z��� �Q+�� RF@0G2
ABa�0%2H;)05*�-I61J>0G2C0@A $'{T65Eb08J=-7:4-7AC:4(70
ΓR F@0G2
Da)056@9DMX?�$'-79'2}$�9>2H;)0
2C6@9�O80@9'2B2C$Γ6G2
B6@9�*Q( M�-I9�O�$�9=2H;)0�E�6@JQ0%E<-7*�0�$'{
AB61E
ΓKL6@9�*Q(70G2
Ca)0%2H;)0
J=-7*�PH?L$'-I9'2^$'{BD
RsS#;)0}E�01OGJ>0@9'2FEAC
6@9�*AD
-I9'2408AFEb0@:C2Γ{l$�A#2H;)0}Eb0@:H$�9�*z2C-IJQ0
682 2H;)0,?L$'-I9'2CEK619�*
LKsAC0@E<?L0@:C2C-�y�0G(�MsR>!l{
M619�*
N6@AC052H;)0,?�AC$1��0@:C2C-+$�9)E�$�9'2C$
KL$'{
A619�*
BK�AC08E�?�01:C2C- y�08( M#K'Eb;)$1"³2H;�682
ML = LK = KNR
A��
p&\�^O_ R �0�O���o����#+>����� ���D2���C� �}#��JH �*#$1 �Q�Z�9�$��)$1��o��>?� �5�6� RF@0G2
AKJ ∈ Mn×n(
R)K@"%;)08AC0
J-IEN2H;)03J=6824AF- ®z6@(.(�$'{'"%;)$�Eb030@9'24AF-+08Ev6@AC0
1K
619�*w(70G2b ∈ R5RwZ10[2
B = bJK^6@9�*w{l$�A
k = 1K2K. . .
KnKN*�089�$D240Qa@M
Ak
2H;)0J>6G24AC- ®V$)aD2C68-79�0@*%{+AC$�J
Aa@M�AC08?)(I6D:4-I9�O3086D:<;�08(70@J>0@9'2T-I9�AC$1"
k"�-.2H;�2H;)0fy'6@(.&L0
bR
»�AC$@y�032H;�682det(A + B) det(A − B) = (det A)2 −
(
n∑
k=1
det Ak
)2 R
p&\�^Sa R �0�O���o����#+>����� ���D2���C� �}#��JH �*# � �Q+ � ��Z�6�� &Y��� H��1��Q�Z�9�$��)$1��o��>?� �5�6� RF@0G2
A1A2 · · · An
a)0�6,:4$�9Dy�0<®,?�$'(�M�O8$�9#KL6@9�*>(+0[2Pa)0�619DM>-I9'2408AC-7$�Af?L$'-I9'2
$'{s2H;)0�?�$'(�M�O8$�9�Rz��$�Ak = 1
K2K. . .
KnK�(70G2
Gk
a�0V2H;)05:H089'24AC$'-+*X$'{#2H;)0�?L$'( M�O8$�9A1A2 · · · Ak−1Ak+1 · · · An
]I2H;)0u?L$'( M�O8$�9p$)aD2C68-79�0@*�a@M�AC08JQ$8y�-I9�O¤y�0@A�240H®Ak{+AC$�J
A1A2 · · · An
`�Rv!l{Bk
-IEv2H;)0VAC0G¬�0@:C2C-+$�9>$'{Gk
2H;�AC$'&LO8;�2H;)0V?L$'-I9'2PK'?�AC$8y�0
2H;�682N2H;)0�(.-79�08EAiBi
6@AC0%:4$�9�:4&�A4AC089'2v{l$�Ai = 1
K2K. . .
KnR
p&\�^&s R �0�O���o����#+>����� ���D2���C� �}#��JH �*#$1 �Q�Z�9�$��)$1��o��>?� �5�6� RF@0G2
na)0V6V?L$�EH-.2C- y�0z-79'2C01O808Av6@9�*
x6VAC0@68(L9�&�JQa)0@A�R »�AC$@y�032H;�682
bxc2 +
⌊
x +1
n
⌋2
+ · · · +
⌊
x +n − 1
n
⌋2
= bnxc2
-I{#6@9�*>$�9�(�M5-.{ bnxc = bxc K1"%;)08AC0 b·c *)0@9�$D2C0@Ev2H;)0%OGAC08682C0@E42�-79'2C01O808A�{7&�9�:C2C-+$�9�Rp&\�^ � R �0�O���o����#+>��� �0# �,#$� �}E�� ��� 1�� ����6�T�7����)#��0�s�,�C��1 ���� ���4�5+5�1D8 �m1��o� � RF@0G2�6u:H$�9Dy�0H®_°'&�6D*�AC-.(76G2408AF6@(
APQC;�6Gy�0�-�2CE=EH-+*)0@E
APK
PQKf619�*
QC2C6@9�O80@9'2s24$�6�J=-I9�$�A^:4-IAC:4&)(76@A�61AC:ABC
682#2H;)0�?�$'-79'2FEAKBK@619�*
CK@AC0@E<?L0@:C2C-�y�0G(�MsRF@0G2
Ea�0 2H;)0�?)AC$1��01:C2C-7$�95$'{
B$�9'2C$
ACRF@0G2T6�Eb08J=-7:4-7AC:4(70f"�-�2H;
PQ61E�*�-I61J>0G2C0@A
:4&�2AC
682H6@9�*
KK@"�-.2H;
Ha�0[2�"%01089
A619�*
KR
�X-.2H;)$'&�2�&�EH-79�O=24AF-+O8$�9�$�J>0G24A�M�Kv?)AC$@y�0>2H;�682BE
a'-7E�01:C2FE∠PEQ
619�*x2H;�6G2PH
a�-IEb0@:C2CE∠APB
Rp&\�^ � R �0�O���o����#+ ��� ����� ���� �5����#$1 ���4�5+ #����61 ��6�JHO#$�7�����9� ��# � �5#�HJ����+5�� ���H;�Q�����1�� �6�Q+������41���� R
�D-I9�*>2H;)05(I61ACO808EH2f-I9'240@O80@AkE<&L:<;=2H;�682f{l$�A 6@(.(�?�$�E<-�2C-�y�0�AC0@68(�9�&�JQa)0@AFE
aKbK
cK1"%0V;�6Gy�0
(
a3 + 3)(
b3 + 6)(
c3 + 12)
> k(a + b + c)3R
p&\�^ � R �0�O���o����#+>������2�s�B�,#y8��� �H�,�����s#*1��}�5�,)#$�6�O����� ��� ���,���# ������1 �o��>?� �5�6� RF@0G2
fa)0,6>:H$�9Dy�0H®={7&�9�:C2C-+$�9w$�9¤619X-79'2C0@A�y'68(
IR=��$�A
i = 1K2K. . .
KnK�(+0[2
ai ∈ IRf�f08��9�0
a =1
n
n∑
i=1
ai
Rf»�AC$@y�032H;�682
n(n − 2)
2f(a) +
n∑
i=1
f(ai) ≥ n
2(n − 1)
∑
i6=j
f
(
a +ai − aj
n
) R
A!�
p&\ n ^ R �0�O���o����#+>������2�s�B�,#y8��� �H�,�����s#*1��}�5�,)#$�6�O����� ��� ���,���# ������1 �o��>?� �5�6� R»�AC$@y�032H;�682bK'{l$�A�6@(.(LAC0@68(�9�&�J>a�08AFE
a1Ka2K. . .
Kan ∈
[
1√2,√
2] K1"%0V;�6Gy�0
n∑
i=1
3
ai + 2ai+1
≥n∑
i=1
2
ai + ai+1
K
"%;)0@AC0z2H;)0�E<&La�Eb:HAF-7?'2CEf61AC032C61/10@9=JQ$)*'&)(+$nR
p&\ n&n R �0�O���o����#+>���/��?�������� >@�0�2�Y���O���o����#$� RF@0G2
ABCDa�0V65°'&�6D*�AC-.(76G2408AF6@(�"%;'-7:<;=-IEf-79)E�:HAC-7a�0@*=-79>6�:4-7AC:4(70
ΓR}�1&�AF2H;)08A
E<&�?)?L$�E�0X2H;�6G2ABCD
-.2FEb0G(I{ ;�61E56@9_-I9�:4-7AC:4(70�R F@0G2EF
a)0X2H;)0�*�-I61J>0G2C0@A%$'{Γ"%;'-+:<;Q-IE3?�0@AC?L089�*�-7:4&)(76@Af24$
BDK)"�-.2H;
E( M�-I9�O5$�9>2H;)0,E<61J>05EH-+*)0,$'{
BD6@E
ARF@0G2
BD-79'2C0@AFE�01:C2
EF6G2
M619�*
AC682
SR
»�AC$@y�032H;�682AS : SC = EM : MF
R� ��+ � S#;'-IEz?)AC$)a�(70@J�:46@JQ0�-79'2C$>2H;)0=� 9�- y�[email protected]�M�$'{B�#01O[-I9)6)� E�WY682H;¤U#0@9'24AF68(
"%01a�E<-�240'K)a�&�2N2H;)0V9)6@JQ0V$'{L2H;)0V?)AC$�?L$�E�0@Av;�61EfEH&La)E�01°'&L0@9'2C( M>a�0@0@9,(+$�E42[R p&\ n \ R �0�O���o����#+ ���3!�����2#������ �=&h2� � �$% �Q� �J�O#��O� � �Q+ � �4�� H�� �"HJ� ��������H������C:���2��Q����#$�s� 1��7����)#��0�s�,�6��� �0��B��� �#9H4���+HZ� + # 8����6���Z�� ����1 �Q� � HO#H�,�����1}�$������� R
F@0G2a1Ka2K. . .
Kan
a�0�AC0@68(s9�&�JQa)0@AFE3E<&L:<;=2H;�6G2ak ≥ 1
{l$�A1 ≤ k ≤ n
R»�AC$@y�032H;�682
n∏
k=1
a( 2k
n(n+1))1/2
k ≤ exp
√
√
√
√
n∑
k=1
ln2 ak
R
R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R Rp n �S\ R#�#0@J=?)(I6D:40@J>0@9'2[R �0�s���o����2#c���� �Y� ���5+�� �\�,�617� ��/!����$#$1D8 �m1��o� � R
Z1$'-.2aK
b0G2
c*)0@EX9�$�JQa�AC0@E¤A�E010G(7EX?L$�EH-.2C-.{+E¤°�&L0G(+:4$�9�°�&L08EGK}0G2>Eb$'-�2
p&�9
9�$�JQa�AC0VA�E0@08()240G(�°�&L00 < p < 1
R]76G`XW_$�9'24AC0@Av°�&L0
a
(b + c)p+
b
(c + a)p+
c
(a + b)p≥ 1
2p
(
a1−p + b1−p + c1−p) R
]la1`¤W_$�9'24AC0@Av°�&L0'K)EH-p = 1/3
K)68(+$�AFE
a
(a + b)p+
b
(b + c)p+
c
(c + a)p≥ 1
2p
(
a1−p + b1−p + c1−p) R
]l:F` ? Us$�9���A4J>0@Av$'&=-79���A4J>0@AB°'&L0
a√a + b
+b√
b + c+
c√c + a
≥ 1√2
(√a +
√b +
√c) R
A;A
p n �Sa R��#08J>?�(761:H08JQ089'2[R �0�s���o����2#3��� � ����� �� � �5����#$1 2#��4�5+ �6��7�61 2�-HO���#��#9HO���Q+5�����s# ��6�JHO#$�7� ���9� ��# ��1 � �6�Q+������41���� R
� E0@EH-+OG9�$�9)E%?)61Aan
(70X:<;'-I¶#AC0X*�08EV&�9�-.2�E08E�*�0(4n)(3n)(2n)n R�S�AC$'&'y�0@A�24$'&�E
(+08E 0@9'2C-70@AFEf?L$�EH-.2C-.{+En240G(7E °'&L0 n
∑
i=1
ai ≥ 4nR
p n � � R#�#0@J=?)(I6D:40@J>0@9'2[R �0�s���o����2#c���� � �+H;��#H���Q���6���A�B�,#*1����0�5#���1 � �4�� HJ# RZ1$'-.2
p = 2n + 1&�9=9�$�JQa�AC0�?�AC08J=-70@A�K)0[2�Eb$'-�2
s&�9>089'2C-+08Av°'&L08(7:H$�9�°'&L0z2C08(
°�&L01 ≤ s ≤ n
R#W_$�9'24AC0@AB°'&L0Hq]76G`
4sn−s∑
k=0
(
2s + 2k − 12s − 1
)
≡ 1 (mod p)K
]la1`4s
n−s∑
k=0
(
2s + 2k2s − 1
)
≡ −1 (mod p)R
p&\�^ n R �0�O���o����2#y��� � �\E �QEY-/#��Q+ #��0������1 �7� �O+ #�� -h�A�B��1h8�� > ����#H�A�+H4�s�����61 ��� #$�> 2#�>@����s#/+ # � ��J�4���y��E �Q�6�> � �6� R
�D2F619'2Q*�$�9)99E0 *�08E¤9�$�JQa�AC0@E�089'2C-+08AFE¤?L$�EH-.2C-.{+Em
0[2nK}$�9³:4$�9)E<-7*9?08AC0Y(+08E
?L$'( M)9 �$�J>0@E,A�E010G(7E�&�9�-.2F6@-IAC0@EP (x) =
m∑
i=0
aixi0G2
Q(x) =n∑
j=0
bjxj6Gy�0@:,*)0@E
:H$)0��X:4-70@9'2FE 9�$�9�9 E01OG6G2C-I{+E8R F86°'&L0@E42C-+$�9�0@E42_*)0³E�6[y�$'-7AxE<-P0G2
Q?�08&'y�089'2
E�6G2C-7EH{+6@-IAC0z(76%:4$�9�*�-�2C-+$�9
P (x)Q(x) =
m+n∑
k=0
xk R
]76G`XW_$�9'24AC0@A,°�&L0uEH-m
0G2nE�$�9'2�24$'&�EQ*)08&D®Y-IJ>?)6@-IAFEGK�-I(}9#� M¢6u?)61EQ*�0w2C08(IE
?L$'( M)9 �$�J>0@E8R]la1`¤W_$�9'24AC0@Av°�&L0VEH-
m = nK'-I(L9#� M=6%?)61E *�03240G(7Ef?�$'(�M�9 �$�JQ08E@R
]l:F`XW_$�9'24AC0@A=°�&L0Y?�$'&�A524$'&�2mKf-.(fM�6u&�9�0u-79���9�-.2�E0Y*�0�y'6@(708&�AFEX*�0
n?�$'&�A
(+08Eb°'&L08(.(+08E *)0z2C08(IE ?�$'(�M�9 �$�JQ08E 0<®@-7E424089'2[R]l*1`¤W_$�9'24AC0@AN°�&L0�*�619)E�2C$'&�2403?�68-7AC0z*�0}240G(7Ev?L$'( M)9 �$�J>0@E[K1(70@EB:4$)0��Q:4-+089'2CEvEb$�9'2
0$'&1R
]lU#$�J>?)61AC08A¢6@&«?)AC$)a�(@?0@J>0 �1 1 *�619)E��)*1"V61AC*�µ8Rw²L61A4a�086@&sK�Wu&�A4AF6[M«Z�R�s(I61J>/1-I9#K�6@9�*Q�X-I(.(I-I61J«ª�R µ8RTW_$�Eb08AbK � �,)# - ���+5�O# + �����C�Q#�>?�����&H����(8-�����B�,#$��;5# � K$ ?&
m = n = 5R `
p&\�^S\ R �0�s���o����2#c����c&fE !E0�>@# #$� ��1������,�6����>?>@#*�A1������9��%�Q�2� RZ1$'-.2
Γ&�9�:40@AC:4(70f*)0BAF6GM�$�9
rKA&�93?L$'-I9'2s°'&L08(7:H$�9�°'&L0BE<&�A
ΓK[0G2
t(76�2C6@9�O80@9'2C0?6
Γ0@9
ARTZ1$'-�2
B0[2
C*)08&D®Q?�$'-79'2FE�E<&�A
tK�*)0=?�6@AF230G23*#� 6@&�24AC0>*�0
AK�2C08(IEV°'&L0
AB = mr0[2
AC = nrKm0G2
n*)08&D®=9�$�J>a)AC08EzA�E010G(7E�?L$�EH-.2C-.{+E@R�W_$�9'24AC0@A}°�&L0
cot ∠APB + cot ∠APC65&�9�0%y'68(+0G&�A�:H$�9)E42C6@9'2405?L$'&�AB24$'&�E�(70@E3?�$'-79'2FE
PEH&�A
ΓK)*'-7E42C-79�:C2FE *�0
AK�0[2B*9E0G2C0@ACJ=-I9�0@AB:40G2F2C03y'68(+0G&�AB0@9,{l$�9�:C2C-7$�9>*�0
m6@9�*
nR
A!D
p&\�^Sp R �0�s���o����2#c����c&fE !E0�>@# #$� ��1������,�6����>?>@#*�A1������9��%�Q�2� RZ1$'-.2
AB(+0,*�-I61JG?0G24AC0=*�� &�9w*)0@J,-+P<:H08AC:4(+0
ΓRsZ1$'-.2
D&�9�?L$'-I9'2�°�&L0G(+:4$�9�°�&L0
E<&�AL(I6v2F619�O8089'240 ?6Γ0@9
B0G2�EH-.2C& E0B*�&�J��0@J>0f: �$D2�E0B*�0
AB°�&L0
ΓK[0G2�E�$'-.2
C(+0v?L$'-I9'2
J=-.(I-708&5*)0BD
RDZ1$'-�2K0G2
L(70@E�*)08&D®@-@?08JQ08E�?L$'-I9'2CE�*#� -79'2C0@AFE�01:C2C-7$�9�AC08E�?�01:C2C-.{+Ev*)0@E
Eb0@OGJQ089'2CEAC
0G2AD
6Gy�0@:ΓR�Z8-
M0G2
NE�$�9'2B(+08E�?)AC$1��01:C2C-7$�9)E�AC0@E<?L0@:C2C-�y�08E3*�0
A0[2
BEH&�A
KLK�J>$�9'24AC0@AB°'&L0
ML = LK = KNR
p&\�^O_ R �0�s���o����2#c���� � � �\2����� �}#��JH �*#$1��Q�4�2���5)�1 ���0�>?� �5�,# RZ1$'-.2
AKJ ∈ Mn×n(
R)K�$ ?&
J0@E42^(76zJ=6824AF-+:40�*�$�9'2T24$'&�E�(+08ESE08(@E0@J>0@9'2FEBE�$�9'2E0@OG6@&D®�?6
1KD0G2^Eb$'-�2
b ∈ R5R�»�$�E�$�9)EB = bJ
0[2bKD?�$'&�Ak = 1
K2K. . .
KnKD*9E08E<-7OG9�$�9)E
?�6@AAk
(763J>6G24AC-7:H0z$)a'2C0@9�&L0H?6z?�6@AF2C-IAN*�0A0@9�AC0@J=?)(I6 �:H619'2
k?)61A
bRLW_$�9'24AC08A�°'&L0
det(A + B) det(A − B) = (det A)2 −
(
n∑
k=1
det Ak
)2 R
p&\�^Sa R �0�s���o����2#c���� � � �\2����� �}#��JH �*#/#�� ��� �Z��� �@&Y��� H��1��Q�4�2���5)�1 ���0�>?� �5�,# RZ1$'-.2
A1A2 · · · An
&�9 ?L$'( M�O8$�9�0u:4$�9Dy�0<®10'KB0G2�Eb$'-�2P&�9 ?L$'-I9'2�-I9'2�E0@AF-+0G&�A
61A4a�-�24AF6@-IAC0>*�&w?L$'( M�O8$�9�0�R�»�$'&�Ak = 1
K2K. . .
KnK#Eb$'-�2
Gk
(70>:40@9'24AC0>*)0>OGAF6Gy'-.2�E0*�&X?L$'( M�O8$�9�0
A1A2 · · · Ak−1Ak+1 · · · An
]I(+0%?�$'(�M�O8$�9�0�$)a'2C0@9�&¤0@9X089�(+0by'6@9'2B(+0Eb$�J=JQ0[2
Ak
*�0A1A2 · · · An
`�RsZ8-Bk
*9E0@EH-+OG9�05(I6=A�E0G¬�0H®1-7$�9w*)0Gk
?�6@A�AF6@?�?�$�AF2?6PK�J>$�9'24AC0@AB°'&L0�(+08E *)AC$'-�2408E
AiBi
Eb$�9'2v:H$�9�:4$'&�AF619'2C0@E ?�$'&�Ai = 1
K2K. . .
KnR
p&\�^&s R �0�s���o����2#c���� � � �\2����� �}#��JH �*#$1��Q�4�2���5)�1 ���0�>?� �5�,# RZ1$'-.2
n&�9=9�$�JQa�AC0V?�$�E<-�2C-I{#0G2
x&�9>9�$�J>a)AC0VA�E010G(FR�W_$�9'24AC0@AB°'&L0
bxc2 +
⌊
x +1
n
⌋2
+ · · · +
⌊
x +n − 1
n
⌋2
= bnxc2
E<-�0[2vEb0G&)(+08JQ089'2vE<- bnxc = bxc K)$ ?& b·c *9E08E<-7OG9�0�(I6�{l$�9�:C2C-+$�9 ..............
..
..
..
...
..
.
..
?)61A�2C-+0V0@9'2C-@?0@AC0..............
..
..
..
...
..
.
..
Rp&\�^ � R �0�s���o����2#c���� �0#���#�� �}E�� ��� 1 � ����6�0�}�5�,)#$�6�O�����1����� �6�Z��+5��1\8 �m1��}� � R
Z1$'-.2APQC
&�9 °'&�6D*�AC-.(76G2�?0@AC0w:4$�9Dy�0<®10w*)$�9'2V(+08E,: �$D2�E0@EAP
KPQ
0G2QCEb$�9'2�AC0@E<?L0@:C2C-�y�08JQ089'2N2C6@9�O80@9'2FE ?6�&�9>6@AC:f:4-IAC:4&)(768-7AC0�J=-I9�08&�A
ABC68&D®%?L$'-I9'2CE
AK
B0[2
CR�Z1$'-�2
E(I65?)AC$1��01:C2C-7$�9¤*)0
BEH&�A
ACR9F@05*)0@J,-+P<:H08AC:4(+0�*�05*'-76@JG?0[24AC0
PQ:H$'&�?�0AC
089H0[2B0@9
KK�6Gy�0@:
H0@9'24AC0
A0[2
KR
Z@619)EX24AF-+O8$�9�$�JGE0G24AF-+0'K}JQ$�9'24AC08A¤°'&L0BE
0@E42>&�9�0¢a'-7E<Eb0@:C24AC-7:H0 *�0_(�� 6@9�O[(+0PEQ
0[2B°'&L0PH
08EH2N&�9�0%a�-IE�E�01:C24AF-+:40V*�0z(\� 619�O[(70APB
Rp&\�^ � R �0�s���o����2# ��� � ����� �� � �5����#$1 2# �4��+ ��� ���61 2�-HO���# ��#9HO���Q+5�����s# ��6�JHO#$�7�� �2��$# �1 � ����+��$���41 ��� R
S�AC$'&'y�0@Av(+0V?�(I&�E�OGAF6@9�*Q0@9'2C-70@Ak240G(s°'&L0%?L$'&�A�2C$'&�E (70@E 9�$�J>a)AC08E}A�E0@08(IE ?L$)P
E<-�2C-I{+EaKb0G2
cK)$�9=6@-�2(
a3 + 3)(
b3 + 6)(
c3 + 12)
> k(a + b + c)3R
A)
p&\�^ � R �0�s���o����2#c���� ��9�O�A��# 8��� �H�,�����s#*1��7����)#��0�s�,�2#/+ # ������,#(�����A1 ���0�>?� �5�,# RZ1$'-.2
f&�9�0%{l$�9�:C2C-+$�9X:4$�9Dy�0<®10�EH&�AB&�9>-79'2C0@A�y'68(I(70
IRT»�$'&�A
i = 1K2K. . .
KnK
Eb$'-�2ai ∈ I
R�ªf9>*9E0G�L9�-�2a =
1
n
n∑
i=1
ai
R�W_$�9'24AC0@AB°'&L0
n(n − 2)
2f(a) +
n∑
i=1
f(ai) ≥ n
2(n − 1)
∑
i6=j
f
(
a +ai − aj
n
) R
p&\ n ^ R �0�s���o����2#c���� ��9�O�A��# 8��� �H�,�����s#*1��7����)#��0�s�,�2#/+ # ������,#(�����A1 ���0�>?� �5�,# RW_$�9'24AC0@A^°�&L0'K1?L$'&�As24$'&�EN(70@E�9�$�J>a)AC08E�A�E0@08(IE
a1Ka2K. . .
Kan ∈
[
1√2,√
2
] K$�9=6
n∑
i=1
3
ai + 2ai+1
≥n∑
i=1
2
ai + ai+1
K
$ ?&=(+08EB-79�*'-+:40@EfEb$�9'2v:468(+:4&)(@E0@EfJQ$)*'&)(+$nR
p&\ n&n R �0�s���o����2#c����h� �y���s���o���$#5��m�������� >@# RZ1$'-.2
ABCD&�9Y°�&�61*)AF-I(I682�?08AC0>-79)E�:HAC-�2�*)6@9)EV&�9Y:40@AC:4(70
ΓR^ªf9xE<&�?)?L$�E�0X*�0
?)(.&�EB°'&L0ABCD
?�$�E�E"?01*)0}(I&)-7P<J��0@J>0}&�9,:40@AC:4(70}-79)E�:HAC-�2[R'Z1$'-�2EF
(703*�-I61JG?0G24AC03*�0Γ?�0@AC?L089�*�-7:4&)(768-7AC0G?6
BDKEEH-.2C& E0=*'&xJ��0@J>0>: �$D2�E0=*)0
BD°'&L0
ARBD
:H$'&�?�0EF
*'-7E�$�9)E 0@9M
0[2AC
0@9SR
W_$�9'24AC0@AB°'&L0AS : SC = EM : MF
R� ��+ � U#0>?)AC$)a�(@?0@J>0Q0@E42361?)?�6@AC&uEH&�A�(+0=E<-�240,-I9'2408A49�0[2�:H089'24AF6@(�*)0�WY6G2H;x*�0
(\� � 9�-�y�08AFE<-�2�E0%*)0=�#01O[-I9)6)K�J=6@-IEB(+0�9�$�J¥*'&Q?)AC$�?L$�E�08&�A�6�*'-7E<?�6@AC&>?)61A�(I6VE<&)-�240)R p&\ n \ R �0�O���o����2#���� ��!�����2#=����� �(&h2� � �$% �Q� �J�O#��O�f# � � �Z� �JH�� � HO� �����6�$H��,���:C:���2������Q#��O� 1�}�5�,)#$�6�O����2# �0�������#�H;����� Q��#/+ # 8����6���,� ;��#*1 �Q�� HJ#*������#*1 ���J���;��# R
Z1$'-.2a1Ka2K. . .
Kan
*)0@EN9�$�J>a)AC08E�A�E010G(7ET2C08(IE�°�&L0ak ≥ 1
?L$'&�A1 ≤ k ≤ n
RW_$�9'24AC0@AB°'&L0
n∏
k=1
a( 2k
n(n+1))1/2
k ≤ exp
√
√
√
√
n∑
k=1
ln2 ak
R
A�©
� � ��� ���}� � �� �y���O����,#$> � �f#�)#��m�o#��J>?���#$�7�����@H��,���$# +QE��2��# #+ �������Y� � ���v�:���9�:�0�,#$�9�$# +
�,� HO�����s��+ #��m�,�����D����A�+HZ�����,��� ��#����$��Z�����,����� ���m��#�� ���-�O��;9�7� � ��� ���2���0���O����,#$> �E��6D/@&La � R ¹ (I- M�0by#K�²L61/1&XZG2C6G240�� 9�- y�[email protected]�M#K�²L61/1&sK ¹ ¼<08AHa�6@- �l619Q;�61E�a�AC$'&LO8;D2
24$>$'&�Av682F2C0@9'2C-7$�9>6�*�-B�X:4&)(�2\M5"�-�2H;52H;)0�{l0@6G2C&�AC01*>E�$'(I&�2C-7$�9,2C$ �1�'©"D � �1�1�D q A� � RS#;)08AC0�-7E�6,:4(I6@-IJ«J>61*�0V2H;�682
ax − bx-7E�E424AC-7:C2C(�M=-79�:HAC0861EH-79�OV{l$�A
a > b > 06@9�*
x ≥ 0r@2H;'-IEf:4(768-7Ji-7Ev9�$D2v:4$�A4AC0@:C2[RTS�;)0�?)AC$)a�(70@J�J>$)*�08AF682C$�AbKD&�?�$�9,AC0by�-70["$'{�2H;)0
{l0@6G2C&�AC01*QE�$'(I&�2C-7$�9#KL;�6@E}-I9�{l$�A4J>01*>JQ0�2H;�682v-79X086D:<;Q$'{�2H;)05:C61E�0@EB"%;)08AC0ax − bx-7E &�E�01*=-I9=2H;)0%E�$'(I&�2C-7$�9#K)-�2B-IE -I9�*�0@01*>EH24AF-+:C2C( M,-79�:HAC0861EH-79�O�RB��0%6@?L$'(7$)O[-.¼<0�{l$�AB6@9DM
:H$�9�{7&�EH-+$�952H;'-IE J=6GM=;�6Gy�0VO80@9�08AF682C01*sR
p n ^ n R � �1�1�D q A;A K AL©$ �0�O���o����#+>�����\E �mEA��E0� �2���U�;�1��Q���;�������s#*1 ���Q+ �6� RS#;)052�"%$w*'-7E42C-79�:C23:H0by'-76@9)E
AP6@9�*
AQ$'{ 4ABC
E<682C-IE<{.MX2H;)0>01°'&�682C-7$�9AQ2 = AP 2 + |AC − AB|2 R]+6[`X!l{
BP = CQK�E�;)$1"�2H;�682
APa'-7E�01:C2FE
∠BACR
]+aD` ? !l{AP
a�-IEb0@:C2CE∠BAC
K'?�AC$8y�0%$�AB*'-7E<?�AC$8y�0z2H;�6G2BP = CQ
R�������������� ���@��� �9�Q�0�1 �0�� �@8Q��>?> � �5����� 8Q��B�,#*;�#*1 � �6���,#$� - ��)#$��1 � ��1p�}� � �7� �Q+!���#*�7�$H;�������2��#$�Z;�1��Q���2�s��+ #$1 � ��1��o� � R
!l{b = c
KGE42C6G2408JQ089'2#]+6[`L$'{82H;)0B?)AC$)a�(70@J³-IE#-79�:4$�A4AC0@:C2[R ¹ Es6 :4$'&�9'2408AC0<®@6@J>?�(+0'K:<;)$)$�Eb0>619DMu:40by�-I619)E
AP6@9�*
AQEH&L:<;�2H;�6G2
∠BAP = ∠CAQ 6= 12∠BAC
RS#;)089
AP6@9�*
AQ61AC0w*�-IEH2C-I9�:C2bKBE<-I9�:H0
∠BAP + ∠CAQ 6= ∠BACR�Z8-I9�:H0
4ABC-7E EFM�J>J>0G24AF-+:C6@(L&�9�*�08AfAC0G¬�0@:C2C-+$�9=-79,2H;)0�a'-7E�01:C2C$�Af$'{
∠BACK)6@9�*>E<-I9�:H0
2H;'-7E%AC08¬L01:C2C-7$�9x-79'2C0@AC:<;�6@9�O80@EP6@9�*
QK�"%0XE�010=2H;�682
AP = AQRxS�;)0@AC0G{l$�AC0�K
AQ2 = AP 2 = AP 2 + (b − c)26@9�*
BP = CQR
¯V089�:H0'K�"%0�"�-.(I(�61E<E<&�J>0V2H;�6G2b 6= c
R F@0[2x = CQ = PB
R S�;)0@9#K�a@M=2H;)0F86G" $'{#Us$�EH-79�08EGK@"%0%;�6[y�0AQ2 = b2 + x2 − 2bx cos C = b2 + x2 − x
a(a2 + b2 − c2)
6@9�*AP 2 = c2 + x2 − 2cx cos B = c2 + x2 − x
a(a2 + c2 − b2)
RS#;)08AC08{l$�AC0'K
(b − c)2 = AQ2 − AP 2 =(
1 − 2x
a
)
(
b2 − c2) K
"%;'-+:<;>EH-7J=?)(.-I�L0@EB2C$a(b − c) = (a − 2x)(b + c)
R�¯�0@9�:40�K b
c=
a − x
x=
CP
P B
K619�*52H;)0�?�AC$)$'{�-IE :H$�J=?)(70G2C0�R
��0VE�;)$1"±9�0<®82�2H;�6G2B?)61A�2f]+aD`T-7Ef9�$D2N24AC&L0%]+a�&�2B68(7J>$�EH2^24AC&L0�`�R�F@0G2x = CQ
RZ8-79�:40
BP =ac
b + c
K1"%0VO80G2
AP 2 = c2 + BP 2 − 2cBP cos B = c2 +(
ac
b + c
)2
− c(a2 + c2 − b2)
b + c
R
A)¨
¹ (7E�$�KAQ2 = x2 + b2 −
(
a2 + b2 − c2
a
)
xR
Z8&La)E42C-.2C&�2C-I9�O32H;)08Eb0z-79'2C$AQ2 = AP 2 + (b − c)2
K1"%0VO80G2
x2 −(
a2 + b2 − c2
a
)
x +c(
(b2 − c2)(b + c) + a2b)
(b + c)2= 0
K"%;'-+:<;5{+6D:C2C$�AFEB-79'2C$
(
x +ac
b + c
)
(
x − a2 + b2 − c2
a+
ac
b + c
)
= 0R
¯V089�:H0'Kx = BP
$�Ax = 2b cos C − BP
R � $D240}2H;�682T2H;)0@E�0�2�"%$�E�$'(I&�2C-7$�9)E^M�-708(7*2�"%$=?L$�EH-.2C-7$�9)Ef{l$�A
Q2H;�6G2f6@AC0%EFM�J>J>0G24AF-+:�"�-�2H;=AC0@E<?L0@:C2v2C$52H;)0%6@(�2C-.2C&L*)0�$�9
BCR
��$�A 6=:4$�9�:HAC0G2C0,0<®@6@J>?�(+0'K�-.{a = 5
Kb = 4
K�619�*c = 3
K�2H;)0@9BP = 15/7
6@9�*CQ = 15/7
$�ACQ = 149/35
R��I TZG/TZGJI {JR d ~ r��F g � � g �h� � F [ � v � X j P� b ��NHM {ORZW TUM_L_ryG��DF SOW�S��URJ{ GJbYF SOW�S��URJ{ GJb ^ GOT�NHM S/S N�dk RZW � R�a G { M N SZu/� i�� ^ � �5^ � � ��b��N'M {JRZW TUMBL r G�� ^ R M W KHL b ^ R M W KHL b [HRO~sS N�GON u wyj � k g*[?^ �0���Qj [H['g b
� G*K R N'b � W�S N �UR,u �Q� [ �'k g$� D6� v i�� F4b � W T4K4I M N R N a r l N S TUMAK l bj_NHN*T ~sW K ��� b'��KJTZL W M S4u x g � g*� � ] �\i0i b^ MBGJI S �N'M {JRZW TUMBL rJb [OS w M W�S d S b � �5b � F�� u �'jU� � ��i v � � ��b � G l PS N R��TZL M b � G lmS N'M SZu*S N�doL q R V W G4V GOT RZWZ] �������������������� ���"!#�%$'&)(%*,+.-/+�0'1�243�576�8 9:8#;:<=1�1'>@?�ACB.D.EGF�3�+�<H<=1�EIAJFLK"1M 1�FLK"1N*�EOD�>"2�0:�
P:QNRD S QTRVU�Q W#XZY�[,\@]^XO_^R`]^abRVU�Qdc,XOY@Q BC
]^a 4ABC� P:Q'R
E e _@Y F S QTRVU�Q\@fg]�h�Q:igRgXZ]b_�cj]^aB]b_^R�]
AC e _@Y C]b_^R�]
AB! fgQ.ck\@Q�igRgX lbQ.m n � P:Q'R
P S Q�RVU�Q�\%]^XI_^R]^aoXO_^RgQ:f�ckQ�igRgXO]b_=]^aAD e _@Y EF
��p U�]�q7RVU e R ! XIa AD =√
32
BC! RVU�Q:_
PXIcTRVU�QWrXOY@[V\%]^XI_^Rs]^a
AD�
t �vu"+�<w-/+�0VxyF�1z+�{%0VxZ<#x|EZD:*`0'+:E�}bFOxG+�>b0 3�5#~1�+���*,1'5d� 8��"D�>"2^D.E|EOA/��D�<=2@1'>�Auv��A��o;'���D�>"2��'xyFg}�B^�:+�>�D�*�}.A`u"+�<��D�>"1@�0.FOxIA�+�<HD:>bxOD:A/1g�.-�D�>"2@1�2�3�5#FLK�1�1:2�xyF�+�*��� Q�q�X|mImQ.WH\bmZ].n�YbXIfgQ�igRgQ�Y=mOQ:_@��RVUbc ! c�] e cdR�]�R e�� Q e iVi�]^�@_^R�]^a e m|m/\%]bc,c,X S mOQi e c�Q.c �o�|� R�fgX e _@�'mOQ�c e RgXIc,a|n^XO_@�`RVU�QjU�nb\%]�RVU�Q.c�Q.c]^a:RVU�Q�\�fg] S mOQ:W�_@Q:Q�Y�_@]�R S Q e i��bR�Q^�X|R�W e n#U e lbQ e _r] S Rg�@c�Q e _@�'mZQ e R B
]bf e R C� �d� �@f`\@fg]�]^a�q�X|mIm%Q.cVR e�S mIXIc�U S ]�RVU�RVU�QfgQ��^��XOfgQ:YrXIWH\bmIXOi e RgXZ]b_ e _@Y#X�Rgcji�]b_�lbQ:f�ckQ ���XIf�cVR/q�Q q�X|mIm@\@fg].lbQ�RVU�Q�aL]^m|mZ]�q�XI_@� � i�a �`�g�:�����H�b�:�^ N&��V�
DP
P A=
1
2
(
BF
F A+
CE
EA
) � ��¡��
...
...
..
...
...
..
...
...
...
..
...
...
...
..
...
...
...
..
...
...
..
...
...
...
..
...
...
...
..
...
...
...
..
...
...
..
...
...
...
..
...
...
...
..
...
...
...
..
...
...
..
...
...
...
..
...
...
...
..
...
...
...
..
...
...
...
..
...
...
..
...
...
...
..
...
...
...
..
...
...
...
..
...
...
..
...
...
...
..
...
...
...
..
...
.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
....................................
....................................
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
A
B CD
E
F
P
S
A �
F@0G2Sa)0V2H;)0�?L$'-I9'2 6G2v"%;'-7:<;=2H;)0%(.-79�0
EFJQ0@0G2FE 2H;)0%(I-I9�0
BCR�]\!l{s2H;)08Eb0
(I-I9�0@E}6@AC0%?�6@AF6@(.(+0G(\K�2H;)089=2C61/10S2C$Qa)0%682v-79���9�-.2�M�R `�F@0[2
x =BF
F A
619�*y =
CE
EA
R²�M¤W_089�08(I6@&�E[�DS#;)0@$�AC0@J 61?)?)(.-+0@*,-I9 4ABC
K1"%0V;�6Gy�0BF
F A· AE
EC· CS
SB= −1
r2H;�682�-IEGK
xCS
BS= y
R^�¤AF-.2C-I9�OCS = BS − BC
K1"%0V$)a'2F6@-I9
xBC
BS= x − y
R ] � `� $1"³"%0�6@?�?�(�MXW_0@9�0G(768&�EG�DS�;)01$�AC08J -79 4ABD
2C$=O80G2BF
F A· AP
P D· DS
SB= −1
r2H;�682�-IEGK DP
P A= x
DS
BS
RT�¤AC-�2C-79�ODS = DB + BS
6@9�*,&�EH-79�O�] � `HK1"%0VO80G2
DP
P A= x
DB + BS
BS= x
BC
BS
DB
BC+ x = (x − y)
DB
BC+ x
= xDB + BC
BC− y
DB
BC= x
DC
BC+ y
BD
BC
R
Z8-79�:40D-7Ef2H;)0�J=-7*�PH?L$'-I9'2 $'{
BCK�"%0�;�6Gy�0 DC
BC=
BD
BC=
1
2
KL6@9�*X] � `�{l$'(.(+$1"VE-7J=JQ0@*�-I682C08( M�R
�X-.2H;%2H;)0z&�EH&�6@(�9�$D2C6G2C-+$�95{l$�ANE<-7*�08EB619�*,619�O[(70@Ev-79�24AF-76@9�O[(+0ABC
K@"%0z��9�*#K&�E<-I9�OV] � `HKD2H;�6G2
DP
PA=
1
2
(
a cos B
b cos A+
a cos C
c cos A
)
=1
2· 2ac cos B + 2ab cos C
2bc cos A
=1
2·(
a2 + c2 − b2)
+(
a2 + b2 − c2)
b2 + c2 − a2=
a2
b2 + c2 − a2
R
��0V9�$1"&�E�032H;)03"%08(.(+P</D9�$1"V9,AC08(I682C-7$�94AD2 = 2b2 + 2c2 − a2 2C$=$)aD2C68-79
DP
PA=
2a2
4AD2 − a2
K
{+AC$�J�"%;'-+:<;%"%0�E�01032H;�682DP/PA = 1
-I{#6@9�*>$�9�( M,-.{AD =
(√3/2
)
aR
. .4R ��������������I��� ��� �9�Q�0�1 �0�� � 8Q��>?> � �5�����?8Q��B�,#*;�#*1 � ���7��#��c- �5)#���1 � ��1B�}� � RF@0G2
a = BCKb = CA
KL6@9�*c = AB
R3Z8-79�:404AD2 = 2b2 + 2c2 − a2 K2H;)0,O[- y�0@9¤:4$�9�*�-�2C-+$�9
AD =(√
3/2)
a-7E�01°'&)-�y'68(+089'2B24$
b2 + c2 = 2a2 R�F@0G2 Qa�0,2H;)0>J,-+*)P<?�$'-79'2z$'{AD
R>��05"�-.(I(^?)AC$@y�0,2H;�6G2QKEKF61AC0Q:4$'(I(.-79�0861A�K#"%;'-7:<;
-7J=?)(.-+08Ev2H;�682Q-IEv2H;)0V?L$'-I9'2
P*�0G�L9�0@*,-7952H;)0�?�AC$)a'(+08JxRF@0G2 4ABC
a�0}2H;)0z:4$)$�AC*�-I9)682C0�24AF-76@9�O[(+0z$'{�2H;)0z;)$�J>$)O80@9�0@$'&�EBa)6@A�M�:H089'24AC-7::H$)$�AC*'-79)6G240BEFM)E42408J�Kb"�-�2H;
A = (1 : 0 : 0)KB = (0 : 1 : 0)
K86@9�*C = (0 : 0 : 1)
R
D1�
S#;)089D = (0 : 1 : 1)
KQ = (2 : 1 : 1)
KE = (b − c cos A : 0 : c cos A)
KL6@9�*F = (c − b cos A : b cos A : 0)
Rv¯V089�:H0'K
det
QEF
= det
2 1 1b − c cos A 0 c cos Ac − b cos A b cos A 0
= (b2 + c2 − 4bc cos A) cos A
=[
2(b2 + c2 − 2bc cos A) − b2 − c2]
cos A
= (2a2 − b2 − c2) cos A = 0K
"%;'-+:<;,JQ08619)Ev2H;�6G2QKEKF6@AC0%:4$'(I(.-79�0861A�R
��I TZG(TZGJI {JR d ~ rcwyj � k g*[h^ �0���Qj [H['g b � G*K R NHb � W�S N �6R,uo� koj�x � � � �sjBF4b0w M T�TZG*K W M�F'G*KHL q RZW NFHL S L R ��NHM {ORZW TUM_L_r4b�DJG4V$I M N'b w i�b � F�� uh� j � k0� � n j ] k g FsF4b ��S N � q�G x S IBGOT � RZW d R TUb � ��b � F�� u��� [ �sk g*� DU� v i �FJb�� W T4K4I M N R N a r l N S TUMAK l b j_NHN*T ~sW K ��� b��KJTZL W M S4u ���Qj � k�j�wh� g �'�7���5bQ� SJ�sS L T4K � M� MBL rJb�i5T SJ�sS b"D S V S N u DOi g$[ F � k [ i5F ^ g*��X b ^�S rJTUMBd R b v �*b��FZ� u S N�d:L q R V W G4V GOT RZWZ] �4q RZW�R � S ToGON RM N � G W6W�RO� LTZGJIBKHL MBGON ]
����� �� ���������r�:���%!��%$'&w(%*V+.-/+�0N1:2 3�5 � x��gK"1�E��"D�FOD.x|EIE�1'A`�/+%}b1'>�A���*�D�>��V1��P:QNR
ABC S Q e _ e i��bRgQ�[ e _@�'mOQ�Y R�f�X e _@�'mZQHq�X|RVU i�XOfgi��@WHiVQ._^R�fgQ O� P:QNR�RVU�QmIXI_@Q:c
AO!
BO! e _@Y CO
WHQ�Q'RzRVU�Q i�XIfgi�mZQ.cBCO
!CAO
! e _@Y ABOaL]bfzRVU�Qc�Q:iV]b_@YTRgXIW�Q e R A′
!B′! e _@Y C′
! fgQ.ck\@Q�igRgX lbQ.m n � P:QNR |XY Z| Y@Q._@]�R�Q�RVU�Q \%Q.fgXIW�Q'R�Q.fe _@Y [XY Z]RVU�Q e fgQ e ]^a%RVU�Q R�fgX e _@�'mOQ XY Z
� fg]:lbQ RVU e R� e � BC
|BCA′| +CA
|CAB′| +AB
|ABC′| = 1�
� S � [BCO] · [BCA′] + [CAO] · [CAB′] + [ABO] · [ABC′] = [ABC]2�
;b+:Ey}�FOx�+�>C3�5r6�8 9:8%;�<=1:1N>�?�A"B.D.EGF�3�+�<H<=1�EIAFLK�1 M 1�FLK�1'*�EZD:>�2�0.�p XO_@i�Q 4ABCXOc e i��bR�Q:[ e _@�'mZQ:Y ! OXOc XO_4RVU�Q XI_^R�Q.fgXO]bf e _@Y ∠COA = 2B
�RVU�Q:_∠A′OC = 180◦ − 2B
�4p XIWrX|m e fgm n !∠A′OB = 180◦ − 2C
��� U�Q.c�Q�Ryq�] e _@�'mZQ.cc,�@W Rg]∠BOC = 2A
�� Q�i e �@ckQ OBA′CXOcCiGnbi�mIXOi ! RVU�Q e _@�'mOQ:c e R O e _@Y e R A′ e fgQc,�@\�\�mOQ:WHQ:_^R e f�n ! ∠A′BC = ∠A′OC! e _@Y
∠A′CB = ∠A′OB��� U^�@c ! RVU�Q e _@�'mZQ.cw]^a
4A′BC e fgQ∠BA′C = 180◦ − 2A
!∠A′BC = 180◦ − 2B
!∠A′CB = 180◦ − 2C
�� QjY�Q:_@]�RgQ RVU�Qji�XOfgi��@Wrf e Y^XI�@c]^a 4A′BC S nρ e _@Y e \�\�m nwRVU�QrP e q ]^a p XI_@Q:c R�]#�.QNR
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...............................................................................................................................
..............................................................................................................................................................................................................................
.......................................................................................................................................................................................................................................................................................................................................................................................................................
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
...............................................................................................................................................................................................................................................................................................................................................................................
..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............
..
...
...
...
....
...
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
....
...
...
...
...
...
....
...
...
...
...
...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
..
..
..
..
..
.
A
A′
B C
O
R R
ρ sin 2AR sin A
ρsin
2C ρsin
2B
D �
BC
|BCA′| =2ρ sin 2A
2ρ(sin 2A + sin 2B + sin 2C)=
sin 2A
sin 2A + sin 2B + sin 2C
R�#0@EH&)(.2B]+6[`^9�$1"{l$'(.(+$1"VEv-IJ>J>01*'-76G240G(�MsR
��$�A>?�6@AF2X]+aD`4Kz2H;)0 F86G"¡$'{%Z8-I9�0@E¤6@?�?�(I-701*�2C$¢2H;)0 :4$�J>J>$�9�E<-7*�0BC
$'{4ABC
619�* 4A′BC]�"�-.2H;�:4-7AC:4&�J=AF6D*'-I-
R619�*
ρKsAC0@E<?L0@:C2C-�y�0G(�M1`B-7J=?)(.-+08E32H;�6G2
2R sin A = 2ρ sin 2AK)$�A
ρ =R
2 cos A
R ] � `��0V9�$D2C0z2H;�6G2
[BCO] · [BCA′] =(
12BO · BC cos A
) (
12BC · BA′ sin 2B
)
= R2 sin A cos A · 2ρ2 sin 2A sin 2B sin 2CR
!�9)Eb08AF2C-I9�O32H;)0�y'6@(.&L0%$'{ρ{+AC$�J¥] � `T-7952H;'-IEB(76@EH2v01°'&�682C-7$�9>O[-�y�08Ef&�E
[BCO] · [BCA′] = 12R4 sin 2A sin 2B sin 2C tan A
RªB{�:4$'&�AFEb0'K�2H;)05619)68(+$)O8$'&�Ez01°'&�682C-7$�9)E�;)$'(+*Q{l$�Af2H;)0=$D2H;)08A}?�68-7AFEz$'{^61AC0861E8R�Z8-79�:40tan A + tan B + tan C = tan A tan B tan C
K1"%0z�L9�*�2H;�682∑
� ����� � �[BCO] · [BCA′] = 1
2R4 sin 2A sin 2B sin 2C(tan A + tan B + tan C)
= 12R4 sin 2A sin 2B sin 2C tan A tan B tan C
= (2R2 sin A sin B sin C)2 = [ABC]2R
��I TZG/TZGJI {JR d ~ r �F g � � g �h� � F [ � v � X j P� b ��NHM {ORZW TUM_L_ryG��DF SOW�S��URJ{ GJbYF SOW�S��URJ{ GJb ^ GOT�NHM S/S N�dk RZW � R�a G { M N SZu DJi k v X�] k g � � g$� b X�W�S N�d R x W�S M W M R b?� ^*u �Q� [ �'k g$� DU� v i �FJb � W T4KJI M N R N`a r l N S TUMAK l b�j N*NHT ~sW K �A� bH��KJTZL W M S4u DJi g*[ F � k [ i5F ^ g*��X b ^�S rOTUM_d R b v �*b �FZ� u x g � g*� � ] �mi i�b ^ M_GJI S�N'M {JRZW TUMBL rJb [OS w M W�S d S b � ��b�� F�� u�[ j�sk�i ��b�x�GJI �c� G lml KJNHM_L_r � GJI I R�aHR bQ�7M N�L RZW k S'{OR N'b � [ b�� F�� u�'jU� �j ��i v � � ��b � G l PS N R��TZL M b � G lmS N'M SZuS N�dmL q R V W G4V GOT RZWZ]
������#������������� �� b!#�%$'&)(%*,+.-/+�0'1�243�576�8 9:8#;:<=1�1'>@?�ACB.D.EGF�3�+�<H<=1�EIAJFLK"1M 1�FLK"1N*�EOD�>"2�0:�
��_ 4ABC! mZQ'R
D!E! e _@Y F S Q�RVU�QwWrXOY@[V\%]^XI_^Rgc ]^aoRVU�QwcVXZY�Q:c BC
!CA
!e _@Y AB
! fgQ.ck\@Q�igRgX lbQ.m n �`p U�]�q RVU e R ! XIa AD =√
32
BC! RVU�Q._
∠BEC = ∠AFC�
;b+:Ey}�FOx�+�>C3�5r(%1�FG1'*o8��w+�+�A��@xG+:EOD��o>^xG�:1'*�0VxyFZ5:A� :D � xZ*�Db2^D.A�u�� A��/;�� �
..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................
......................................................................................................................................................................................................................
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
..
A
B CD
EF
G
DD�
��0VE424AC0@9�O�2H;)089,2H;)0�?�AC$)a'(+08J {+AC$�J � -.{�� 2C$ � -I{#6@9�*>$�9�( M,-.{��DRF@0G2Ga�032H;)0%:40@9'24AC$'-7*�RN� E<-I9�O ¹ ?L$'(.(+$�9�-.&�EG�1S�;)01$�AC08J�K1"%0VO80G2
BE2 = 12
(
c2 + a2)− (
12b)2
= 12c2 + 1
2a2 − 1
4b2 = 1
4
(
2c2 + 2a2 − b2) R
S#;)08AC08{l$�AC0'KG"%0�;�6[y�0BG ·BE = 2
3BE2 = 1
6
(
2c2 + 2a2 − b2) R � $D2C0}68(7E�$32H;�6G2
BF · BA = 12c2 R
� $1"�2H;)0z{l$'(I(7$1"�-79�O3E42C6G2408JQ089'2CEf61AC0V01°'&)-�y'68(+089'2 q• ∠BEC = ∠AFC
r•
°�&�61*)AF-I(I682C0@AF68(AEGF
-IE :�M�:4(.-+: r• BG · BE = BF · BA
r•
12c2 = 1
6(2c2 + 2a2 − b2 KD2H;�682�-IEGK b2 + c2 = 2a2 r
• 2AD2 = 2a2 − 12a2 ]7&�EH-79�O ¹ ?�$'(I(7$�9�-I&�E[�1S#;)0@$�AC0@J�` r
• AD =(√
3/2)
aR
S#;'&�E[K∠BEC = AFC
-I{#6@9�*>$�9�(�M5-.{AD =
(√3/2
)
BCR
��I TZG/TZGJI {JR d ~ r �F g � � g �h� � F [ � v � X j P� b ��NHM {ORZW TUM_L_ryG��DF SOW�S��URJ{ GJbYF SOW�S��URJ{ GJb ^ GOT�NHM S/S N�dk RZW � R�a G { M N SZu wyj � k g*[ ^ �Q���0j ['[Hg b � G*K R NHb � W�S N �6R,uf� k�jAx � � � �'j_FJb\w M T�TZG*K W MoFHG*K'L_q RZW N?F'L S L R�N'M {JRZW TUMBL rJb6DJG4V$I M N'b*w i�b �FZ� u$� k0� �5[Hg F ��] n�jAwyj v}v j g bJ��N aHR I_GQF'L S L R ��NHM {ORZW TUM_L_r4bHF S N0��N a'R IBGJb��z�b� F�� u � j � k � � n@j ] k g FsF4b ��S N � q�Ghx S IBGOT � RZW d R TUb � �5b �FZ� u DOi�k v X�] k g � � g$� b X�W�S N�d R x W�S M W M R b� ^*u �Q� [ �'k g$� DU� v i �FJb � W T4KJI M N R N a r l N S TUMBK l bQj N*NHT ~sW K �A� b���KJTZL W M S4u7X�g i ��� �5g �:� ] �s� v n � ['[ bk SJl d R NHb � ��b�� F�� u � g$g `����0j [ � ��b�koGON a ��GON a b � qHM N SZu � g n � [ � vD] w � � �,�*b�n�G {ORZW b�xJ��b�� F�� uDOi g$[ F � k [ i�F ^ g$��X b ^�S rJTUMBd R b v �*b �FZ� u ^ i ^ F g*� � g �*b [HR GON'M S b v D,b � F�� u ���0j � koj0wh� g �'�7���5b� SJ�sS L T4K � M � M_L_r4bi�T SO�OS b D S V S N u[ j�skDi���b�x�GJI �D� G lml KON'MBL r � GJI I R�aHR b �}M N�L RZW k S'{OR N'b � [ b �FZ� u �sj�� � ��i v � � ��b � G l PS N R��TZL M b � G lmS N'M SZuS N�dmL q R V W G4V GOT RZWZ]
����������������������@ ^!b���.& (%*,+.-/+�0'1�2 3�5��bD.0,x|EG1#u��� *�FG+�D,1�Ad�v>bx��:1N*y0,xGFL5r+�{`(�EG+:x�1'0NFOxOA�+�<HD:>bxOD@�P:QNR
a!b!c!d S Q�\%]bcVX|RgX lbQ�fgQ e m"_b�@W S Q.f�c �� e � fg].lbQ�RVU e R�RVU�QzaL]^m|mZ]�q�XI_@��XI_@Q��^� e m|X|R�n�U�]^mOY�c�aL]bf 0 ≤ x ≤ (
5 − √17)
/2 e _@Ye mOck]�aL]bf x = 1�
∑
� � ��� � �a
a + (3 − x)b + xc≥ 1
�
� S � ? fg].lbQ�RVU�Q e�S ]:lbQ�XO_@Q:�b� e mIX�RGnwaL]bf 0 ≤ x ≤ 1�
;b+:Ey}�FOx�+�>CF�+�� e ��3�5#FLK�1T-�*V+.-/+�0N1'*gA/1g�.-�D�>"2@1�2�3�5#FLK�1�1:2�xyF�+�*��P:QNRy = 3 − x
��� U�Q._ ! S n#cVR�f e XZ�.U�RgaL]bfGq e fgYHi�]bWH\b�bR e RgXZ]b_�c ! q�Q��%_@YwRVU e Ra
a + by + cx+
c
c + dy + ax=
A
A + B
!
D"�
"%;)0@AC0A6@9�*
B6@AC0%*)08��9�01*=6@Ef{l$'(.(+$1"VE�qA = a(c + dy + ax) + c(a + by + cx)
= (a2 + c2)x + (ad + bc)y + 2acK
B = (a + by + cx)(c + dy + ax) − A
= (ab + cd)xy − ac(1 − x2) + bdy2 RZ8-7J,-I(I61AF(�M�K b
b + cy + dx+
d
d + ay + bx=
C
C + D
K1"%;)08AC0
C = (b2 + d2)x + (ab + cd)y + 2bd619�*D = (ad + bc)xy − bd(1 − x2) + acy2 R
S#;'&�E[K'2H;)0VO[-�y�089=-I9�01°'&�6@(.-.2�M>:C619>a)03"VAF-.2F2C0@9=6@EA
A + B+
C
C + D≥ 1
K"%;'-+:<;5-7E 0@°�&)- y'6@(70@9'2N2C$
AC ≥ BDR ] � `
F@0G2E = 2ac(x + 1) + (ad + bc)y
619�*F = 2bd(x + 1) + (ab + cd)y
RZ8-79�:40
A − E = (a2 + c2 − 2ac)x = (a − c)2x ≥ 06@9�*C − F = (b2 + d2 − 2bd)y = (b − d)2y ≥ 0
K"%0%;�6[y�0
A ≥ E619�*
C ≥ FRv¯�0@9�:40�KD24$5?�AC$8y�0%] � `HK'-.2�E<&��Q:H08EB2C$,?�AC$8y�0z2H;�682
EF ≥ BDR ] � `
WY6D/@-79�ON2H;)0fEH&La)E42C-.2C&�2C-7$�9)Ep = ac
Kq = bd
Kr = ad+bc
KG619�*s = ab+cd
K"%0};�6[y�0
E = 2p(x+1)+ryKF = 2q(x+1)+sy
KB = sxy−p(1−x2)+qy2 K619�*
D = rxy − q(1 − x2) + py2 R � $1"EF = 4pq(x + 1)2 + rsy2 + 2(ps + qr)(x + 1)y6@9�*BD = rsx2y2 + pq(1 − x2)2 + pqy4 − (pr + qs)(1 − x2)xy
+ (ps + qr)xy3 − (p2 + q2)(1 − x2)y2 R²�M>*'-7AC0@:C2Ba�&�2N240@*�-7$'&�EfJ>6@9�-7?�&)(76G2C-+$�9)EB&�E<-I9�O
y = 3 − xK1"%0VO80G2
2(x + 1)y − xy3 = y(2(x + 1) − x(3 − x)2)
= y(2 − 7x + 6x2 − x3) = (1 − x)(2 − 5x + x2)y619�*(1 − x2)2 + y4 − 4(x + 1)2
= y4 + (x + 1)2((1 − x)2 − 4) = y4 + (x + 1)2(x2 − 2x − 3)
= y4 + (x + 1)2(x + 1)(x − 3) =(
(3 − x)3 − (x + 1)3)
y
= (26 − 30x + 6x2 − 2x3)y = 2(13 − 15x + 3x2 − x3)y
= (1 − x)(13 − 2x + x2)yR
D�A
¯V089�:H0'K�] � `T-IE 01°'&)-�y'68(+089'2N24$(p2 + q2 + rs)(1 − x2)y2 + (pr + qs)(1 − x2)xy
+ (ps + qr)(1 − x)(2 − 5x + x2)y
≥ 2pq(1 − x)(13 − 2x + x2)yR ] � `
!l{x = 1
K)01°'&�6@(.-.2�M>;)$'(7*)EB-79>] � `�Rv!l{x 6= 1
KD2H;)089>*�- y�-7*�-I9�OVa@M(1 − x)y
O[- y�0@E(p2 + q2 + rs)(1 + x)y + (pr + qs)(1 + x)x
+(ps + qr)(2 − 5x + x2) ≥ 2pq(1 − x)(13 − 2x + x2)R ] A `
Z8-79�:400 ≤ x ≤ 1
2
(
5 − √17) Ks"%0X$)a'2F6@-I9
2 − 5x + x2 ≥ 0Rw��0>6@(IEb$x;�6[y�0
r = ad + bc ≥ 2√
pq6@9�*
s = ab + cd ≥ √pqRTS�;'&�EGK
p2 + q2 + rs ≥ 2pq + 4pq = 6pqK
pr + qs ≥ 2(p + q)√
pq ≥ 4pqK
6@9�*ps + qr ≥ 2(p + q)
√pq ≥ 4pq
RUs$�9)E�01°'&L0@9'2C( M#K
(p2 + q2 + rs)(1 + x)y + (pr + qs)(1 + x)x + (ps + qr)(2 − 5x + x2)
≥ 6pq(1 + x)(3 − x) + 4pq(1 + x)x + 4pq(2 − 5x + x2)
= 2pq(9 + 6x − 3x2 + 2x + 2x2 + 4 − 10x + 2x2)
= 2pq(13 − 2x + x2)K
0@E42C61a�(.-7E�;'-79�O�] A `�R!+2�{l$'(I(7$1"VEv2H;�6G2B] � `HK�619�*>;)089�:H0%] � `HK);)$'(7*#K):4$�J>?�(+0[2C-79�O�2H;)0�?�AC$)$'{CR��I TZGYTZGJI {JR d ~ r ��� [ �sk g*� D6� v i�� F4b � W T4KJI M N R N a r l N S TUMBK l b0j_NHN*T ~sW K ��� b5�QKOTZL W M S
(V SJW L�� ���GONHI r
)]v G(TZGJIAK'L M_GON LBGmV SOW L�� ��� ? � S T W�RO�6R M {JR d ] �Jq�KOTUb�M_L W�RZlmS M N*TDG4V R N ] �4q R V W G4V GOT RZW T�q�G � R dL_q S L b�M N�� S'� L b L_q R M N R�� K S I M_L_r
EF ≥ BDM ToL W K R GON'I rDM �
x = 1G W
0 ≤ x ≤ 12
(
5 −√
17) b ~ KHL
� G lml:R N�L R dDL_q S L LBG}T�q�G �AC ≥ BD
�AG W0 ≤ x ≤ 1
M T0dOM � K4IBL RJ{JR N�M NYL_q R}�BS T Rd = 0
] ��TUM N a7S� G l V�K'L RZW bHq R q S T {JRZW M R d}L_q R L W K'L_q�G���L q R5a M {JR N7M N R�� K S I MBL r �0q R N 1
2
(
5−√
17)
< x < 1�AG W�lmS N*r
l M I I M_GON*T�G�� �AS T R T�M N �0qHM � qabbbcbdSJW�R V GOTUM_L M {OR'] D S N�G*KOT�V W G4V GOT R d�T RJ{ORZW�S I W�R I S L R d7V W G ~ I RZl T ]
������z� ���������r�:�@ ^!����.&�(%*V+.-/+�0N1:2�3�5 � xZK`�DNEL5 �o1N>��� �1'A �"*�D:0'+b�'A`�/+�<HD�>^xZD�� fg]:lbQ RVU�Q�aL]^m|mZ]�q�XI_@� XOY@Q._^RgX|RgXOQ:c �
� e � n∑
k=1
k+1∑
i=1
(
ki − 1
)2(2kk
)
22k
(
2k2i − 2
)
(2i − 1)
=22n+1
(
2n + 1n
) − 2�
� S � n∑
k=1
k+1∑
i=1
(
ki − 1
)2(2kk
)
22k
(
2k2i − 2
)
i
=
(2n + 3)
(
2n + 2n + 1
)
22n+1− 3
�
D"D
��������������I��� ��� �9�Q�0�1 �0�� � 8Q��>?> � �5�����?8Q��B�,#*;�#*1 � ���7��#��c- �5)#���1 � ��1��}� � R]+6[`v��$�A |t| < 1
K)*�0G�L9�0
f(t) =arcsin t√
1 − t2619�*
g(t) =∞∑
k=0
22k
(2k + 1)
(
2kk
) t2k+1 R
!+2B-7E�0861EFM,2C$5y�0@AF-I{.M,2H;�682 a)$D2H;f(t)
6@9�*g(t)
E<682C-IE<{.M,2H;)0V{l$'(I(7$1"�-79�Oz-I9�-.2C-I6@(�y'68(I&L0?�AC$)a'(+08J q
(1 − t2)dy
dt− ty = 1
619�*y(0) = 0
RB¯�0@9�:40�Kf(t) = g(t)
R� E<-I9�O�2H;)0%a'-79�$�J,-768(�E�0@AF-+08E 1√
1 − 4x2=
∞∑
j=0
(
2jj
)
x2jK1"%0VO80G2
1
2arcsin(2x) =
∫ x
0
dz√1 − 4z2
=
∞∑
j=0
1
2j + 1
(
2j
j
)
x2j+1 K
619�*>;)089�:H0'K1
2f(2x) =
( ∞∑
j=0
(
2j
j
)
x2j
)( ∞∑
j=0
1
2j + 1
(
2j
j
)
x2j+1
)
=
∞∑
k=0
(
k∑
j=0
1
2j + 1
(
2j
j
)(
2k − 2j
k − j
)
)
x2k+1 R ] � `ªf9,2H;)0V$D2H;)0@Av;�619�*�K
1
2g(2x) =
∞∑
k=0
24k
(2k + 1)
(
2kk
) x2k+1 R ] � `
Z8-79�:40 12f(2x) = 1
2g(2x)
K#2H;)0=?L$1"%08A�Eb08AC-70@E3-I9x] � `f619�*w] � `fJ=&�E423a�0,-7*�089'2C-+:C6@(�RUs$�J=?�6@AC-I9�OV:4$)0��Q:4-+089'2CE[K1"%0���9�*52H;�6G2
k∑
j=0
1
22k(2j + 1)
(
2j
j
)(
2k − 2j
k − j
)
=22k
(2k + 1)
(
2kk
)
=22k+1
(
2k + 1k
) − 22k−1
(
2k − 1k − 1
)
R
²�M52C08(70@E�:H$�?�-79�ODK@"%0%$)aD2C68-79n∑
k=1
k∑
j=0
1
22k(2j + 1)
(
2j
j
)(
2k − 2j
k − j
)
=n∑
k=1
22k+1
(
2k + 1k
) − 22k−1
(
2k − 1k − 1
)
=22k+1
(
2n + 1n
) − 2K
"%;'-+:<;5-7E 0@°�&)- y'6@(70@9'2N2C$�2H;)0%O[- y�0@9,-7*�089'2C-.2�M�R
D1
]la1`f²�M52H;)0Va�-I9�$�J=-I6@(LE�0@AF-+08E 1√1 − 4x
=∞∑
j=0
(
2jj
)
xjK1"%0VO80G2
1 − √1 − 4x
2x=
1
x
∫ x
0
dz√1 − 4z
=
∞∑
k=0
1
j + 1
(
2j
j
)
xj R¯V089�:H0'K
∞∑
k=0
k∑
j=0
1
j + 1
(
2j
j
)(
2k − 2j
k − j
)
xk
=1
2x
(
1√1 − 4x
− 1
)
=
∞∑
j=0
1
2
(
2j
j
)
xj−1 =
∞∑
k=0
(
2k + 1
k
)
xk R
Us$�J=?�6@AC-I9�OV:4$)0��Q:4-+089'2CE O[-�y�08Ek∑
j=0
1
22k(j + 1)
(
2j
j
)(
2k − 2j
k − j
)
=1
22k
(
2k + 1
k
)
=
(2k + 3)
(
2k + 2k + 1
)
22k+1−
(2k + 1)
(
2kk
)
22n−1
R²�M52C08(70@E�:H$�?�-79�ODK1"%0V:4$�9�:4(I&L*)0z2H;�682
n∑
k=1
k∑
j=0
1
22k(j + 1)
(
2j
j
)(
2k − 2j
k − j
)
=n∑
k=1
(2k + 3)
(
2k + 2k + 1
)
22k+1−
(2k + 1)
(
2kk
)
22n−1
=
(2n + 3)
(
2n + 2n + 1
)
22n+1− 3
K"%;'-+:<;5-7E 0@°�&)- y'6@(70@9'2N2C$�2H;)0%O[- y�0@9,-7*�089'2C-.2�M�R
��I TZG/TZGJI {JR d ~ rywyj � k g*[ ^ �0���Qj [H['g b � G*K R N'b � W�S N �6R,u �Q� [ �'k g$� DU� v i �FJb � W T4KJI M N R N`a r l N S TUMAK l b5j N*NHT ~sW K �A� b*�QKOTZL W M SZu�[ � PjBF [ i x g FJb � MBG\d R D S N R M W GJb ^�W�S �6M I u�S N�dmL_q R V W G4V GOT RZWZ]
��������T�g�:���������� b!.� �:& (%*,+.-/+�0'1�2�3�5 ��x��kF�+�*�����<HD�>bA��w1'0NFG1'*�>�~vDNEIx|EG1:1�u"+:EIE�1��^1'A� 0k*gD^1�Ey�
P:QNRA1B1C1 e _@Y A2B2C2 S Q�Ryq�]�R�fgX e _@�'mOQ:c=q�X|RVU A1C1 = A2C2
�p �@\@\@]bc�Q�RVU e R`RVU�Q�XI_^R�Q.fgXO]bf e _@�'mZQ S XIc�Q:igR�]bf�c A1D1 e _@Y A2D2 e fgQzQ��^� e m �� e � �LaoRVU�Q e m|RgX�Rg�%[email protected] B1H1 e _@Y B2H2 e fgQrQ:�b� e m ! ckU�]�q7RVU e R�RVU�QwR�f�X e _@�'mZQ.c e fgQi�]b_@�Nfg�%Q._^R �� S � �LavRVU�Q�XO_^RgQ:f�XZ]bf e _@�'mOQ S XIc�Q:igR�]bf�c B1E1 e _@Y B2E2 e fgQ#Q:�b� e m ! c�U�]�q RVU e R�RVU�QR�f�X e _@�'mZQ.c e fgQziV]b_@�Nf��%Q:_^R �
DD©
��������������I��� �o��� �Q�� ��� �Z�1��}�5�,)#$�6�O����� �����}#*���*���61��}#*���*���61��5#���� �Q��� R!�9 4ABC
K8(+0[2aKbKca�0f2H;)0 (+089�O�2H;�E�$'{
BCKCA
KAB
K@AC0@E<?L0@:C2C-�y�0G(�MsRF@0[2ha�0�2H;)0,(70@9�O�2H;�$'{�2H;)0,6@(�2C-.2C&L*)0
BH619�*X(+0[2
`a
6@9�*`b
a)0�2H;)0,(70@9�O�2H;�E�$'{T2H;)0-79'2C0@AF-+$�Af619�O[(705a�-IEb0@:C24$�AFE
AD6@9�*
BEKLAC08E�?�01:C2C- y�08( M�R ¹ (IEb$�K)(+0[2
x619�*
ya)0%2H;)0
(+089�O�2H;�E^$'{82H;)0BE�01OGJ>0@9'2FECD
6@9�*BD
KGAC08E�?�01:C2C- y�08( M#KG6@9�*�(+0[22α
a)0�2H;)0fJ>0@6@E<&�AC0$'{
∠BACRv!�952H;'-7EfEb$'(.&�2C-+$�9#K � -I9�:HAC0@6@E<-I9�OD�3J>0@6@9)E � E424AC-7:C2C(�M�-79�:HAC0861EH-79�OD�DR
]+6[`^��0VE�;)$1"2H;�6G2 4ABC-IEf&�9�-+°'&L08( MQ*)0G2C0@ACJ=-I9�01*Q]I&�?=24$,6�:H$�9�OGAF&L0@9�:40[`
a@MbKhK#619�*
laRV��0,6@E�EH&�JQ0
b619�*
h2C$¤a�0=:4$�9)EH2F619'2[R�� E<-I9�O
c =h
sin 2α
6@9�*2H;)03"%08(.(+P</D9�$1"V95{l$�A4J,&)(76
`a =2bc
b + ccos α
K1"%0V$)a'2F6@-I9
1
`a
=1
2bh
(
2b sin α +h
cos α
) R!+2�-IE50861EFMw2C$xEb0@0>2H;�6G2
1/`a
-7E%6@9u-I9�:HAC0@6@E<-I9�O>{7&�9�:C2C-+$�9Y$'{αR ¯V089�:H0'K
`a
-IE�6@9-79D��0@:C2C-�y�0u{7&�9�:C2C-7$�9�$'{
α6@9�* 2H;)0@AC0G{l$�AC0�K
α-IEQ&�9�-7°�&L0G(�Mp*�0[2408A4J,-79�0@*�a@M
bK
hK
619�*`a
RTS#;)089 4ABC-7EB&�9�-7°�&L0G(�M>*)0G2C0@ACJ=-I9�01*>a@M
bK`a
K�619�*αR
]la1`^��0�Eb;)$1"³2H;�682 4ABC-IEf&�9�-7°�&L0G(�M>*)0G2C0@ACJ=-I9�01*>]7&�?,2C$,6�:H$�9�OGAF&L0@9�:40[`
a@MbK`a
K1619�*`b
R���0�6@E�EH&�JQ0b6@9�*
`a
24$%a�03:H$�9)E42C6@9'2[R^U�(70@6@AC( M#Kx-7E�6@9%-I9�:HAC0@6@EbP
-79�O {7&�9�:C2C-7$�9,$'{αRv�DAC$�J
`a =2bc
b + ccos α
K8"%0�E�010 2H;�682c =
b`a
2b cos α − `a
-7E�6@9-79�:HAC0861EH-79�O�{7&�9�:C2C-+$�9X$'{
αR3�'AC$�J y
x=
c
b
K�"%0%Eb0@0�2H;�682y =
cx
b
-7E}6@9Q-I9�:HAC0@6@E<-I9�O{7&�9�:C2C-+$�9>$'{
αK�619�*�2H;)0@9=Eb$�-7E
a = x + yR � $1",K
`2b = ac
(
1 − b2
(a + c)2
)
-7E�619�-79�:HAC0861EH-79�O,{7&�9�:C2C-7$�9u$'{αKTE<-I9�:H0>0@61:<;�{+61:C24$�A�-IE@R>S�;'&�EGK
`b
-7E�619�-79D��0@:C2C-�y�0{7&�9�:C2C-+$�9,$'{
α6@9�*%2H;)0@AC0G{l$�AC0�K
α-IE�&�9�-+°'&L08( M,*�0[2408A4J,-79�0@*=a@M
bK`a
KD619�*`b
RTS�;)0@94ABC
-IEB&�9�-+°'&L08( MQ*)0G2C0@ACJ=-I9�01*>a@MbK`a
K�6@9�*αR
��I TZG/TZGJI {JR d ~ rywyj � k g*[ ^ �0���Qj [H['g b � G*K R N'b � W�S N �6R,u �Q� [ �'k g$� DU� v i �FJb � W T4KJI M N R N`a r l N S TUMAK l b j N*NHT ~sW K �A� bh�QKOTZL W M SZu DOi g$[ F � k [ i5F ^ g*��X b ^�S rOTUM_d R b v �*b � F��
(V SOW L � ��� GONHI r
)ux g � g$� � ] �mi i�b ^ MBGJI S �N'M {JRZW TUMBL rJb [JS w M W�S d S b � ��b � F�� u�[ j%sk�i ��b7x�GJI � � G lml KJNHM_L_r � GJI I R�aHR b�}M N�L RZW k S'{OR N'b � [ b��FZ� u�S N�dmL q R V W G4V GOT RZWZ]
D S N�G*KJT V GJM N�L R doG*K'L�L_q S L�L_q R L �7G S T�T RZW L M_GON*T M NDL_q'M TV W G ~ I RZl b S T � R I I S T lmS N$r}GHL q RZW TUM l M I SOWGON R TUb � M_L_q\TZGJIBKHL MBGONHTUb �AS N ~*R �AG*KJN�doM N������ S N�d������ ]� / /"��/")�3./�������� �������������������� �����$n�M R ��GONHTZL W K � L MBGON { GON(n W�R M RO�A�*R N��! !��"�#��� �$ �%���'&(����$ �*)*�!+%���!����)��,�-�%�!��.*/,0�1�����32��4��� #���562��*� ��5�7�� X�R G l:R L W M �AS N S L W�R K a KJI NHM �sS �98 ����5!:!" �<;���54#�7!�� �!�4+,5*= ������.*.,>�1 �����?z� �g�:����� ���@ ^!�� �:& (%*,+.-/+�0'1�2 3�5 9:}:D�>A@ �o+�0 �V+ �/+�<=1N*V+ � �D:**B"}^1 VA�o>^xG�:1'*�0VxG2^D^2�2@1 �bD.E|EZD^2@+:EIx�2�A �bD.E|EZD^2@+:EIx�2�Ao;'-�D.xO>o�
P:QNRABC S Q e R�f�X e _@�'mZQ#XI_ q�U^XOi,U e _@�'mOQ:c B e _@Y C e fgQ S ]�RVU e i��bR�Q � P:Q'R
H S Q RVU�Q=\%]^XI_^R�]b_ c,XOY@Q BCc,�%i,U RVU e R AH ⊥ BC
� P:QNRr!
r1
! e _@Y r2 S QRVU�Q#XI_@i�XOfgi�mOQ:c�]^a/R�f�X e _@�'mZQ.c ABC!ABH
! e _@Y AHC! fgQ.ck\@Q�igRgX lbQ.m n �#p U�]�q�RVU e R
r + r1 + r2 − AHXIcd\%]bcVX|RgX lbQ ! _@Q:� e RgX lbQ ! ]bf<C,Q:fg] e i�iV]bfgY^XO_@� e c ∠A
XOc ] S Rg�@c�Q !e i��bRgQ ! ]bf�fgXO�.U�RV[ e _@�'mZQ:Y �
D1¨
��������������I���h!���#y-/���:��s+�1 �0���H�6���,#(�'1 � ��1��}� � R......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.................................................................................................
.
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...................................................................
...........................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................
r
rr
A
B C
Ir
d d
c − d
c − d
b − d
b − d
......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
.
..
.
..
.
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.................................
......................................................................................................................................................................
...................................................................
.
.
..
.
..
.
.
..
.
..
..
..
..
..
.
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
...............................................
..........................................................................................................................................................................................................................................
..................................................................................................
.................................................
......................................................................
A
B CH
r1
r2
bc
c′ b′F@0G2da�0f2H;)0 (70@9�O�2H;%$'{D2H;)0f2C6@9�O80@9'2�{+AC$�J
A2C$32H;)0 -79�:4-IAC:4(+0)R^Z8-I9�:H0f2C6@9�O80@9'2FE
*)AF6["V9x24$x6¤:4-7AC:4(70Q{+AC$�J§6@9_0H®@2C0@AC9)6@(�?L$'-I9'2%;�6[y�0¤01°'&�6@(N(+089�O�2H;�KT"%0XJ,&�EH2V;�6[y�0a = (c − d) + (b − d)
r@"%;)0@9�:40�Kd =
b + c − a
2
RZ8-7J,-I(I61AF(�M�K'{l$�A^24AC-I619�O[(70@E
AHB6@9�*
AHCK1"%0V;�6Gy�0
r1 =c′ + h − c
2
Kr2 =
b′ + h − b
2
K"%;)0@AC0
h = AHKc′ = BH
K�6@9�*b′ = CH
R^Us$�9)E�01°'&L0@9'2C( M#K
h − r1 − r2 =b + c − a
2= d
R� $D2C0z2H;�6G2
∠A-IE $)a'2C&�E�0�K�AF-+O8;D2bK)$�A�6D:4&�2C0V6D:4:H$�AC*'-79�Oz61E
d < rKd = r
K)$�Ad > r
R� ��+ �������"� � HO��>?>@#$�7��� �#086D*)0@AFE�"%;)$5��9�*,6V?�AC$)$'{#a@M=?�-+:C2C&�AC0z&�9)E�6G2C-7EH{.M�-I9�OzE�;)$'&)(+*:H$�9�:40@9'24AF6G2405$�9X08-�2H;)0@Af$'{#2H;)0V2�"%$>AF-+O8;D2v24AF-76@9�O[(+08E 2H;�6G2fEb;�6@AC0V2H;)0G-7A ;1M�?L$D2C0@9�&�E�0AI
6@9�*u;�6Gy�0>(701OGE%$'{v(70@9�O�2H;d619�*
rq5�1- ®>2H;)0X:4-7AC:4(70,"�-�2H;wAF6D*'-I&�E
r24$xa�0=2H;)0
-79�:4-IAC:4(+05$'{T6Vy'61AF-761a�(70V24AC-I619�O[(70ABC
rL6@EAJQ$8y�0@E�6["V6GM>{+AC$�J¦2H;)0,:4-IAC:4(+0'K
d-79�P
:HAC0@6@Eb08Ef{+AC$�J024$�-79���9�-.2�M�RB¯�$1"V61AC*,E<&�?)?)(.-+08E ;'-7E $1"V9=6@(�2408A49)6G2C-�y�0�61ACO[&�J>0@9'2v-I9
;'-7EB�L9)68(�?)61AF61OGAF61?�;sR S�;'&�EGK• ∠A
-7E $)aD2C&�Eb0z-.{#619�*>$�9�( M5-I{d < r
r12H;�6G2�-7E[Kh − r1 − r2 < r
K• ∠A
-7Ef6VAF-+O8;D2v619�O[(70�-.{#6@9�*>$�9�(�M5-I{d = r
r12H;�6G2�-7E[Kh − r1 − r2 = r
K• ∠A
-7Ef61:4&�240z-I{#6@9�*>$�9�( M,-.{d > r
r12H;�682�-IEGKh − r1 − r2 > r
RS#;)0}AC0@EH&)(.2s{l$'(.(+$1"VET-IJ>J>01*'-76G240G(�MsRvWwMV6@ACO[&�JQ089'2#"V61ENO[&)-+*)01*%a@M � S�AC-I619�O[(70@E�"�-�2H;2H;)0Q��-+O8;D2 ZG2C&)¶^�3{+AC$�J C:�� ���4� > K � q ] � � � ¨ `HKL?)?TR �1� B;�;� R ¹ AC08(I682C01*>?�AC$)a'(+08J¥-IE� A!�D�1� -79 �$H;���������H���#��JHO# � �Q+ � ���C��#$>?�����+H0� K�� k q � ] ���� � `4K)?TR � �© qv»�AC$8y�0z2H;�6G2
r + r1 + r2 ≤ h sin AR
��$�A�619=68(.2C0@AC9)682C- y�0z2C$�2H;)0�?�AC$)$'{sa@M=?�-+:C2C&�AC0'K1"%0�&�E�0tan
A
2=
r
d
RTS�;)0@9
h − r1 − r2 = d = r cotA
2
K"%;'-+:<;x-7E%(+08E�EV2H;�6@9#KB0@°�&�68(�24$�Kv$�A%OGAC08682C0@Az2H;�6@9
r61:H:4$�AC*�-I9�OX6@E
∠A-IE,$)aD2C&�Eb0'K
AC-7O8;D2bK�$�A�61:4&�240)R
D �
��I TZGhTZGJI {JR d ~ r wyj � k g*[f^ �0���Qj [H['g b � G*K R N'b � W�S N �6R,um� k0� �5[Hg F ��] n�jBwyj v}v j g b���N a'R IBGFHL S L R �N'M {JRZW TUMBL rJb�F S Nm��N aHR I_GJb���z�b�� F�� u �Q� [ �'k g$� DU� v i �FJb�� W T4KJI M N R N a r l N S TUMBK l b�j_NHN*T ~sW K ��� b��KJTZL W M S4u X�g i ��� �5g �y� ] �s� v n � ['[ b\k SOl d R N'b � ��b � F�� u ���0j � kojmwh� g �'�7���5b:� SO�OS L T4K � M � MBL rJbi5T SJ�sS b2D S V S N u � g n � [ � vD] w � � �,�*b�n�G {ORZW b�xJ��b��FZ� u DJi g*[ F � k [ i�F ^ g$��X b ^�S rOTUM_d R b v �*b�� F�� un ] D ] F'w g*g*v ��b� S IBL ~ G lml:R I b�L q RQvoR L q RZW I S N�dsT u x g � g*� � ] �mi i�b ^ M_GJI S ��NHM {ORZW TUM_L_r4b [JS w M W�S d S b � �5b� F�� u[ j'sk�i ��b x�GJI �m� G lml KJNHM_L_r � GJI I R�a'R b5�}M N�L RZW k SH{JR NHb � [ b�� F�� u �sj�� �s ��i v � � ��b � G l PS N R��TZL M b� G lmS N'M SZuS N�dmL_q R V W G4V GOT RZWZ]
������z� �g�:����� �����b!�� �:& (%*,+.-/+�0'1�2 3�5 9:}:D�>A@ �o+�0 �V+ �/+�<=1N*V+ � �D:**B"}^1 VA�o>^xG�:1'*�0VxG2^D^2�2@1 �bD.E|EZD^2@+:EIx�2�A �bD.E|EZD^2@+:EIx�2�Ao;'-�D.xO>o�P:QNR
ABC S Q e R�fgX e _@�'mOQdXI_�q�U^XZi,U e _@�'mZQ.c B e _@Y C e fgQ S ]�RVU e i��bR�Q ! e _@YzmZQ'Ra!b!c S Q�RVU�Q�mZQ._@��RVUbc�]^a�RVU�Qjc,[email protected]`]b\@\@]bc,X�R�QsRVU�Q lbQ:fyRgXZi�Q:c A
!B!C! fgQ:c,\%Q:igRgX�lbQNm�n �
�Laha
XIcjRVU�Q e m�RgX|Rg�%Y�Q�aZfg]bW AR�]
BC! \@fg].lbQ�RVU e R 1
h2a
−(
1
b2+
1
c2
) XIcd\%]bcVX|RgX lbQ !_@Q�� e RgX�lbQ ! ]bf C,Q.fg] e i�iV]bfgY^XO_@� e c ∠A
XIcj] S Rg�@ckQ ! e i��bR�Q ! ]bf�fgXO�.U�RV[ e _@�'mZQ:Y �;b+:Ey}�FOx�+�> 3�5 u�Kbx�-�uv}�*�FOxI0kA � xI0k0'+%}�*�xj;b+%}bFLK�1'*�>�;bFZDbFG1 �o>^xG�:1'*�0VxyFZ5:As9N+.-%E|xZ>bA � �A�/;�� �P:QNR
S =1
h2a
−(
1
b2+
1
c2
) �`p XI_@iVQha = c sin B
! q�QzU e lbQ
S =1
c2 sin2 B−(
1
b2+
1
c2
)
=1
c2 sin2 B−(
sin2 C
c2 sin2 B+
1
c2
)
=1
c2 sin2 B
(
1 − sin2 C − sin2 B)
=1
c2 sin2 B
(
cos2 C − sin2 B)
=1
2c2 sin2 B
(
[1 + cos(2C)] − [1 − cos(2B)])
=1
2c2 sin2 B
[
cos(2B) + cos(2C)]
=1
c2 sin2 Bcos(B + C) cos(B − C)
= − 1
c2 sin2 Bcos A · cos(B − C)
�p XO_@i�Q
B e _@Y C e fgQ S ]�RVU e i��bRgQ ! q�QHU e lbQ −π/2 < B − C < π/2�vq�U�Q:_@i�Q !
cos(B −C) > 0� � U^�@c ! RVU�Q cVXZ�N_w]^a
SXOcRVU�Q ]b\@\@]bc,X�R�Q ]^a^RVU�Q cVXZ�N_w]^a
cos A! e _@YRVU�Q:fgQNaL]bfgQ !
SXOcd\%]bcVX|RgX lbQ ! _@Q�� e RgX�lbQ ! ]bf C,Q.fg] e i�iV]bfgY^XO_@� e c ∠A
XIc�] S Rg�@c�Q ! e i��bR�Q !]bf f�XZ�.U�R ���� ����������� �������������������! "��#��%$%�'&)(�*+�,.-�/�01���23�40151�6�78�9�239;:4���,<�9�239;:4���,>="���;/�0 9�9/�
? ��2A@B�BC����0 /D9�E� �FHGI=��! �=��J ���,�-K/�0���23�40�5 �L�7>="�B0 2BM�5 ,�="�B0 2BM�5 ,N#�����9/���/�EPO6(3+ ? �D#Q=��J�;�!(A#�#���, )�DM���/�,R��239�/TS4�UER �(V+ ? �! �WI(4X ? �����,J �9�/TS3YT��Z�9[�����]\T��2; D���4,'+���,^-"�B��EK_�FH� ? F!`'�J �WH,!ZT��2U5 9[�����4,$aOb,�-"�B��E"`!�!#�� ? �� ^_4�%$cFa-K�,)-K2V�[M[� 0 /���/�C��d]/D9�40AMde,�( /D/�����2BM�SAf�,��!M���5 230 9�EK\"�DW8-�#B�b$8X�Og-� %��GD,W'�����23,"Z��,%-K���)E)WaX _�XD��O6����$^��,�h[9[��5A�D��d]di�B� ,"51YT�R$^��51YT��23� 9/� ��VE)&)�TFH �&)�i���B�!Z�J��(AW'(��[,D��C�230 /�01�,&�2;����S4�UE8ZK�K���� jG)X'`]F%F),R=�0��� 9L-�/�01���23�40151�[,^#9bOb0 239� �9[,8+���,]-"�B��ER#�(!h ? Fa-T,aZT�� f6+T��d]dNM/�0151�+T�� �1�BC��B,�`a0 /�5���2 ? 9����/�,)��#,J-K���)EH��(;�T-ih�\TFH$%�J !-�,�+T��d *9/���k��5 0 ,) )��d]9/�0 9[E�9/� ]51YT�JlT2;�[l"������2�X
D�
p n&n ^ R � �@�D�1 q A) K A � �0�s���o���$# + ��� !�� ��%�}��� HO� �o��>@#��O� � 2�� Q�5# �O1�}�5�,)#$�6�O�,+5�5+ + # ����B�6�5+ ��A��+�1 ����B�6�5+ ��A��+�1}�$������� R
F@0G2mb
a)0}2H;)03(70@9�O�2H;5$'{�2H;)03JQ0@*�-I619%2C$�EH-+*)0b-I9 4ABC
KD6@9�*5*�0G�L9�0mcE<-IJ=-.(76@AC( M�R,»�AC$@y�0%2H;�6G2
4a4 + 9b2c2 − 16m2bm2
c
-7E3?�$�E<-�2C-�y�0'K#9�0@OG682C- y�0�Ks$�AB¼<08AC$6D:4:H$�AC*'-79�Oz61Ef619�O[(70
A-IEf6D:4&�2C0�K)$)aD2C&�Eb0'K)$�A�AC-7O8;D2HP<6@9�O[(+0@*�R
��������������I���h!���#H�7��H;�5�,���"��#���;1��Q���9�O�,+ #*1 � ��1��}� � R� E<-I9�O«2H;)0 "%0G(I(7P�/19�$1"V9¸{l$�A4J,&)(76@E
mb = 12
√2a2 − b2 + 2c2
6@9�*mc = 1
2
√2a2 + 2b2 − c2
K1"%0V;�6Gy�0
4a4 + 9b2c2 − 16m2bm2
c
= 4a4 + 9b2c2 − (2a2 − b2 + 2c2)(2a2 + 2b2 − c2)
= 4a4 + 9b2c2 − 4a4 − 2a2[
(2b2 − c2) + (2c2 − b2)]
+ (b2 − 2c2)(2b2 − c2)
= −2a2(b2 + c2) + 9b2c2 + (2b4 + 2c4 − 5b2c2)
= −2a2(b2 + c2) + 2(b2 + c2)2
= 2(b2 + c2)(b2 + c2 − a2)R
U�(70@6@AC( M#K2(b2 + c2)
-IEB?L$�EH-.2C- y�0�KDEb$V2H;�6824a4 + 9b2c2 − 16m2
bm2c
;�61E�2H;)0zE�6@JQ0E<-7OG9X6@E
b2 + c2 − a2 K�"%;'-+:<;=-7E}?L$�EH-.2C- y�0�K�9�01OG6G2C-�y�0'K�$�Av¼<0@AC$=-.{∠BAC
-IE�61:4&�240'K$)a'2C&�E�0�K)$�A�AC-7O8;D2bK�AC0@E<?L0@:C2C-�y�0G(�MsR
��I TZG/TZGJI {JR d ~ r �F g � � g �h� � F [ � v � X j P� b ��NHM {ORZW TUM_L_ryG��DF SOW�S��URJ{ GJbYF SOW�S��URJ{ GJb ^ GOT�NHM S/S N�dk RZW � R�a G { M N SZu/� i�� ^ � �5^ � � ��b��N'M {JRZW TUMBL r G�� ^ R M W KHL b ^ R M W KHL b [HRO~sS N�GON u wyj � k g*[?^ �0���Qj [H['g b� G*K R N'b � W�S N �UR,u n�j6i v}vogy^ �Qj [Hg �*b g*[ F'j gy� � wyx ^ g$['[ b S N�d � k � �5['g F ��] n�jBwyj v}v j g b ��N a'R IBGFHL S L R �N'M {JRZW TUMBL rJboF S N ��N a'R IBGJb(��z�b � F�� u�� koj�x � � � �'j_FJb}w M T�TZG*K W M�FHG*K'L_q RZW NyF'L S L R ��NHM {ORZW TUM_L_r4bDJG4V$I M N'b?w i b �FZ� u/� j � k0� � n j ] k g FsF4b ��S N � q�G�x S I_GOT � RZW d R TUb � �5b � F�� u DJi g k�i��Q� � n�bx�G W L S I R TUb v w b�� F�� u ��� [ �sk g*� DU� v i �FJb � W T4K4I M N R N a r l N S TUMAK l b j N*NHT ~sW K �A� b'��KJTZL W M S4u X�g i ��� �5g �� ] �'� v n�� [H[ b�k SOl d R N'b � ��b��FZ� u ���Qj � k�j�wh� g �s�}�Q��bQ� SJ�sS L T4K � M � M_L_r4b0i�T SO�OS b2D S V S N u � g n � [ �vD] w � � �,�*bQn�G {JRZW b�xJ��b �FZ� u n ] D ] F'w g*g*v ��bb S IBL ~ G lml:R I b�L_q RYvoR L_q RZW I S N�dsT u x g � g$� � ] �\i0i b^ MBGJI S �N'M {JRZW TUMBL rJb [OS w M W�S d S b � ��b��FZ� u�[ j�sk�i ��b�x�GJI �c� G lml KJNHM_L_r � GJI I R�a'R bQ�7M N�L RZW k SH{JR NHb � [ b� F�� u �'jU� �j ��i v � � ��b � G l PS N R��TZL M b � G lmS N'M SZuS N�dmL q R V W G4V GOT RZWZ]
�� � ��� ���������r�:���b!����.&�(%*V+.-/+�0N1:2�3�5 � xZK`�DNEL5 �o1N>��� �1'A �"*�D:0'+b�'A`�/+�<HD�>^xZD��P:QNR
ak
!bk
! e _@Y ck S QdRVU�Q�mZQ._@��RVUbcj]^a@RVU�Q�cVXZY�Q:c�]b\�\%]bcVX|RgQ RVU�QdlbQ.f�RgXOiVQ.c Ak
!Bk
! e _@Y Ck
! fgQ.ck\@Q�igRgX lbQ.m n ! XI_rR�f�X e _@�'mZQ AkBkCk
! aL]bfk = 1
!2!. . .
!n� �La
rk
XIcRVU�Q�XO_�f e YbX|�@cT]^a%R�f�X e _@�'mZQ AkBkCk e _@Y#X|a Rk
XIcsX|R�cTi�XIfgi��@WHf e YbX|�@c ! \@fg].lbQ�RVU e R
6√
3
(
n∏
k=1
rk
)1n
≤(
n∏
k=1
ak
)1n
+
(
n∏
k=1
bk
)1n
+
(
n∏
k=1
ck
)1n
≤ 3√
3
(
n∏
k=1
Rk
)1n �
�
��������������I����� �&H;�Q#*���Q���6���A�B�,#*1��o�0��#$��1 � �Z� �JHO# RS#;)0VAF-+O8;D2�-I9�01°'&�6@(.-.2�M=J=6GM=a�0�"VAC-�2F24089>01°'&)-�y'68(+089'2C(�M=6@E
(
n∏
k=1
ak
Rk
)1n
+
(
n∏
k=1
bk
Rk
)1n
+
(
n∏
k=1
ck
Rk
)1n
≤ 3√
3R ] � `
� E<-I9�OB2H;)0�F86G"�$'{�Z8-79�08E�6@9�*�2H;)0@9�2H;)0 ¹ W B fW�!�9�0@°�&�68(I-�2\M�K8"%0 ��9�*�2H;�682s2H;)0 (+0G{I2E<-7*�0V$'{s] � `T-IE 01°'&�6@(�2C$(
n∏
k=1
2 sin Ak
)1n
+
(
n∏
k=1
2 sin Bk
)1n
+
(
n∏
k=1
2 sin Ck
)1n
≤ 2
(
1
n
n∑
k=1
sin Ak +1
n
n∑
k=1
sin Bk +1
n
n∑
k=1
sin Ck
)
=2
n
n∑
k=1
(sin Ak + sin Bk + sin Ck)R
� $1"¦"%0Q&�Eb0>2H;)0>"%0G(I(7P�/19�$1"V9u-I9�01°'&�6@(.-.2�Msin A + sin B + sin C ≤ 3
√3/2
K"%;'-+:<;3-IE#y'68(I-7*3{l$�A#619�6@AHa'-.24AF6@A�M324AF-76@9�O[(+0�"�-�2H;z6@9�O[(+08E
AKBK86@9�*
CR:� ��+QE � ��$�AL2H;)0
:H$�9Dy�089�-+089�:H05$'{�619DMQAC086D*)0@AFEf"%;)$>61AC0%9�$D2f{+6@J=-.(I-I61AN"�-�2H;=2H;'-IE�AC08E<&)(�2bK�"%0%E<&�?)?)( M6w?)AC$)$'{ q=EH-79�:40Q2H;)0�EH-79�0X{7&�9�:C2C-7$�9Y-7E5:H$�9�:C6Gy�0�$�9u2H;)0X-79'2C0@A�y'68(
[0, π]K^"%0�;�6[y�0
sin A + sin B + sin C ≤ 3 sin(
13(A + B + C)
)
= 3 sin(π/3) = 3√
3/2R
S#;'&�E[K(
n∏
k=1
ak
Rk
)1n
+
(
n∏
k=1
bk
Rk
)1n
+
(
n∏
k=1
ck
Rk
)1n
≤ 2
n
(
n · 3√
3
2
)
= 3√
3R
� $1"�:H$�9)EH-+*)0@A#2H;)0 (+0G{I2�-79�0@°�&�68(I-�2\MsR^� EH-79�OB2H;)0 ¹ W B BW�!�9�0@°�&�68(I-�2\M�K8"%0�O80[2(
n∏
k=1
ak
)1n
+
(
n∏
k=1
bk
)1n
+
(
n∏
k=1
ck
)1n
≥ 3
(
n∏
k=1
akbkck
)13n R ] � `
��0�"�-I(.(TJ>61/10,&�Eb0>$'{NEb$�J>0=J>$�AC0=EH2F619�*�61AC*�AC0@EH&)(.2FE�61a�$'&�2 24AC-I619�O[(70@E8RtF@0[2sa�0
2H;)0¤E�0@J,-7?�0@AF-7J>0G2C0@A�RYS�;)0@9x2H;)0¤6@AC0@6�$'{�2H;)0Q24AF-76@9�O[(+0Q-7E�O[-�y�089_a@M_a�$D2H;rs
6@9�*√
s(s − a)(s − b)(s − c)]�¯V08AC$�9#� E3�'$�A4J,&)(76[`�RTS�;)0@AC0G{l$�AC0�K
r =
√
(s − a)(s − b)(s − c)
s
≤√
1
s
(
(s − a) + (s − b) + (s − c)
3
)3
=
√
1
s
(
s
3
)3
=s
3√
3
R
]\¯�0@AC0="%0Q;�6Gy�0>&�Eb0@*w2H;)0 ¹ W B BW !�9�0@°�&�68(I-�2\Mx6DOG68-79�R `¢S�;'&�Es ≥ 3
√3rR���0
6@(IEb$5;�6Gy�0abc = 4Rrs
619�*R ≥ 2r
]��'&)(70@A�� E�!�9�0@°�&�68(I-�2\M1`�RNUs$�J>a�-I9�-79�O36@(.(�2H;)0@E�0
'�
AC0@EH&)(.2FEGK1"%0V$)a'2F6@-I9abc ≥ 4(2r)r(3
√3r) = (2
√3r)3
K1"%;'-+:<;%"%0z&�Eb0z-I9>] � ` q(
n∏
k=1
ak
)1n
+
(
n∏
k=1
bk
)1n
+
(
n∏
k=1
ck
)1n
≥ 3
(
n∏
k=1
(2√
3rk)3
)13n
= 6√
3
(
n∏
k=1
rk
)1n R
��I TZG/TZGJI {JR d ~ r �F g � � g �h� � F [ � v � X j P� b ��NHM {ORZW TUM_L_ryG��DF SOW�S��URJ{ GJbYF SOW�S��URJ{ GJb ^ GOT�NHM S/S N�dk RZW � R�a G { M N SZu�� koj�x � � � �sjBF4b�w M T�TZG*K W MF'G*KHL q RZW NYF'L S L R �N'M {JRZW TUMBL rJb D4G4V$I M NHb�w i b �FZ� u �Q� [ �'k g$�DU� v i �FJb � W T4K4I M N R N a r l N S TUMAK l bfj N*NHT ~sW K �A� b(�QKOTZL W M SZu x g � g$� � ] �\i0i b ^ M_GJI S ��NHM {ORZW TUM_L_r4b [OSw M W�S d S b � �5b � F�� u:[ j skDi���b:x�GJI �@� G lml KON'MBL r � GJI I R�aHR b/�}M N�L RZW k S'{OR N'b � [ b � F�� umS N�d L_q RV W G4V GOT RZWZ]
�� ��� ? � ��������� �����b!�� �:& (%*V+.-/+�0N1:2�3�5 � +�K@D:<H<=1�2=�sD.0k0VxIEOD:A ;bFL*gD.0.3�+%}:* �.A��*�D�>��V1��
P:QNRMABC S Q e RgQNR�f e U�Q�Ybfg]b_ ! e _@YrmZQ'R M ′ S Q e _�nH\%]^XI_^RsXI_rRVU�QzXO_^RgQ:f�XZ]bf]^a 4ABC� ��Q._@]�R�Q RVU�Q e fgQ e ]^a 4XY Z S n [XY Z]
� fg].lbQ�RVU e R
(MM ′)2 = MA2 [BM ′C]
[ABC]+ MB2 [CM ′A]
[ABC]+ MC2 [AM ′B]
[ABC]
−(
AB2 [BM ′C][CM ′A]
[ABC]2+ BC2 [CM ′A][AM ′B]
[ABC]2
+ CA2 [AM ′B][BM ′C]
[ABC]2
) �u"+�<H<=1N>�F�� � U^XIcfgQ:cV��m|RvaL]bf e RgQNR�f e U�Q�Ybfg]b_zXIc���c,XIWrX|m e f��R�] p R�Q�q e fyR�� c � U�Q:]bfgQ:W aL]bfe R�f�X e _@�'mZQ � �La M ′ = G
! RVU�Q�iVQ._^R�fg]^XZYr]^a 4ABC! RVU�Q._wRVU�QzfgQNm e RgXZ]b_ S Q�i�]bW�Q.c
MG2 =1
3(MA2 + MB2 + MC2) − 1
9(AB2 + BC2 + CA2)
!q�U^XZi,UwXOc�q�QNmIm � _@]�qz_ � 0,0N1'>vFOxOD.E|EL5�FLK"1�0kD�<=1`0N+:Ey}�FOx�+�>w3�5 � x��gK"1�E �"D�FOD.x|EIE�1'A"�/+%}b1'>�A���*gD:> ��1��bD�>"2 Nx:B.K�+%}.A(%+:EZ? u"+�<H<�}:>bxGFL5Hu"+:E|EG1 �b1�A ��xZ>�FG1'*T��Db�:1'>�A � .Aj�/;�� �
p XO_@i�QM ′ XIc�XI_#RVU�Q�XI_^R�Q.fgXO]bfs]^a 4ABC
! q�QzU e lbQ M ′ = αA + βB + γCaL]bfjc�]bWHQw\@]bc,X�RgX�lbQ�fgQ e mv_b�@W S Q:f�c α!β!γcV�%i,UHRVU e R α + β + γ = 1
�d� igRg� e mIm n(α, β, γ) e fgQ RVU�Q e fgQ e m"i�]�]bfgYbXI_ e RgQ:cj]^a M ′ fgQNm e RgX�lbQ Rg] (A, B, C) e _@Y e fgQz�'X lbQ:_S n
α =[BM ′C]
[ABC]
!β =
[CM ′A]
[ABC]
!γ =
[AM ′B]
[ABC]
� ��¡��
;�
S#;)089 −−−→MM ′ = α
−−→MA + β
−−→MB + γ
−−→MC
K'619�*�"%0V;�6Gy�0
(MM ′)2 =−−−→MM ′ · −−−→
MM ′
= α2MA2 + β2MB2 + γ2MC2
+2αβ−−→MA · −−→
MB + 2βγ−−→MB · −−→
MC + 2γα−−→MC · −−→
MA
= α2MA2 + β2MB2 + γ2MC2
+ αβ(MA2 + MB2 − AB2)
+ βγ(MB2 + MC2 − BC2)
+ γα(MC2 + MA2 − CA2)
= (α + β + γ)(αMA2 + βMB2 + γMC2)
− αβAB2 − βγBC2 − γαCA2 RZ8-79�:40
α + β + γ = 1K1"%0%:4$�9�:4(I&L*)0z2H;�682
(MM ′)2 = αMA2 + βMB2 + γMC2 − (αβAB2 + βγBC2 + γαCA2)K
619�*52H;)0�?�AC$)$'{�-IE :H$�J=?)(70G2C0%a@M>] � `^61a�$8y�0�R��I TZGoTZGJI {JR d ~ r(�Q� [ �'k g$� D6� v i�� F4b � W T4K4I M N R N a r l N S TUMAK l b�j N*NHT ~sW K �A� b*�QKOTZL W M SZu�S N�d DOi g$[F � k [ i5F ^ g*��X b ^�S rJTUMBd R b v �*b � F�� ]
�� �� ��T�������������%$�!N��� � ¡^$:¡b!�¡^$��^& (%*V+.-/+�0N1:2w3�5 9:}�D:> @ �o+�0 ��+��/+�<=1N*V+ � �D:**B"}^1 VA�o>^xG�:1'*�0VxG2^D^2�2@1 �bD.E|EZD^2@+:EIx�2�A �bD.E|EZD^2@+:EIx�2�Ao;'-�D.xO>o�P:QNR
ABC S Q e R�f�X e _@�'mZQ e _@YwmZQ'R a S Q RVU�Q�mZQ._@��RVUr]^a@RVU�Q�c,XOY@Q�]b\@\@]bc,X�R�Q RVU�QlbQ:fyR�Q��A� �La
ma
XIcTRVU�Q�mOQ:_@��RVU=]^avRVU�QwWHQ�Y^X e _ aZfg]bW ARg]
BC! e _@Y X|a R
XIcTRVU�Qi�XOfgi��@Wrf e Y^XI�@c�]^a 4ABC! \�fg]:lbQ#RVU e R ma − R
XIc�\%]bcVX|RgX lbQ ! _@Q:� e RgX lbQ ! ]bf�C,Q.fg] !e i�iV]bfgY^XO_@� e c ∠AXOcj] S Rg�@c�Q ! e i��bR�Q ! ]bf�f�XZ�.U�RV[ e _@�'mOQ�Y �u"+�<C3:xZ>�D�FOx�+�>�+�{.0,xO<#xIEOD�*@0N+:Ey}�FOx�+�>�0�3�5 �/+�5��"D�*�3�D:*gD.A�v>bx��:1N*y0,xGFL5d+�{ �o1�xO*�}�FZA �o1�xO*�}�FZA
^1�3�D:>�+�>^�vD:>�2���x �gK@D:*,2 � 8v��1'0,0kA��D�>��gK�+H(bDNEG+�0 �%1'*,2@1�0kA`u�� Aj�/;�� �� Q=mZQ'R
M S Q RVU�QCWrXOY@[V\%]^XI_^Rw]^a BC e _@Y�i�]b_�c,[email protected]�RVU�Q=R�f�X e _@�'mZQ OMAq�X|RVUHc,[email protected] = ma e _@Y AO = R
��� iVi�]bfgYbXI_@��R�]����%i�mIXOY ! RVU�Q�fgQ.m e RgX lbQ�c,X C,Q:c]^a%RVU�Q:ckQ�Ryq�]wc,[email protected]@Q.\%Q._@Y�cT]b_wRVU�Q�c,X C,Qz]^a%RVU�Qz]b\@\@]bc,X�R�Q e _@�'mZQ.c �������� �
AXOcj] S Rg�@c�Q �
`Q:fyR�Q��A
� ]b_ RVU�Q i�XIfgi��@W�i�XIfgi�mZQ � XIcc�Q.\ e f e RgQ�YwaZfg]bW�RVU�Qzi�XIfgi��@W�i�Q:_^R�fgQ O S n�RVU�Qi,U�]bfgYBC
� p XO_@i�QOM ⊥ BC
!∠OMAXOc�] S Rg�@c�Q^��q�U�Q:_@i�Q ! RVU�Q#]b\�\%]bcVX|RgQwc,XOY@Q R
XIcmZ]b_@�.Q.f�RVU e _rRVU�Q e Y�h e iVQ._^R�c,XOY@Q ma
�^RVU e RsXIc !ma−R < 0
q�U�Q:_AXIc] S Rg�@c�Q ! e ci�m e XIW�Q:Y � .
..
..
..
.
..
..
.
..
..
.
..
..
.
..
.
..
..
..
..
.
..
..
..
..
..
.
..
.
..
.
..
.
..
.
..
..
..
.
..
.
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.................................................................................................................................
...................................................................................................................................................................................................................................................................................................
...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...................................................................................................................................................................................
.
.
..
.
.
..
.
..
.
..
.
.
.
.
..
.
..
.
..
.
.
..
.
.
A
B CM
O
ma R
������� �A = 90◦
��zQ.fgQ
BCXOc e YbX e WHQNRgQ:fk��RVU^�@c ! ma = R
! e _@Y ma − R = 0�
�A
/l� ��c�� RA-7Ef61:4&�240)R
S#;)0�?)AC$�?L$�E<6@(v-IE5-I9�:H$�ACAC01:C2 qma − R:46@9�a�0 ?L$�EH-.2C- y�0�K3¼<08AC$�K%$�AQ9�0@OG682C- y�0Y"%;)089
A-7E56D:4&�2C0�KB61E�{l$'(I(7$1"VE@R�F@0[2
B′C′ a�0X2H;)0?L08A4?�0@9�*'-+:4&)(I61A3a'-7E�01:C2C$�A�$'{OM
R,��$�AA$�9
2H;)0w(+$�9�O�61AC:=$'{ 2H;)0u:4-7AC:4&�J>:4-7AC:4(70ua�0[2�"%01089B′ 6@9�* C′ K#"%0Q;�6Gy�0
∠MOA > ∠OMAr
"%;)0@9�:40ma − R > 0
RQ��$�AA6G2
B′ $�A�682C′ K�"%0XO80[2 ma − R = 0
R_�D-I9)6@(.(�M�K�"%;)089A(I-70@EX$�9�08-�2H;)0@A,Eb;)$�A�2=6@AC:
B′B$�A
C′CK
"%0,Eb0@0�2H;�682ma − R < 0
R � $D240�2H;�6G2 2H;)0?�AC$�?�$�E�68(¢a�0@:H$�J>0@E�:4$�A4AC0@:C2±{l$�A�24AF-76@9�O[(+08E
.
.
..
..
.
..
..
.
..
..
.
..
..
.
..
..
.
..
..
..
..
.
..
..
..
..
..
.
..
.
..
.
..
.
..
..
..
.
..
.
..
.
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
...............................................................................................................................
...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..........................................................................................................................................................................................................................................................................
................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
..
..
.
..
.
..
..
.
..
..
.
..
.
...................................................................................................................................................................................
..
.
..
.
..
.
..
.
..
.
.
.
.
.
..
.
..
.
..
.
..
.
.
.
.
..
.
.
..
.
..
.
..
.
.
........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ ........
A
B
B′
C
C′
M
O ma
R
ABC-79}"%;'-+:<; ���A� 619�O[(70@ET61AC0f6D:4&�2C0$r�2H;)0@9
A"�-I(.(19�0@:H08E�E<61AF-I( M3(I-70 $�932H;)0 6@AC:
B′C′"%;'-+:<;�K'6@EN"%0�;�6Gy�0}�+&�E42�Eb0@0@9#K1{l$�AC:H08Ema − R > 0
K'61EB:4(768-7J>01*�-I9�2H;)0z?�AC$�?�$�E�68(FR��I TZG(TZGJI {JR d ~ rcwyj � k g*[h^ �0���Qj [H['g b � G*K R NHb � W�S N �6R,uo� koj�x � � � �sjBF4b0w M T�TZG*K W M�F'G*KHL q RZW NFHL S L R ��NHM {ORZW TUM_L_r4b DJG4V$I M N'b w i�b � F�� u ��� [ �sk g*� DU� v i �FJb � W T4K4I M N R N a r l N S TUMAK l bfj_NHN*T ~sW K ��� b��KJTZL W M S4u X�g i ��� �5g �m� ] �s� v n � ['[ b�k SOl d R N'b � �,b��FZ� u�� � g n � [ � vD] w � � �,�*b�n�G {ORZW b�xJ�5b�� F�� u� x g � g$� � ] �mi i�b ^ M_GJI S �N'M {JRZW TUMBL rJb [JS w M W�S d S b � �5b � F�� u0[ jbsk�i ��b0x�GJI � � G lml KON'MBL r � GJI I R�aHR b�}M N�L RZW k S'{OR N'b � [ b �FZ� u �sj�� �� ��i v � � ��b � G l PS N R��TZL M b � G lmS N'M S
(� MBL q\L �7G0V W GHG�� T��AG W L_q R G ~ L6KJT R `
S N a I Ro�AS T R)u�S N�dmL_q R�� V W G4V GOT RZWZ] �4q RQS TZL RZW M T � d R TUM a N S L R T�TZGJIAK'L M_GON*T�L q S L � RZW�R}� G W6W�RO� L b ~ KHL �0q�GOT R
S N S I rOTUM ToG���L q R}S'� K'L R ` S N a I RD�BS T R � S T0M N � G l V$I R L R'] j_N S d*dOMBL MBGON � P� ��[ ��� ��i vog ��$v P�*b ^ M am��S V$M_dsTUbwyj b�� F��hV W G { MBd R d S�� G*KON�L RZW�R��OSJl V$I R T�q�G � M N a L_q S L L q R7� GON � IBKOTUM_GON(LBGDL q R7� G W6W�RO� L R d�V W G4V GOT S I*TZL M I I� S T�� S � R d ] �4q RZW�R � RZW�R L_q W�ROR M N � G W6W�RO� L�T4K ~sl M T�TUMBGONHT ]
����� � ��������� ��������������
� �"!$#&% '(!)#+*-,.'$!/�10$'23% '4�#+*-,57698 :<;9=�57>@?"=A8 :CBED�FHG�I?"67JLKA:M?NBE=<OP?">@?"=A8 :CBEDRQTSU=CBEKM?WVEX G�X9Y[Z7JN\�]9?"=^`_@a�b % '(!)#+*-,.'$!/�10(c _@a ,
d7;LBUeE698 eEfg5U698 :M;9=AD�FHG�I?"67JLKA:M?NBE=ADhO<ij;9eU67JR:<?RBU=CDRQkVEI?";9lE;Lm 6HY"JRB7noI?qprd9=C?[6U?R=C8 KPsutvX SkXEw)J"DhsL?Nm m5U698 :M;9=ADx?">@?"=A8 :C8yFWGxI?[67J[KC:<?RBU=CDhOP?R>�?R=C8 :<8zQ{tvX |$X7Y"J"eE67DR}EG�X 5{XL|~;7;767=C;[\H}ESE=AByK<?HVUX G�X7Y[Z7JN\�]9?"=
% '(!)#+*-,.'$!/�10$'2�% '�4�#+*-,d7;LBUeE698 eEfg5U698 :M;9=AD�FHGxI?[67J[KC:<?RBU=CDhO<i ;9eE67JN:M?NBE=ADRQk�9JR:<=C8 KPsqYRBE=C=z]@pGyJNn987�oJLs"8 m5U698 :M;9=ADx?">@?"=A8 :C8yFWGxI?[67J[KC:<?RBU=CDhOP?R>�?R=C8 :<8zQT�EZL8 m 8 lu�P;9eUf9}"�P?R�/��8 fRZ7J">~}
�"X �1X7tk=C;9DPDP>@J"e{}"�TeE67=C?q�1Z9J[eUf9}U�vJ[;7s[8yY"JN:M;y}9�7]7=ABED���DM8 J