�

���������

���������������������������������! #"%$ �&�

')(+*,(.-0/21435-61.7839(4:;78=5?)

�

���������

� ������������������ �"!$#�%'&(%*)�+-,�%

.0/2143657185*149;:=A@;BC5*>A/D@?EGF2>?905CEH>=E(I018J;BG>0KA/214LM5NFO9PE(IAFQ5R16:AFDEGFO>?9SKT?J6Aegfih-j?kmlnfih-j?oqp0h rSfihtsukgl;fwv�x"y{z}|=~FQ/D/�K01PJ;BG145*149AEH1T:EG>>?9;1J0BG16H@;9?F?14B5'FE?9M>Ae;E(I01�1T:?FEH>?BFO5};903T/ `

?BGB34EG@;L=E(I01 c >?9AEG1T5HE�FO9E(I01�1 c 1TLMK;14B �66?i6 >A/DFO36:~N365=E(I01 �T"A 36BFDEGFQL81N>?9AEG1T5HE*?9;>"EE(I01 �T"6 36BFDEGFQL81Nt>?9AEH145(E ` @;BJ0BG>0K?/O1TL5E(IAFQ5LM>?9AE(I c >?LM1e2BG>?LE(I01 t>A/Q/G¡16¢41 >?9AEHLM>?BG1T9 c ?9AEG1T5HE* �66A£¥¤"6n`8 @;B�E(I?3T9 5N¢4>EG> a 9;:?B§¦1=¨43"K014/D/21A�t>A/Q/G¡16¢41 >?9AEHLM>?BG1T9 c S/2>0> 5�34eQEH14BNE(IAFQ5 c >?9AEG1T5HE�:01T5(F2¢§9;1T:eq>?B�5*1 c >?9;:?36B{?9 `

«¬C®¯-°±¬²n³ ®´µ·¶¬®}¯-³ ¸¹¯º�»g»¼½g»g¾¿nÀtÁ8Â�à ���Ä À¹ÅÇÆ�À¹È�É �-��Ê

Ë ` 6A34/Q@;3§EH1gE(I01eq>A/D/2>6~FQ9;¢"0¢XF ?FQ9;¢mE(I01N3T905H~�14Bu3T53Ne2B3 c EGF2>?9(

1 − 122

)(

1 − 132

)

· · ·(

1 − 120052

) `º `gÌ 983 c FOB c /O1RI?3§AFO9;¢ c 149AEHBG1

O?E(I01 c I0>?BG:?5

AB3T9;:

CD36BG1}J;1TBGJn149;:?F c @0/O3TBnEG>g1T3 c IC>"E(I014B¹369;:g9;1§FDE(I014B c I0>?BG:gJ;3T57571T5E(I?BG>A@n¢4I�E(I01 c 1T9AEHBG1 `� I0>6~�E(I?3§E∠AOD + ∠BOC = 180◦

`.........................................................................................................................................

......................................................................................................................................................................................................................................................................................................

..................................................................................................................................

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

.

..............................................................................................................................................

..

..

..

..

..

..

..

..

..

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

..

.

.

.

.

.

.

..

..

..

..

..

..

..

..

..

..

.

.

.

.

.

.

.

.

.

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

....

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

................... . . . . . . . ..........

A

B

C D

O

Í `� I0>6~ÎE(I?34E}eq>?B}BG1T34/x369;:

y~�I014BG1

x 6= 0 369;: y 6= 0 ?E(I01TBG1�FQ5�9;>M5*>A/D@?EGF2>?9eq>?BE(I01N16ÏA@;34EGFO>?9(x+ y)4 = x4 + y4

`Ð `Ñ 3XÒuÓ>6~ÎL369"?5(FDEGFO>?905 >?93§?14B3"¢41 c 369©3 9?FO¢4I"E�BG143 c ISFQ9>?9;1©LM>T?1©>?9b3

8 × 8 c I0145757K;>?3TBG:;ÔÑ K"Ò�Õ=I?3§EBG1T5(@0/DEM:0>~�1S¢41XEeq>?Bu3T9n× n c I01T5'5*K0>?36BG:0Ô

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

.

��*���

AAKHHY��� ���

AAU

HHj

ÖA×6ØÙÚ¥Û§Ø*ÙØ*ܧÝdÚ§Þßà Ü4áDâ*×XÝN×4ߥãÝG×TØ8ãH×4ßXä6ØÚ§Þ$ßXÜæå*ç§èXédßêãH×4á Þ ÝÚ§ÞRÝqë�ÚãHì4íTßXîØ7ã©×6Ú4îðïá ñÚ4ܧÝ{ߥò ò óSÞOÚ§ò òDÚ¥ë Ø¥ôdõ¥óÚ4Ü6Ø�Û§Ø*î2ÝðáDö{ߥò ò óA÷0Ú4î�Û§áDöðØÛ§Ø*îqãßAø¼ `� I0>6~E(I?3§EgKTAFO9AE5gFO905(F2:01�3BGFO¢4I"E(HEHBFO3T9;¢X/21�~FE(I�5(F2:01T5

6c L=

8c L=?3T9;:10c L�?34E/214365HEEð~�>�>AenE(I01TLùI?3§?13R:AFO5HEG3T9 c 157L3T/D/214B�E(I?369

5c L `

�

¾ `� I?BG1T1MB@;909;1TB5X��¹?1T5X��(@0/2145§369;:��¹14BEHB3T9;:PLM161XEg>?9;1:03X?L8149AE*E(I?34E�¹14BEHB3T9;:tE(I01e23T5(EG1T5HEC>Ae�E(I01E(I?BG161A}:;>0145R3�eO@0/D/uBG1*?>A/Q@?EGFO>?9©>Ae�E(I018EHB3 c' FO92LFQ9?@?EH145369;:E(I?3§E�¹?145M:0>01T5�FERFQ9

8L�FO9?@?EG1T5X�I0>6~ /O>?9;¢P:0>01T5�FEEG3 1��(@0/2145�EG>�:0>R3CeO@0/D/;BG1*?>A/D@?EGF2>?9t6FQe?~�1meO@;BE(I014B9;>"EH1�E(I?34E3T/D/E(I?BG161�B@;909;1TB5�LM161XEN34ECE(I01573TL81J?/O3 c 1�>?9[E(I018EHB3 c'K;1§eq>?BG1��¹?145§"E(I01N5'/O>6~�1T5HE}>AenE(I01CE(I?BG161NBG@;909;14B5§;I?365�:0>?9;13NeO@0/D/¹BG1*?>A/D@?EGF2>?9;Ô

.

..

..

.

..

..

..

.

..

..

..

.

..

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..............................................................

...........................................................................................................................................................................................................................................................................................................

...................................................................................................................................

..

..

.

..

..

..

..

..

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.............................................................................................................................................

...

...

...

...

...

.

..

..

..

..

.

..

..

..

..

.

..

..

..

..

...

...

...

...

......................................................................................................................................................................................................

.

.

.

.

.

..

.

.

.

.

.

..

.

.

.

.

.................................................................................................................................................

Ir

?Ø*î2ÝðîqßXÜ6ôr � í6òDØ7ã

r � Û4Ø7ã� Ý{ßXî2Ý

� ` Ó>6~L369"?9AE3TFQ9�E(I01R:AF2¢XFE734E/214365HE}>?9 c 16Ô

� ` Õ=1MK?@0FD/2:P3 c >?9;1e2BG>?L 3 c FQB c @0/O3TBgJ0FO1 c 1>Ae c 3TBG:;K0>?36BG:I?3§AFO9;¢3�B3":AFQ@;5�>Ae10

c L�KT< c @?EEGFQ9;¢�>A@?EN3�5*1 c EH>?BCe2BG>?LFDE ` 1XEH14BHL�FO9;18E(I01M?>A/Q@;LM1=>Ae}E(I01 c >?9;1M~�1=>0K"EG34FO9S365�3eO@;9 c EGF2>?9d>AeE(I01P3T9;¢X/21AFQ9©B3":AFO3T905§34ENE(I01 c 1T9AEG1TB�>AeE(I015*1 c EH>?B `�t1 c 3T/D/§E(I?34E��>A/Q@;LM1

= 13πr2h

*~�I014BG1rFQ5tE(I01}B36:?FD@;5>AeAE(I01mK03T5*1m>Ae"E(I01 c >?9;1A

hFQ5�E(I01�3T/EGFDEG@n:01m>AeAE(I01 c >?9;1A63T9;:

2πB3":AFO3T905

= 360◦`

..

.

..

.

..

.

..

.

..

.

..

..

..

.

..

..

..

..

..

..

.

..

..

.

..

.

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

........................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................

.........................................................................................................................................................................

............................................................................................................................................................................................................................................

...................

...................

.....................

.....................

¶¬C®N´�¬���²;¸«¬®}¯�°±¬C²¹³®N´µ º�»g»¼½g»g¾����� v����0x� �"!$#&%'�(*)*)�+

Ë ` �,1 c EG@n14B}1XE}:0>?909;1TBu57>A@;5}/O3eq>?BHLM1Ce2B3 c EGFO>?90903TFQBG1(

1 − 122

)(

1 − 132

)

· · ·(

1 − 120052

) `º ` }36905�@;9 c 1TB c /O1=:;1 c 149AEHBG1

O�/O1T5 c >?BG:;145

AB1§ECD5*>?9AECJ;1TBGJn149;:?F c @0/O34FOBG145R1T9AEHBG1T71§/Q/O1T5�1XEC9;18J0365'5*149AECJ;3T5 Ñ 9?F/.- @;9;1N9?Fn//- 34@?EHBG1XÒ�J036B�/O1 c 149AEHBG1 `t >?9AEHBG1"0ÏA@n1

∠AOD + ∠BOC = 180◦`

..

..

..

.

..

..

..

.

..

..

..

.

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...................................................................

.....................................................................................................................................................................................................................................................................................................

..................................................................................................................................

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

.

.

.

..............................................................................................................................................

..

..

..

..

..

..

..

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

.

.

.

.

.

.

.

.

.

.

.

.

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

....

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

................... . . . . . . . ..........

A

B

C D

O

Í ` >?9AEHBG110�ÏA@n1=J;>A@;Bx1XEyB§¦1T14/Q5�>¡@

x 6= 0 1XE y 6= 0 �FD/}92- 1436FQ5(EG1�J;3T5�:;15*>A/D@?EGF2>?9�Jn>A@;B�//-q¦1TÏ?@;3§EGF2>?9(x+ y)4 = x4 + y4

`Ð `Ñ 3XÒª>?LMK?FO1T9 :;1 c 36571T5©@;9c 3XA3T/DF214BJn1§@?E((FD/S3§EEH1§FO9;:?BG1149L8>4

�

¼ `¹ >?9AEHBG1"0}Ï?@7- 1T9J?/O3��c 3T9AE c FO9;Ï�J;>AFO9AE5N¡3C/.- FO9AE§¦14BGFO14@;Bu:�- @;9�EHBGFQ369;¢X/O1BG1 c EG3T9;¢X/21N:;1c��>"E4¦16c L=

8c L 1XE

10c L�AFQ/?AFO905 :014@&3R:?FQ5(E369AE}:01NLM>AFO905 :01

5c L `

¾ `- BG>AFQ5 c >A@;BG1§@;B5M��¹?1T5X��(@0/2145�1§E�¹1TBðEHB369;:P36BGBGF ?1T9AEm@;9�{>A@;B8¡3@;9;1J0FQ5(EG1�:2- 34E(IA/G¦1§EGFQ57LM1 c FOB c @0/Q3TFQBG1 ` 3 c I?3T9AE�Ï?@7- FQ/Q5J;3TBEG1T9AE�EH>A@;5u1T90571TLMK?/O1?AÏA@n1 �¹14BEHB3T9;:t6/O1gJ?/Q@;5uB36J?F2:01C:;145EHBG>AFO5X�e234FDE�/O1�EH>A@;B�:;1=/Q3[J0FQ5(EG1[149

2L�FO9?@?EG1T5M1XE�ÏA@7- �¹?145/.- 1�,1 c EG@n1C149

8L�FO9?@?EG1T5 ` ?9 c >?LMK?FO1T9�:;1 EH14LMJ05 �(@0/2145�e234FDE(FQ//O1�EH>A@;B�:;1 c 1§EEG1[J?FO5HEH1Au5(F�>?9Ç51- 3TJn14B��c >AFDE�ÏA@n1P/O1T5REHBG>AFO536L�FO5-571BG149 c >?9AEHBG149AE¡3}@;9 c 1TBðEG34FO91T9;:?BG>AFDE:01}/Q3J0FQ5(EG13XA369AEL �14L81gÏA@7- �¹?145§4/21 c >A@;BG1§@;B�/O1mJ?/Q@;5/O1T9AE*61T9�3TFE�1�,1 c EG@�¦1 @;9EH>A@;B c >?LMJ?/21XE;Ô

.

..

..

.

..

..

..

..

.

..

..

.

..

.

..

.

..

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.............................................................

............................................................................................................................................................................................................................................................................................................

...................................................................................................................................

..

..

..

.

..

..

..

..

..

.

..

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.....................................................

..............................................................................................................................................................................................................................................

......................................................................................................

.

.

.

.

.

.

..

.

.

.

.

.

..

.

.

.

.................................................................................................................................................

Ir

?Ø*î2ÝðîqßXÜ6ôr � í6òDØ7ã

r � Û4Ø7ã� 8Ø*äTßXî2Ý

� ` t>?L8KAF2149R1'3TFO5HEH1E - FQ/?:;1�9;>?L8K?BG1T5�1T9AEGFO1TB5�FQ9?e(¦1TBF21§@;B5C¡310 000

ÏA@0F c >?9AEGF214909;1T9AE3T@ML8>AFQ905}@;9;1eq>AFO5u/21 c IAF ,tBG17Ô

� `0 9�e236K0BF2ÏA@n1C@;9 c��>?9;1M¡3J;3TBEGFQB�:2- @;9LM>?B c 143T@M:01 c 36BðEH>?9c FOB c @0/Q3TFQBG1�:0110

c L :;1RB3X?9�1T98/D@0F c >A@;J;3T9AE@;95*1 c EH1§@;B `N¦1§EG1TBGLFQ9;1TB�/21?>A/D@;L81[:A@ c��>?9;1[>0KAEG1T9?@i149©eq>?9 c EGFO>?9d:01/.- 369;¢X/O1N3T@ c 1T9AEHBG1�149B3":AFO3T905�:A@8571 c EG14@;B ` 9B36J0Jn1§/MÏ?@n1 �>A/D@;L81= 1

3πr2h

>¡@r1T5HE[/21B3§?9P:01�/Q365CK03T5*1:A@ c��>?9;1?

h1T5HE /O3MI?34@?EH1§@;Bg:A@ c��>?9;1M1XE

2πB3":AFO3T905

= 360◦`

..

.

..

.

..

.

..

.

..

.

..

..

..

..

.

..

..

..

..

..

.

..

.

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.........................................................................................

....................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................

............................................................................................................................................................................................................................................

...................

...................

.....................

.....................

� 143TEn~�1�¢XF ?1}E(I01 5*>A/D@?EGF2>?905EG>gE(I01 �T"6 �nBFDEGFQ5*IC>A/D@;L8KAFO3mt>A/Q/O16¢4145 149?F2>?BÓCFO¢4I 6c I0>0>A/ 3§E(I01TL34EGF c 5 >?9AEG1T5HE �T"A ��� �¥¤ ��� £�?`Ë ` 1XEH14BHL�FO9;1�E(I01�9?@;LMK;14Bg>Ae�5716ÏA@n1T9 c 1T5C>Ae c >?90571 c @?EGF ?1�FQ9AEH1T¢41TB5 ~�I0>?5*1�5'@;LFO5

100`

����������������������� ��!-`¨T1§E

kK01ªE(I01ªnB5HEFQ9AEH1T¢41TB�FO9dE(I01S5*1TÏ?@n149 c 1S369;:

nK;1[E(I019?@;LMK;14B8>AeFO9AEG16¢414B5FO9�E(I015*1TÏ?@n149 c 1 `� I01T9�E(I01N5(@;L >AenE(I01 c >?90571 c @?EGF ?1FQ9AEH1T¢41TB5}FO5

k + (k + 1) + (k + 2) + · · · + (k + n− 1) = nk + 1 + 2 + · · · + n− 1= nk + 1

2n(n− 1) = 100 "E(I?34E�FQ5§

2nk + n(n− 1) = n(2k + n− 1) = 200 0>?B2k + n− 1 = 200

n

` IA@;5X

nL�@;5(E}K01N3�:?F ?FQ5*>?Bu>Ae

200~FE(I

n > 1 " E(I?3§E�FO5Xn ∈ {2 4 5 8 10 20 25 40 50 100 200}.>?905(F2:01TB�E(I01eq>A/Q/O>6~FO9;¢�E3"KA/21?

£

n?Ï?@;3§EGF2>?9�eq>?B

k k

2 2k + 1 = 1009;>�FQ9AEH1T¢41TB�5*>A/D@?EGF2>?9

4 2k + 3 = 509;>�FQ9AEH1T¢41TB�5*>A/D@?EGF2>?9

5 2k + 4 = 40 k = 188 2k + 7 = 25 k = 9

10 2k + 9 = 209;>�FQ9AEH1T¢41TB�5*>A/D@?EGF2>?9

20 2k + 19 = 109;>�FQ9AEH1T¢41TB�5*>A/D@?EGF2>?9

25 2k + 24 = 8 k = −840 2k + 39 = 5 k = −1750 2k + 49 = 4

9;>�FQ9AEH1T¢41TB�5*>A/D@?EGF2>?9100 2k + 99 = 2

9;>�FQ9AEH1T¢41TB�5*>A/D@?EGF2>?9200 2k + 199 = 1 k = −99

IA@;5XE(I014BG1[3TBG1PA?1P5*1TÏ?@n149 c 1458>Ae c >?905*1 c @?EGF?1=FQ9AEH1T¢41TB5R~�I0>?5*1P5(@;L FQ5100

`� >"EG1�E(I?34E E(I015*1TÏ?@n149 c 1�eq>?B

n = 25FQ5gE(I015*1TÏ?@n149 c 1�eq>?B

n = 8~FE(IE(I01

17c >?90571 c @?EGF ?1FO9AEG16¢414B5e2BG>?L −8 EG> 8 t~�IAF c IP36:;:=@;J=EG> 0'1TBG>¹FO9 c /D@n:;1T: `=Ì 9=E(I01573TL81N~N3§?Bn = 40

FO5�E(I01�5716ÏA@n1T9 c 1�eq>?Bn = 5

~FE(IME(I0135c >?90571 c @?EGF ?1NFQ9AEH1T¢41TB5e2BG>?L −17 EH> 17 "~�IAF c I3":0:@;JEG> 0'1TBG>¹AFO9 c /D@n:;1T: `� I014BG136BG1C??15*1TÏ?@n149 c 145�>Ae c >?905*1 c @?EGF?1CFO9AEG16¢414B5u~�I0>?5*15(@;L FQ5

100`

º `¹ac @nK?F c 34/6¢X/O3T575¹E369 ~FE(Im5(F2:01T5->AeT/O1T9;¢¥E(Ig>?9;1}LM1§EHBG1uFQ5J0/Q3 c 16:C>?9g3I0>?BGF 0'>?9AEG34/EG36K?/O1m369;:RI?34/Qe?;/D/21T:C~FE(IC~N3§EH14B `- IA@;5§4E(I01g:01TJAE(IR>Ae?E(I01~N3§EH14B-FQ9NE(I01 E369 Ñ E(I01:?FQ5(E369 c 1�>AenE(I01N5(@;BGe23 c 1N>AenE(I01g~N34EG1TBe2BG>?LE(I015'@;Be23 c 1�>AenE(I01CE3"KA/21¥Ò�FQ5�>?9;1NI?3T/DeL81XEHBG1 `- I01 E369 FO5BG>"EG3§EH1T:�3"K0>A@?E>?9;1g>AeAE(I01g16:0¢41T5�E(I?3§E�FQ5u>?9E(I01 EG36K?/O1m5*>CE(I?3§E>?9;1Ce23 c 1N>Ae;E(I01gE369 L3 145330◦

369;¢X/O1m~FE(I�E(I01gEG36K?/O1 `�� FO9;:�E(I01N:;14J?E(I>Ae;E(I01~N34EG1TBFO9�E(I01gEG3T9 3TeQEG1TB�E(I01NBG>"E34EGFO>?9 `� ��������� ��!������ � ���� !�

� ��!������� ����� ! ���������!������������! � ����� ������� ��� ������� ����!�������"�#%$ �'& �� ��('�"�)��� �*���+�"� ��� ���¥`

¨T1§ER@;5BG1TJ0BG1T571T9AE�3 c BG>?5'55*1 c EGF2>?9d>AeE(I01c @nK;1KT<E(I0157Ï?@;3TBG1ÎFQ9E(I01:?FQ3"¢§B3TL ` a 9"A@n¢4IªE(I01 c 149AEHBG1P>Ae}E(I01=57Ï?@;3TBG1 `¨T1§E@;5:?B3§~�eq>A@;B30◦

¤60◦

¤90◦

EHBGFQ369;¢X/O1T5uFQ905'FO:;1E(I01C57Ï?@;3TBG1?"365u57I0>6~N9 `� I01T9�KT

Í `- I01m/O1T9;¢¥E(I?5}>Ae?E(I01g5(F2:01T5u>Ae;3mEHBGFQ369;¢X/O1g3TBG11313A3T9;:

10`� I01

��! ����� � �������!� ��� � >Ae6E(IAFO5EHBGFQ369;¢X/O1FQ5E(I01 c FOB c /O1u~FE(I c 149AEHBG1FO905(F2:01}E(I01EHBGFQ369;¢X/O1E(I?34E-FQ5�E369;¢4149AEEH>N143 c I�>Ae"E(I01E(I?BG1T1m5(F2:01T5>AeAE(I01EHBFO3T9;¢X/21 `Ñq 1T1CE(I01�:AFO36¢§B36L ` Ò � FO9;:RE(I01NB36:?FD@;5�>AenE(I01CFO905 c BGFOK;1T: c FQB c /21 `

.................................................................................................................................................................................................................................................................................

.

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

.

.

..

.

..

.

..

..

..

..

..

..

..

..

..

..

..

.........................................................................

..

..

..

..

.

.

..

.

..

.

.

.

..

.

.

..

..

.

..

.......................................................

....................................................

� ��������� ��!������ � ���� !�

� ��!������� ����� ! ���������!������������! � ����� ������� ��� ������� ����!�������"�#%$ �'& �� ��('�"�)��� �*���+�"� ��� ���¥`

>?909;1 c EGFO9;¢}E(I01 c 149AEHBG1g>Ae"E(I01�FQ905 c BF2K016: c FQB c /21EG>CE(I01}?1TBðEGF c 1T5�>Ae"E(I01EHBGFQ369;¢X/21A?>?9;1 c 369�5'J0/DFDE�E(I01gEHBFO3T9;¢X/21gFO9AEG>�E(I?BG1T1gEHBGFQ369;¢X/O1T5§nEð~�>N~FE(I�3NK03T5*1>Ae133T9;:>?9;1C~FE(I3RK03T5*1�>Ae

10` A3 c IM>Ae¹E(I0145*1CE(I?BG1T1�9;1X~$EHBFO3T9;¢X/2145u~FQ/D/tI?3X?1E(I01N573TL81I014FO¢4I"E Ñ ~�IAF c IFQ5mE(I01B36:?FD@;5N>Ae�E(I01�FO905 c BGFOK;1T: c FQB c /21¥Ò `� FO9 c 1�E(I01M¢XF ?1T9EHBFO3T9;¢X/21FO5mFQ5*>?5 c 1§/2145§?~�1 c 36936J0J0/ ?BG14L EH>14FE(I01TB BF2¢4I"EEHBFO3T9;¢X/21>0KAE3TFQ9;16:MKT?9;1 L3 1}E(I01}eq>A/Q/O>6~FO9;¢c >?9;:AFDEGFO>?905}EHB@n10

a(b+ c+ d+ e) = 128

b(a+ c+ d+ e) = 155

c(a+ b+ d+ e) = 203

d(a+ b+ c+ e) = 243

e(a+ b+ c+ d) = 275`

� FQ9;:�E(I01A?1FQ9AEH1T¢41TB5 `����������������������� ��!-`

� FQB5(E 9;>"EG1�E(I?34E 5'FQ9 c 1abcdt3T9;:

e3TBG1RFO9AEG16¢414B5C¢§BG1T3§EH14BE(I?3T9

1 " E(I?3§EFO5Xabcde ≥ 2 nE(I01�5(@;L >Ae�3T9"A@;Bm>Ae-E(I014L FQ5C3§E /O1T3T5(E 8 `N I01T9tt5'FQ9 c 1

b(a + c + d + e) = 155 = 5 · 31 A~�I01TBG1 5 3T9;: 31 36BG1�J0BGFQL81A?~�1�L�@;5(EI?3§?1b = 5

369;:a + c + d + e = 31

;5'FQ9 c 1a + c + d + e ≥ 8 `gÌ 9E(I01�573TL81C~N3XAe;E(I01T571BG1T5(@0/DE5

d + e = 36 − a − b − c = 20 `� FQ9 c 1 275 = 11 · 25 3T9;: 16 + d ≥ 18 ;~�1L@;5HEgI?3X?1e = 11

369;:d = 25 − 16 = 9 ` � >"EG1RE(I?3§EE(I01>"E(I014B e23 c EG>?BGFQ9;¢�>Ae

275 = 5 · 55

�

� ������ � ���m�ù

� � ��� � 34E(I014LM3§EGF c 3T/ 3X?9AEGFO9?@n145§"~FE(I�E(I01N573TL81�14LMJ;I?365(FO5X365369�FQ9AEH1T¢§B3T/nJ;3TBE}>Ae �!XV#"%$'&(*),+#-%&(/.10�2�!¥V3-546.7(8)�$'&(8),+�-%&(1./09&XW:$'&"Y3),+#- ` I01 3X?B�FO5 Ì 369 ��369;:01TB �;@;BG¢4I Ñ7; 9?F?14B5'FEAetÕ�34EG1TB/2>0>6Ò `u I01>"E(I01TBC5HEG3�,©L814L8K01TB5�3TBG1 >?9?F 3 � I0K014FQ5 ÑDa 5 c 14905'FO>?9ª>Ae @;B�¨T>?BG: 1 c >?9;:?36B{0>A/ FO5'5'FQ575'3T@n¢§3XÒ(4?BF c �t>0K;14BE Ñ ¨T16>CÓ3X

�

« ÍgÍ�» `¢¡ � �3£ � � ���)��� �*� �b¤ ����� � & ������¥`Ì enFQ5ª3J;>?5'FEGF?1ÇFO9AEG16¢414B*�E(I01

n�� EHBFO3T9;¢X@0/O3TBP9?@;LMK;14B=FQ5S:014;9;16:Î3T5

Tn = 1 + 2 + · · · + (n − 1) + n = 12n(n + 1)` 1§EG1TBGLFQ9;1=34/Q/uJ03TFQB5�>AeEHBGFQ369;¢X@0/Q36Bu9?@;LMK;14B5u~�I0>?5*1N:AF ,1TBG149 c 1FQ5

2008`

« ÍgÍ Ë `¢¡ � �3£ � � ���)��� �*� �b¤ ����� � & ������¥`Ì 9EHB3TJn110'>AF2:

ABCD45'FO:;1

ABFO5�J036B34/Q/O14/TEG>

DC4369;:N:?FQ3"¢4>?9034/O5

AC3T9;:

BDFQ9AEH14B5*1 c Eu34E

P`

Ñ 3§Ò Ì enE(I01N3TBG1T3�>Ae 4APB FO5 4 369;:�E(I0136BG143R>Ae 4DPC FQ5 9 Ñ F ÒJ0BG>T?1CE(I?3§EAP : PC = 2 : 3

Ñ FDFÒ143"J?/O34FO9N~�I6�>Ae0E(I01C36BG143N>Ae 4BPC EG>NE(I013TBG1T3N>Ae 4BPA1TÏ?@;34/O53 : 2

?3T9;:Ñ FQFDFÒ:01§EG1TBGLFQ9;1CE(I01N3TBG1T3�>AenEHB3TJn110'>AF2:ABCD

`Ñ K6Ò Ì etE(I01�36BG143�>Ae 4APB FO5 x 3T9;:E(I01�36BG143�>Ae 4DPC FQ5 y n:;1XEH14BHL�FO9;1NE(I0136BG143�>AenEHB36J;1"0'>AFO:

ABCDFQ9�EH14BHL5�>Ae

x369;:

y`

` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `« Ígº�¾ `M¡ � �3£ � �®��£ ��� ��� ®� ±�����£ � ���b¤ ���"��� &P`

¨�- 143"J0BG1T5'5'FO>?9a89b

:�¦149;>"EH1m@;9�1T9AEGFO1TB�:;1gÏ?@;3§EHBG1 c IAF ,tBG1T5 Ñ ¦1 c BGFE1T9�K03T5*110Ò:;>?9AEu/O1 c IAF ,tBG1R:01T5 LFD/Q/DF214B5�145(E

a c 1§/Q@0Ft:;145 c 149AEG34FO9;145m145(E

8 c 14/D@0Ft:;145m:AF 0(34FO9;1451T5HE

9¹1§E c 14/D@0F�:01T5�@;9?FDE§¦145C1T5HE

b` N¦1§EG1TBGLFQ9;1TB}EG>A@?EH145m/O1T5mJ03TFQBG1T5C:01 c IAF ,BG145g9;>?99?@0/O5

a1XEbEH1§/O5 ÏA@n1

a89b− 5904 = b98a `« Ígº � ` ¡ � �3£ � �®�A£ ���G¦���! �¢¨�� & ��� �!�§®��� � � ����! ��A© ! �1ª"� ��� � ��®� ��� ���� � � � ��������� � � � ����� �6¬R`

t3§E(I01TBFO9;1u5"- 1T5HE¹3 c I01§E§¦1�@;9;1u9;>A@A?14/D/21}¢4>?LMLM1¡31�,t3 c 1TB7X149meq>?BHLM1}:;1uJ;BFO5'L81BG1 c E369;¢X@0/Q3TFQBG1?A:0>?9AE�/2145u:?FQ3"¢4>?9034/2145}:01T5�e23 c 145uL8145'@;BG149AE10 √

61"1§E √

89` 5}@n14/1T5HE�/21m?>A/D@;L81N:;1C/O3�¢4>?LMLM1�:01�¹34E(I014BGFQ9;1�Ô

« Ígº�� `¡ � �3£ � �®��£ �"� � ��� � �®� � ���"��� ���� �"&������ # £�� � !�� � ��! � ����¯���£ ��±������®��� ����� ��&���� �A`

>?9AEHBG1TBCÏ?@n1M5'F�/.- >?9[57>A@;5(EHB34FDEg:2- @;9S149AEGF214BgJn>?5(FDEGFDe�36BHK?FEHB3TFQBG1?t/O3857>?LMLM1:;1 c I?3 c @;9b:01S5*145 c IAF ,BG145©¦14/O1*�¦1b¡3S@;9;1SJ0@0FQ575'369 c 1ªFOLJ;34FOBG1ÏA@n14/ c >?9;ÏA@n1 Ñ J03659�¦1 c 1T5'5734FOBG14L8149AE�/O3NL �14L81¥Ò(>?9M>0K"EGF2149AEEH>A@ �{>A@;B5}@;9ML�@0/DEGFQJ0/O1�:013`

« Ígº]² `M¡ � �3£ � �®��£ ��� ��� ®� ±�����£ � ���b¤ ���"��� &P`N¦1§EG1TBGLFQ9;1TB�/Q3gA3T/O14@;Bu:;1

cos2 1◦ + cos2 2◦ + cos2 3◦ + · · · + cos2 89◦ + cos2 90◦ `« ÍgÍ�» `M¡ � �3£ � �®��£ ��� ��� ®� ±�����£ � ���b¤ ���"��� &P` >AFDE

n@;9b149AEGF214BJn>?5(FDEGFDe `� 9b:�¦14;9?FDE�/21

n ���� 9;>?LMK0BG1[EHBGFQ369;¢X@0/Q3TFQBG1

Tnc >?LL81g/Q357>?LMLM1Tn = 1 + 2 + · · · + (n− 1) +n = 12n(n+ 1)

` ¦1XEH14BHL�FO9;14BEH>A@?EG1T5}/2145J;34FOBG145�:01N9;>?LMK0BG145}EHBFO3T9;¢X@0/O34FOBG145�:0>?9AE�/O3�:?F ,}¦14BG1T9 c 1�1T5HE2008

`

�

« ÍgÍ Ë `M¡ � �3£ � �®��£ ��� ��� ®� ±�����£ � ���b¤ ���"��� &P` >AFDE�@;9REHB36J�¡1"0'1

ABCD:0>?9AE/O1T5 c��>"E4¦1T5

AB1§EDC

57>?9AE�J036B34/Q/G¡14/O1T5}1§E�/2145:?FQ3"¢4>?9034/2145AC

1§EBD

571 c >A@;Jn149AE149P`

Ñ 3§Ò F;/.- 3TFQBG1�:A@EHBFO3T9;¢X/21APB

145(E41XE c 1§/Q/O1N:?@�EHBGFQ369;¢X/O1

DPC145(E

9

Ñ F ÒLM>?9AEHBG1TBuÏ?@n1AP : PC = 2 : 3

Ñ FDFÒ143"J?/QFOÏ?@n14BmJn>A@;BGÏA@n>AF�/O1B36J0Jn>?BðEC:;1455(@;BGe23 c 145N:;145gEHBGFQ369;¢X/O1T5

BPC1XE

BPA145(EC¦1T¢§3T/�¡3

3 : 201§E

Ñ FQFDFÒ:�¦1§EG1TBGLFQ9;1TB�//- 34FOBG1N:?@�EHB3TJ�¡110'1ABCD

`Ñ K6Ò F§/.- 3TFQBG1}:?@mEHBFO3T9;¢X/21

APB145(E

x1XE c 14/D/21}:A@mEHBGFQ369;¢X/O1

DPC1T5HE

y¥:�¦1XEH14BHL�FO9;14B/.- 3TFQBG1N:?@�EHB36J�¡1"0'1

ABCD1T9�eq>?9 c EGF2>?9:;1

x1§Ey`

< =%> @CB�D �HML������3H��CN

« º � ¾ `I¡ � �3£ � � �"� ��� $ ������� ��� � � ��� � �"�

����� � � �����'� �"� � �A`Ì 9BG1 c EG3T9;¢X/21

ABCD;J;>AFO9AE5

E3T9;:

F:?F ?FO:;1�5(F2:01

DCFO9AEG>�E(I?BG1T1�16ÏA@;3T/J;3TBE5

DE = EF = FC3T9;:ªJn>AFQ9AEG5

G369;:

H:AFAF2:015(F2:01

BCFQ9AEH>=E(I?BG1T116ÏA@;3T/tJ;3TBE5

BG = GH = HC`u I01N/QFQ9;1

AEc @?E5E(I01�/DFO9;145

DG369;:

DH3§EJn>AFQ9AEG5

K3T9;:

L0BG1T5'Jn1 c EGF?1§/< ` FOL�FQ/Q36B/AFO9AE5

M3T9;:

N?BG1T5'Jn1 c EGF?1§/< ` I0>6~�E(I?34E

KN ‖ CD `

���&�����! ��� ����������� ��!C«*� ��&� � ��� ��� # £ � ��� ��� �3£ � ��� � � �k¤ � ����� ���������'������������� ��� ��� �¤ � �� ��� ����� ���� ������ �� � �*��� ��� ������ � �m£ ��!�� ¬ � �� � � ��� ��!� ��� � ����! ���� ��� � � �

����*����� � �%� ��� ������� $ ���vª"� � � ��!� ���'�© � # ����!�� ����������!���� !��"����� ����� ! ��I¦ ����� $ � !������� ��! � ����� � ��� ������� � � !�� � �vª"� ��� $ "!

�t1 c E369;¢X/O1ABCD

FO5J?/O3 c 1T:FO9AEG>·E(I01±¹36BðEH145'FQ369 J?/O3T9;1 5'@ c IE(I?34E8FEG58?14BEGF c 145=36BG1©365�eq>A/D/2>6~N5§A(0, 3b)

B(3a, 3b)

C(3a, 0)

369;:D(0, 0)

`� I01 c >0>?BG:AFO903§EH145>AeE(I01BG14LM34FO9?FQ9;¢Jn>AFQ9AEG53TBG1E(I01T9G(3a, 2b)

H(3a, b)

F (2a, 0)

369;:E(a, 0)

`· I01d1TÏ?@;3§EGF2>?905=>Ae/QFQ9;1T5DH

369;:AF

36BG1�>0K"EG34FO9;1T:365y =

b

3ax3T9;:

y = − 3b2ax+3b

BG1T5'Jn1 c EGF?1§/< ` >A/AFO9;¢E(I01T571?u~�1¢41§EE(I01

y c >0>?BG:?FQ9034EG1�>AebE(I014FQB

-

6.....................................................................................................................................................................................................................................................................................................................................................................................

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

..

.

..

.

..

.

..

.

..

..

.

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

AB

CD E F

G

HK

L

M

N

Jn>AFQ9AEC>Ae�FQ9AEH14B5*1 c EGF2>?9tN365

y =6b

11

` FOL�FQ/Q36B/?905N>Ae�/QFQ9;1T5DG369;:

AE36BG1�eq>A@;9;:MEG>K01

y =2b

3ax3T9;:

y = −3bax + 3b

¹BG1457J;1 c EGF ?14/ < `� IAFO5

�4�

ÓN149 c 1A�5(FO9 c 1dJ;>AFO9AE5K

3T9;:N

36BG1i1TÏ?@0FO:?FQ5(E369AE�e2BG>?L 5(F2:01DC

�/DFO9;15*1T¢§L8149AEG5KN

369;:DC

L�@;5(EuK;1NJ036B34/Q/O14/ `��� ���������������������������������� �!��"$#%��& '��(�)+*-, ) .+/0.1,�23�4, &�.65(78.$�1��� #:9-, ) .+;(��)1&

) @�2BA!�!�C5��D�FE4G1��/:G H0#C���= @�)$-'�� #4I�'�J?K1��Ae¹~N3§

� �

��� ���+��������� �� E4�?9%T=:AF2¢XFEG5g369;:8E(I01�5'@;L >Ae-E(I01:?FO¢XFDE5C3TBG1K;>"E(I�16ÏA@;3T/EH>

16` Ó>6~LM3T9"Ae¹5'@ c I�J03T/DFO9;:?BG>?L8145§T~FDE(I01'3 c /D@n:;1T:�365unB5HE}:AF2¢XFE*AFQ5>0KAE3TFQ9;16:3T5}eq>A/Q/O>6~N54Ì eTE(I01}nB5HE

8:?FO¢XFDE5�>Ae4E(I01J03T/DFO9;:?BG>?L81 c >?9AEG34FO9mEð~�>

25X§eq>A@;B

15X§369;:mEð~�>

05XE(I01T9�E(I01N9?@;LMK;14B>Ae¹36BGB369;¢414L8149AEG5�>AenE(I0145*1

8:AF2¢XFEG5}FO5

6(7!)

2!2!4!= 315

`Ì eE(I01nB5HE

8:?FO¢XFDE5�>Ae�E(I01MJ03T/DFO9;:?BG>?L81 c >?9AE3TFQ9ª>?9;1

4-eq>A@;B

15§�3T9;:=E(I?BG1T1

05XE(I01T9�E(I01N9?@;LMK;14B>Ae¹36BGB369;¢414L8149AEG5FO5

5(7!)

4!3!= 175

` IA@;5XAE(I014BG1N36BG1

315 + 175 = 4905(@ c I�9?@;L8K01TB5 `

� �

��� ��� �����X����[��6G-;(� �C5[�UL%*D�:�5�5Q�!� I�E4G(5��!� �CH:�8G�#D� � �1@���=J8,7 '�2 #D= .0����=.$�J8� @�)0���?AR@�)0�H:�?�$�D��H:E �OLD��G15�5�#4H0@�)07�.0� M1@�����[9$�JN� ��O#����:# �U5��4L4I?903T/9?@;L8K01TBR57>A/Q@?EGFO>?905

(α, β)EH>ªE(I0116ÏA@;34EGFO>?9

α2 + β2 = α3 + β3`

���&?£ � � � � � ����������� ��!{«*� ��& � ��� ��!A¨%��! � ����� ������!'� ���!� � �"� � ��� ��!�� � ����¥'��� � # !� ��!�� ��� ��� � �"& ����!�� "� ��© � # `

I01�¢XF ?1T9M1TÏ?@;3§EGF2>?9�FQ5 16ÏA@0FA34/2149AE�EG>(α+ β)2 − 2αβ = α2 + β2 = α3 + β3 = (α+ β)(α2 − αβ + β2)

= (α+ β)[

(α+ β)2 − 3αβ] `

1§EEGFQ9;¢α + β = p

3T9;:αβ = q

0~�1�>0K"EG34FO9p2 − 2q = p(p2 − 3q) ` >A/ ?FQ9;¢E(IAFO5 16ÏA@;34EGFO>?9�eq>?B

q"EG1 E(I?34E

α3T9;:

β36BG1mBG>0>"E5�>Ae?E(I01gÏA@;3":?B34EGF c 16ÏA@;34EGFO>?9

x2 −px+q = 0 ` I01�:AFO5 c BGFQLFQ90369AE}>Ae;E(IAFO5 16ÏA@;34EGFO>?9�FO5D = p2 − 4q `; 5(FO9;¢ Ñ � Ò(6~�1N>0KAE3TFQ9

D =p2(2 − p)3p− 2

` FO9 c 1C~�1g~N369AE

α3T9;:

βEG>MK01�B3§EGF2>?9034/¹9?@;L8K01TB5XA~�1�L�@;5(E}I?3§?1 √

DB3§EGF2>?9034/ `

¨T1§ED = p2k2

0~�I01TBG1k ∈ � 369;: k2 = 2 − p

3p− 2`N >A/AFO9;¢E(IAFO5C1TÏ?@;3§EGF2>?98eq>?B

p

� �

� ��������� ��!���� ����& � � ���x®��& ���]¤ ��� � ! ����©�! �*ª"� � � � � � � ��� � �®��!��� �®� !�'�m£ ����!"!Õ=1~FQ/D/ c >?905(F2:01TBE(I01 EHBFO3T9;¢X/21~FE(IE(I01 E(IAFQBG:?14BEG1'3gFQ9NE(I01�nB5HE�Ï?@;36:0B3T9AE `Ì e

n = 1nFEFQ5 c /21436BuE(I?3§E}E(I01TBG1R36BG1R9;>8Jn>AFQ9AEG5~FE(IMFO9AEG16¢414B c >0>?BG:?FQ9034EG1T5�FO9ME(I01FO9AEG1TBF2>?B>Ae0E(I01mEHBFO3T9;¢X/21 " FDe n = 2 "FE�FO5u34/O57> c /O1T3TB-E(I?34E�E(I01TBG1mFO5}1'3T3 c EG/?9;1g5'@ c IJn>AFQ9AEFO9E(I01RFQ9AEH14BGFO>?B >AeE(I01NEHBGFQ369;¢X/O1?n9036LM14/ <

(1, 1)` ¨T1XE@;5m9;>6~3T575(@;L81NE(I?3§E

n ≥ 3 ` ~�>>AeTE(I01/DFO9;145t~�IAF c I:;1XEH14BHL�FO9;1uE(I01uEHBGFQ369;¢X/O1 3TBG1y =

(

tan π6

)

x =√

33x369;:

y = n −(

tan π6

)

x = n −√

33x " E(IA@;5§nE(I01RE(IAFQBG:M?1TBðEH1438I?3T5 c >0>?BG:?FQ9034EG1T5

(

√3

2n, 1

2n) ` � >?B 1T3 c I8Jn>?5(FDEGF ?1�FO9AEG16¢414B

i;E(I01R9?@;L8K01TB √

3iFO5m9;>"E 3T9FQ9AEH1T¢41TB "I01T9 c 1?XE(I014BG136BG1 b√3ic /Q34EEGF c 1}J;>AFO9AE5-FQ905'FO:;1�E(I01uEHBFO3T9;¢X/21}369;:C>?9mE(I01}/QFQ9;1 y = i `�?

� £

E(I01¨43§~Î>Ae>?5(FO9;145 >?9∠OBC

EH>�5(@ c(c 1T5'5'F ?14/ 0KAE3TFQ9

r2 = s2 + s2 − 2s2 cos 150◦ = s2(

2 + 2 ·√

3

2

) (

r

s

)2

= 2 +√

3

s

r=

1√

2 +√

3=

√

2 −√

3`

��� ���������X���� �� ���4� ���X��� ��� �!��"$# ��& '��(�)�*-, ) . /U.(,X�2 �4, &�. 5(78.$�1��� # 9-, ) . ;1��)(&�#9-,�=)1@�2BA�H:�?�$�D��H:E �OLC��G15�5�#4H0@�)07�.0� M1@

�

Ì en = 1

A~FQ905 KT?BG18FQ9AEH14BG1T5HEGFO9;¢ `Ì e

ABG1TLM>T?1451BBG1 c 1§F?145�3J0FD/218>Ae

43T9;:=~FQ905RKT

A:;>0145792- Eu~N3T9AEuEH>MBG14L8>4?11`CÌ e

ABG1TLM>T?145

4BBG1 c 14F ?1T5�3�J?FQ/O1�>Ae

1n369;:�~FO905KT

A:;>0145792- EN~N3T9AE�EG>BG1TLM>T?1

41§FDE(I014B `ÎÌ e

ABG14L8>4?1T5

3BBG1 c 14F ?1T5�3J0FD/21>Ae

2` ÓN14BG1?

Bc 3T9S>?9?/T?1

1Ñ 9;>"E

3>?B

4Ò(369;:

AE(I01T9BG1 c 14F ?1T53�J?FQ/O1N>Ae

1?3T9;:5*>N~FO905 KT4?FQ9;¢

1`

Õ=I?34E:;>0145E(IAFO5 c 36571uEH1§/Q/T@;57ÔÕ=I0>I?3T5E(I01u~FQ909?FO9;¢}5HEHB34EG16¢*0>?5*145

1>?B

465*I01�/O>?5*145§6KA@?E�FQe05*I01 c I0>0>?571T5

3"57I01~FO905 `- IA@;5§

AI?3T5�3 ~FO909?FQ9;¢�5(EHB3§E(16¢*?9AEHBG>A/Q5�I014B�>6~N9e23§EH1�KT< c I0>0>?5'FQ9;¢}nB5HE Ñ 3T9;:�I0>?Jn1§eO@0/Q/ < c I0>0>?5'FQ9;¢

3Ò `Õ=I01T9

n = 5A- 5u~FO909?FQ9;¢5HEHB34EG16¢*�BG1TLM>T?1

3?/O1T3X?FQ9;¢

B~FE(I

2`Õ=1C57I0>6~�16:�KT0> FO9;¢m3§E-~�I?34E

B3T9;:

Ac 369�BG14L8>4?1mE(I?34E

AL@;5HE�~FQ9 ` �;@?EFQ5E(I01TBG1�3C~N3§Aet/2>0> FO9;¢C34E�E(IAFQ5}E(I?34E}LFO¢4I"EK;1R14365(F214BEH>M¢4149;1TB34/QF 0'16Ô IAFQ5 c >A@0/2:K;1NBG143T/D/T?1E(I?34E

A/2>?571T5~�I01T9 c I0>0>?5(FO9;¢}e2BG>?L3mJ?FQ/O1m>Ae05(F 0'1

2` IA@;5X0FDe

ABG14L8>4?1T5

1AE(I01T9

Bc 369eq>?B c 1

AEH>�/2>?571 `Ì e

ABG1TLM>T?145

3/2143§AFO9;¢

B~FDE(I3;E(I01T9

Bc 369�BG14L8>4?1

3369;:~FO9 `g I?3§E FQ5§nFDe

ABG1TLM>T?145

3;E(I01T9

Bc 369eq>?B c 1

AEG>N/2>?571 `Ì e

ABG14L8>4?1T5

4/2143§AFO9;¢

B~FE(I

2TE(I01T9

B~FQ/D/A/2>?571m3T5E(I01�nB5HEJn14B5*>?9 c I0>0>?5'FQ9;¢ge2BG>?L 3�J?FQ/O1N>Aet5'F 0'1

2~FD/Q/0/2>?571 `

Õ�3TFE � I?3§E - 5�E(I01 1* 16:834EuE(I01T571 c 36571T5 `m I01R;B5(E J?/O3X

�

E(I01MJ;BG>0KA/214L=E(I01T9�EHB{AFQ9;¢57>?L81�/Q36BG¢414B c 3T5*145§-5'@ c I=365n = 8

EG>n = 12

` I0149MBG143":M>?9 `� ��������� ��!

�� BG>?L36K;>4?1?

AI?3T5u3�~FQ909?FO9;¢�5(EHB3§EH1T¢*01T59;>"E}I?3§?1N3g~FQ909?FO9;¢C5(EHB3§EH1T¢*?Bn = 7

` EG3TBEGFQ9;¢�~FDE(IC3 J0FD/21 >Ae?5(F 0'1nAL�@;5(EBG1T:?@ c 1E(I01 J?FQ/O1}EG>N>?9;1 >AeA5'F 0'1

n−1 n− 3 0>?B n− 4 3T9;:�J;3T575}FE�EH> B `uÌ e;E(I01CnB5HEuJ;1TB57>?9�EG> c I0>0>?5*1 Ñ 9;>6~ B Ò�I?3653~FO909?FQ9;¢5(EHB3§EH1T¢*Ae1T3 c IM>AenE(I0145*1N5(F 0'145§6E(I01T9

A~FQ/D/;/2>?571 `Ì 98>"E(I01TB�~�>?BG:?5§AFQe

AI?3T5 3C~FQ909?FO9;¢5HEHB34EG16¢*Aet5'F 0'1

n− 1 n− 3 ?369;: n− 4 AE(I0149 A ~FD/Q/;/O>?5*15HEG3TBEGFQ9;¢�~FDE(I�3NJ0FD/21N>Aet5(F 0'1 n ?5(FO9 c 1 B c 369FOLJ0/O1TLM1T9AE

A- 5�5HEHB34EG16¢*?BtE(I01m5'LM34/Q/O1TBJ0FD/21�3T9;:C~FQ9t69;>NL34EEG1TB¹~�I?3§E

A:;>0145 `Ì e¹>?9;1C>?B�L8>?BG1C>Ae?E(I0145*1gJ?FQ/O1g5(F 0'145�36BG1g5(@ c INE(I?3§E�E(I01m;B5(E�J;1TB57>?9�:0>01T5�9;>"EI?3§?1g3 ~FO909?FQ9;¢�5(EHB3§EH1T¢*A@0/O:RBG1T:?@ c 1mEG>E(IAFQ5u5(F 0'1A§~�IAF c I�J0BG1*?1T9AE5

Be2BG>?L K;1§FO9;¢C36K?/O1CEG>�~FQ9 `� IA@;5§AI014B5*1§/Qe;~FD/Q/?~FQ9 `

� BG>?Ln = 8

Ac 3T9MBG1T:?@ c 1EG>

750>?B

4`� FO9 c 1CE(I01;B5(EuJ?/O3X

� �

����� � �b � � ��

� � ����� ���� �����

�

�������� �R���R�0���ÓN>6~ùE(I01EGFOLM1571614LM5EG>��?

���

º `[Ñ 3XÒN.;BG>T?1E(I?3§ECFQ9ª3MEHBFO3T9;¢X/21E(I0185'@;L >AeE(I01:?FQ5(E369 c 1T5Ne2BG>?L E(I01 c 1T9AEHBG>AFO:EH>E(I01�E(I?BG1T185(F2:01T5CFQ5�¢§BG1434EG1TB�E(I?3T9ª>?BC1TÏ?@;34/�EH>E(I?BG1T1�EGFOLM1T5gE(I01MB36:?FD@;5�>AeE(I01FO9 c FQB c /21 ` 1§EG1TBGLFQ9;1g~�I01T9M143T3 c E}1TÏ?@;34/QFEA/2:?5 `Ñ K6Ò}1§EG1TBGLFQ9;1gE(I01CJn>AFQ9AEG5uFQ9�3gEHBGFQ369;¢X/O15(@ c I�E(I?34E�E(I01C5'@;L >Ae;E(I01:?FQ5(E369 c 1T5e2BG>?LE(I01N5(F2:01T5}FQ5LFQ9?FOL3T/ `Í `�� >?B�3NJn>?5(FDEGF ?1CFO9AEG16¢414B

nA/21XE

ψ(n) =n∑

k=1

gcd(k, n)`

Ñ 3§Ò�.;BG>T?1gE(I?3§Eψ(mn) = ψ(m)ψ(n)

eq>?Bm369;:

nBG14/Q34EGF ?14/ ?5'FEGF?1FO9AEG16¢414B5 `

Ñ K6Ò=.;BG>T?1 E(I?34E*Teq>?B1*?1TB{?5'FEGF?1�FQ9AEH1T¢41TBaTE(I01C1TÏ?@;3§EGF2>?9

ψ(x) = axI?3T5u3§E/214365HEu>?9;1N57>A/Q@?EGFO>?9 `

� BG>?L ÌGi I0>?BðEg¨§FQ5(E �66 � ` Ð ` ¨T1§E

Sn = {12. . .

n} ` ¨T1XE f : S1600 → S1600 K;1N5(@ c IRE(I?34E

f (2005)(x) = xeq>?B

x = 12. . .

1600

3T9;:f(1) = 1

` Ñ � ÒÑ 3§Ò�.;BG>T?1gE(I?3§E

fI?3T5 3§E�/O1T3T5(E�369;>"E(I014B�&3616:�Jn>AFQ9AE `

Ñ K6Ò=1§EG1TBGLFQ9;1RE(I0>?5*1n > 1600

5(@ c I8E(I?34E�369"<f : Sn → Sn

5'34EGFQ5'eDR�361T:�Jn>AFQ9AEG5 `¼ `ta[c FOB c /O1

γ3T9;:N3�/QFQ9;1

`I?3X?1m9;>Jn>AFQ9AEG5�FQ9 c >?LMLM>?9 ` ¨T1XE

ABK;1E(I01m:?FQ36LM1§EG1TB>Ae

γJn14BHJ;1T9;:AF c @0/Q36B�EG>

`n~FDE(I

Bc /O>?5*14BEH>

`E(I?3T9

A` ¨T1XE

CK013MJn>AFQ9AEC>?9

γ:?F ,-14BG1T9AENe2BG>?LA3T9;:

B`Ç I01�/DFO9;1

ACFQ9AEH14B5*1 c EG5

`34ED`Ç I01/QFQ9;1

DEFQ5EG3T9;¢41T9AEEG>

γ34EE�~FDE(I

B3T9;:

E>?9[E(I01�5'36LM1�5'FO:;1�>Ae

AC` I01/DFO9;1

BEFO9AEG1TB571 c E5`34EF03T9;:

GFQ5}E(I01�>"E(I01TB�FQ9AEH14B5*1 c EGF2>?9M>Ae¹E(I01/QFQ9;1

AF~FDE(I

γ` ¨T1XE

HK01�5

G~FE(I8BG1T5'Jn1 c E}EH>

AB` .;BG>4?1�E(I?34E

FC¹3T9;:

H3TBG1�>?9E(I01N573TL81C/DFO9;1 ` � BG>?L ÌG I0>?BðEm¨§FO5HE �T" � `

¾ ` ¨T1XENK01�3�J;>?5'FEGF?1FQ9AEH1T¢41TB `�a /2K01TBðEH>�369;: �n36BHK03TB3C~NBGFEH1N3�9?@;L8K01TB>?9�E(I01K?/Q3 c' K;>?3TBG:8EG3 FQ9;¢�EG@;BG905§t3 cHc >?BG:AFO9;¢NEH>ME(I01Req>A/Q/O>6~FO9;¢�BG@0/O1T5§ a /2K01TBðEH>5HEG3TBE5CKT?9�E(I01RKA/O3 c' K0>?36BG: " 5'@nK?5*1TÏ?@n149AEG/

�6

� 143TE*t~�18¢XF ?1�E(I01 �c� � � � >?BHL·>AeE(I01 � FO9034/ �t>A@;9;:ª>Ae�E(I01 (F( ( Ì �¹@;575(FO3T9 34E(I014LM3§EGF c 3T/ /

� �

� `�� � ! $ � � � �vª! m 1T3§EH1T:=34Em3BG>A@;9;:EG36K?/O13TBG1100

BG1TJ0BG1T571T9AE34EGF ?1T5N>Ae25

c >A@;9;EHBGFO1T5 Ñ eq>A@;BgJ;1TB57>?905e2BG>?L·1T3 c I c >A@;9AEHB{Ae �(@0/ < �66A£0`a ¢§3TFQ9RE(I?3T9 5¢4>NEG> � 1§/QF 3$�t1 c FO>eq>?B c >A/Q/O1 c EGFQ9;¢mE(I014L eq>?B�@;5 `+ -�*�� -�3 « -�+�� 5 « -�+6* ¶ -0/ 9�/ «�� * -�

¿nÀ � ÀtÁ��6À � DÈ � Æ � À¹Å G�8�?y ��� � (*)*)�+

Ë `a 4ABC FQ5�¢XF ?1T9~FDE(I=5'FO:;1�/2149;¢¥E(I?5 a b �369;: c `a Jn>AFQ9AE P /QFO1T5CFQ905'FO:;14ABC 4369;:gE(I01�:AFO5HEG3T9 c 145�e2BG>?L P EH>mE(I?BG161 5(F2:01T5�3TBG1 p q T3T9;: r 4BG1T5'Jn1 c EGF?1§/< `.;BG>T?1gE(I?34E

R ≤ a2 + b2 + c2

18 3√pqr

~�I01TBG1

RFO5uE(I01 c FOB c @;LB3":AFQ@;5�>Ae 4ABC ` Õ=I01T9M:;>0145 16ÏA@;3T/DFDE{A/O:;Ôº ` @;J0Jn>?571E(I?34E

GFO5 3�¢§B3TJnI8>Ae->?BG:01TB

n~�IAF c I8:0>01T5 9;>"E c >?9AE3TFQ983 c >?LMJ?/21XEH1¢§B36J;IM>Ae>?BG:01TB

k3653N5'@nK0¢§B36J;I ` .;BG>T?1gE(I?34E

Gc >?9AE3TFQ905 3§EuLM>?5(E

k − 2k − 1

· n2 − r2

2+

(

r

2

)

16:0¢41T5X6~�I01TBG1n ≡ r (mod (k − 1)) 3T9;: 0 ≤ r ≤ k − 2 `Í `� I01 ÌG ~FD/Q/AE3 1CJ0/Q3 c 1>?9RE(I01 � � � � 3T9;: � � � � >Ae��(@0/ < �66A£0` .;BG>4?1mE(I?34E�E(I015'@;L∑

1≤i

j

k≤3

i6=j 6=k

csc13(

2iπ

7

)

csc14(

2jπ

7

)

csc2005(

2kπ

7

)

FO53NB34EGFO>?903T/n9?@;LMK;14B `Ð ` ¨T1XE

a mod b:01T9;>"EG1

a − b⌊

a

b

⌋

" E(I?34EuFO5X?E(I01�BG14LM34FO9;:01TB~�I01T9 a FO5�:?F ?FO:;1T:KT<b` � FO9;:�3T/D/n5*>A/D@?EGF2>?905uFO9J;>?5'FEGF?1CFQ9AEH1T¢41TB5

(x, y, z)EG>RE(I01�1TÏ?@;3§EGF2>?905

xy mod z = yz mod x = zx mod y = 2`

¼ ` � F ?1T9 4ABC u3 c FOB c /O1 Γ ~FE(I c 149AEHBG1 O J;3T57571T5NE(I?BG>A@n¢4I B 3T9;: C 3T9;:FO9AEG1TB571 c E5m5'FO:;145AC

3T9;:AB

34EJn>AFQ9AEG5D369;:

EnBG1457J;1 c EGF ?14/ < ` ¨T1§E

FK01�E(I01FO9AEG1TB571 c EGFO>?9b>Ae

BD3T9;:

CE` I01ª/QFQ9;1

OFFO9AEG1TB571 c E58E(I01 c FQB c @;L c FQB c /21>Ae

4ABC 3§E P ` .;BG>4?1�E(I?34E�E(I01mFO9 c 1T9AEHBG1>Ae 4PBD c >AFO9 c FO:;145�~FDE(IE(I01gFQ9 c 149AEHBG1>Ae 4PCE `

�"�

¾ ` � FQ9;:3T/D/nJn>?5(FDEGF ?1FQ9AEH1T¢41TB5n ≥ 3 ~FDE(IRE(I01eq>A/D/2>6~FQ9;¢CJ;BG>?J;1TBðE?5(FDEGF ?1©FQ9AEH1T¢41TB

Mn5'@ c IiE(I?34E*Neq>?B8369"?5(FDEGF ?1ÇBG143T/9?@;L8K01TB5

a1a2. . .

an6E(I01eq>A/Q/O>6~FO9;¢C1TÏ?@;34/QFEA/2:?54

a1 + a2 + · · · + ann√a1a2 · · · an

≤ Mn(

a2

a1+a3

a2+ · · · + an

an−1+a1

an

) `

Õ=18I?3§?18I?36:[57>?L81BG1T36:;14B5 c I?3T/D/2149;¢41�E(I01�A3T/D@n147

365CE(I01/O1T3T5(EmJn>?5'5'FOK?/O1R3T905H~�14B�EH>J;BG>0KA/214L �8>Ae¹E(I01Rt3§EH1T¢4>?B{< � �¹14/Q36B@;5 34E(I014LM3§EGF c 3T/ / ?5'FEGF?1NFO9AEG16¢414B5

abcdefgh369;:

n5'34EGFQ5'eD?575(F2KA/21g571§E�>Ae

ab. . .

hÑ 5'LM34/Q/O1T5HE�FQ9RE(I01g5*14905*1>Aeu/21436F c >0¢§B3TJnIAF c >?BG:;14BGFQ9;¢8>Ae}Jn>?5'5'FOK?/O1857>?BEG16:

ab. . .

h` Ó1�¢4149;1TB3§EH1T:ª3T/D/

n = ab + cdeq>?B

a = 4EH>

50;~FE(I

bcd5'LM34/Q/O1TBuE(I?369

a369;:�:?FQ5(EGFQ9 c E*¹369;:/2>0> 1T:Peq>?BC3T9

n~FE(I=Eð~�>=BG1TJ0BG1T571T9AE34EGFO>?905N@;5'FQ9;¢

8:AFO5HEGFO9 c E1§/214L8149AEG5 `P FO9 c 1

50 > 31369;:

n > a�E(I014BG1 c 3T909;>"E�K;1�3T9"A/Q@?EGFO>?9S5'LM34/Q/O1TBmE(I?369

31` I01c I01 c' FO9;¢ FO9"?>A/ ?16:�36K;>A@?E

700000

c 36571T5X4E3 FO9;¢�3gL�FO9?@?EG1C>?BEð~�>�>?9R3gJ;1TB57>?903T/c >?LJ0@?EG1TB `¬ ��� � �������1ª"� �*� � � ��� � ��� ��! ����¡ � � � &)��� ����� ¤ � ��������� ��! � ����� � �� � � ����� � ��� ��������'� ���� � �¡ ������!�� !

I01N3T905H~�14BFO531 = 8 · 2 + 3 · 5 = 6 · 4 + 1 · 7 ` >�J0BG>T?1NL�FO9?FQLM34/QFE6~�E(I?3§E¥

Ñ FÒ I01LFQ9?FOL�@;LùJn>?5'5'FOK?/O1mA34/Q@n1N>AeS = ab+ cd+ ef + gh

FO560`

Ñ FDFÒ Ì 9E(I01 c 36571C~�I01TBG1S = 60

?E(I01EH14BHL5 FO9�E(I01�5'@;L 3TBG181418;369;:

20` I014BG14eq>?BG1ATE(I01TBG1mFQ5�9;>C~N3§NBG1T3TBHB3T9;¢41mE(I014LFO9AEG>�J;34FOB5u¢XF ?FQ9;¢ E(I01m573TL815'@;L5 `

>J;BG>4?1ªK0>"E(I c /Q3TFQLM5X}/21XE�@;5365'5'@;LM1�E(I?34Eabcdefg 3T9;:

hLFQ9?FOL�F 0'1mE(I015'@;LS`�Ì et3T9"Ae;E(I01N14FO¢4I"Eu9?@;LMK;14B5 �©/O1§E�@;5 c 34/Q/0FDE

x�©FO5¢§BG1T3§EH14BE(I?369

8tE(I01T9[>?9;1M>Ae�E(I019?@;LMK;14B5e2BG>?L {1 2 3 4 5 6 7 8} L@;5HEgBG14LM34FO9@;9?@;5*1T: �©/O1§E�@;5 c 3T/D/;FDE

y` �t14J0/Q3 c FQ9;¢

x~FDE(I

y:01 c BG1436571T5

S?3 c >?9AEHB36:?F c EGF2>?9 `

�*�

IAFQ5 L81436905}E(I?3§E {a b c d e f g h} = {1 2 3 4 5 6 7 8} ~�IAF c I36BG1C¢§BG>A@;J;16:�FO9AEG>�eq>A@;B�J03TFQB5E(I?3§E�¢XF ?1�E(I01�EH14BHL5 `ta 5'5'@;LM1�E(I?3§E8FQ5uJ03TFQBG16:~FE(I5*>?LM1§E(IAFQ9;¢R>"E(I01TBuE(I?3T9

1n573X<

8FO5mJ03TFQBG16:�~FDE(I

w`� 3§EG@;B3T/D/

� �

� ���1ª"��� ��� ¤ � ����� � $ ������� � � � ����� � ��!�� ��� ��! � ���§¡�� � � � � $ ��� ! � � � � ��!� ¤ ��� ����! � �¦���� � � � � ��� � !������ ����� !�! � � ����� � � ��� � # �*��� !�� � �'����������� # ! � � � �¤ ��! �"� ��� ��� ����� ! ��©�! �*ª"� � � ���� ��«"� ����! � ��� � ����! � ��� � ������ �"� �"�������g¬ !¤ �"� �*���¨ �vª"� ���,¡ # �© � # ��¨ ��! � � �� � ����� ��� � ����! �� � ����£ � � # & � � � ��� ! � ��� ������� � ����£ � ����� ��� � � ! � � ! ��� ��� �M¯ª���! �"� �"�

���&®� ! �

�

������� ����& ��! ��� !�� ��( ���������*ª"� $ ������� � � � � � � ��� � ��� ��!"!Ì e-~�1R/21XE

FK01RE(I01�36BG143M>Ae 4ABC 3T9;:�/O1§E ha K;1RE(I01�34/DEGFEG@n:;1Re2BG>?L A E(I01NJ;BG>?J;>?5*1T:�FQ9;16ÏA@;3T/DFDE{A/Q/O>6~N5�e2BG>?LE(I01>0KT?FO>A@;5�FO9;1TÏ?@;34/QFE<

ha ≤ ma`

¬ ��� ���������*ª"� �*���bª"� � � � ��! ��« $ ��� ! � � � � ��! ! >AFDE

S/.- 3TFQBG1�:A@EHBFO3T9;¢X/21 `¹Ì /¹1T5HE}K?FO1T9 c >?909?@ÏA@n1

S = 12(a+b+c)r = 1

4

√

(a+ b+ c)(a+ b− c)(a− b+ c)(−a+ b+ c) `�- 34@?EHBG1NJ;3TBE*0:�- 3TJ;B§¡1T5}/O3/O>AF¹:;1455'FQ9?@;5§?>?93 sin(A)

a=

1

2R

` ¨�- FO9�¦1T¢§3T/DFDE§¦1N:�¦1T5(FOB§¦1611T5HE}:;>?9 c ¦1TÏ?@0F A3T/O1T9AEG1¡3

a4 ≥ (a+ b+ c)(a+ b− c)(a− b+ c)(−a+ b+ c) ` Ñ � Ò B* c >?LMLM1∠A ≥ 90◦ 0>?93 a2 = b2 + c2 − 2bc cos(A) ≥ b2 + c2 ` .A36B/O3N5'@0FEH1A0:�- 3TJ;B§¡1T5//- FQ9�¦16¢§34/QFE4¦136BFDE(I?Lª¦1§EGF c > ¤ ¢;¦16>?Lª¦1§EHBF2ÏA@n1?

(b+ c)2 = b2 + c2 + 2bc ≤ 2(b2 + c2) ≤ 2a2 `"Eu34FO905(F(a+ b+ c)(a+ b− c)(a− b+ c)(−a+ b+ c)

= (a2 − (b− c)2)((b+ c)2 − a2) ≤ a2 · a2c 1mÏ?@0FAJ;BG>A@A?1mÏ?@n1 Ñ � Ò1T5HE¹0B34F21 `" 9�J;14@?E�9;>"EG1TB�ÏA@7- FD/6

�6£

� ���1ª"��� ��� ¤ � ����� � $ ������� � � � ����� � ��!�� ��� ��! � ���§¡�� � � � � $ ��� ! � � � � ��!� ¤ ��� ����! � �¦���� � � � � ��� ��! � ������� �®��¦���� �¨k®� � � � $ ��� ��� � ���©�! �*ª"� � � ��������¡ ��� � ���� � ! � ������� �� ����� ��! �����$ ��� ��� � ��!��"� �m£ ����! � � ����¥'��� � # ! ��� ! � ��� �!� � �"& ����!�� "� �© � # � � ����! ! ��� ����� � � ��� �# �*��� !�� � �'� �"� � ��� � �"������� ¬ !]¤ ��� ��"�b¨ ��ª"� ���b¡ # �© � # �g¨ ��! � � � � � ���!� ��� ����� ! ��� ���a£ � � # & � � � ��� ! � ���������!� � ����£ � ��� � ��� � � ! � ��!�� � ��� �¯vª���!���� ��� ���&®� ! �

�

�����������& ��! ��� ! � �%( ���������*ª"� �*� � � ��� � ��� ��! ����� � ��� !

Õ=1RJ;BG>4?1�E(I01R5'/DF2¢4I"EG/ ?9;¢414B BG145'@0/E}E(I?3§Ex3 + 2px2 + 2p2x + p = 0c 3T909;>"EI?3§?1[34ER/214365HEREð~�>Ç:AFO5HEGFO9 c E�BG1T34/�BG>0>"EG5X eq>?B�369"N:?FQ5(EGFQ9 c E>?9;1T5 `� I0149

x3 + 2px2 + 2p2x+ p

= x3 − (α+ β + γ)x2 + (αβ + βγ + γα)x− αβγeq>?Bu3T9"01�� c FO1T9AE5�¢XF ?1T5

2p = −(α+ β + γ) Ñ � Ò2p2 = αβ + βγ + γα

Ñ{� Ò369;:

p = −αβγ ` Ñ/� Ò Ï?@;3TBGFQ9;¢ Ñ � Ò}

�6

� 143TE�~�1EG@;BG9EH>�E(I01Óg@;9;¢§36BFO3T9 � 34EGFO>?903T/ /

�"

AM2 = CA2 + CM2 − 2CA · CM · cos γ1

>?Bc2 = 4b2 + 4m2c − 8bmc cos γ1

` FOL�FQ/Q36B/?L~�IAF c I�~�1NI?3X?1

cos γ1cos γ2

=a(a2 + 3b2 − c2)b(3a2 + b2 − c2)

`¨T1§EE(I01R/QFQ9;1RE(I?BG>A@n¢4I

MJ;1TBGJn149;:?F c @0/O3TB EG>

CMFQ9AEH14B5*1 c E

BC3§ED3T9;:

AC34EE`a J;J?/Ae FQ9;1T5uEH> 4CDE T~�1N>0KAE3TFQ9

CD

CE=

sin ∠CED

sin ∠CDE=

cos γ1cos γ2

=a(a2 + 3b2 − c2)b(3a2 + b2 − c2)

` Ñ{� Ò�?< Ñ � Ò�3T9;: Ñ{� Ò(6~�1NI?3§?1

CX

CY=

a(a2 + 3b2 − c2)b(3a2 + b2 − c2) =

CD

CE

` FO9 c 1 4CXY 3T9;: 4CDE 3T/Q5*>RI?3§?1mE(I01C369;¢X/O1C3§E C FO9 c >?LL8>?9t4~�1C5*1T1mE(I?3§EE(I01*

���

~�I01TBG1�K0>"E(I�5'@;LLM3§EGF2>?905}FQ9�E(I01/Q365HE�/DFO9;136BG1Ce2BG>?Lk = 0

EH>k = n

`; 5'FQ9;¢CE(I01N5(E369;:?36BG:�~�14/D/2 9;>6~N9L81XE(I0>0:t"~�1R:AF ,1TBG149AEGFO3§EH1E(I01�K?FQ9;>?LFQ3T/1'36J;3T905'FO>?9

(1 + x)n =

n∑

k=0

(

n

k

)

xk

EH>>0K"EG34FO9n(1 + x)n−1 =

n∑

k=1

k

(

n

k

)

xk−1`

@0/DEGFQJ0/

� �

Í ` >?905(F2:01TB�E(I01gE(I?BG161N:?FQ5 �{>AFQ9AEu3TB c 5 >Ae¹3 c FOB c /O1N:;1XEH14BHL�FO9;1T:MKTAFO9AE5 >?9E(I01 c FOB c /O1 ` � >?B�1T3 c I�>Ae-E(I0145*1�36B c 5§t:0B3X~ 3 c FQB c /21 c 149AEHBG16:�3§EE(I01�LFO:;(Jn>AFQ9AEm>AeE(I01R36B c 3T9;:8J;3T575(FO9;¢CE(I?BG>A@n¢4IE(I01�149;:;(Jn>AFQ9AEG5m>AetE(I01R36B c§` .;BG>T?1E(I?34EuE(I01NE(I?BG1T1c FOB c /O1T5 I?3X?1N3 c >?LMLM>?9Jn>AFQ9AE `� ��������� ��!����¨ ! ��! ��& �"��! ��¯���� � ����& & � �����*���b¬ � �*��� � ����!���� !

I01�eq>A/Q/O>6~FO9;¢NE(I016>?BG14L±FQ5�~�14/D/2 9;>6~N9¨T1§EIK;1�E(I01FO9 c 1T9AEHBG18>Ae 4ABC `M I01T9�E(I01/QFQ9;1T5AI

BI

3T9;:CI

FO9AEG1TB571 c EE(I01 c FOB c @;LMc FOB c /O1Req>?BE(I01�571 c >?9;:8EGFQL81RFQ98E(I01�LFO:;(Jn>AFQ9AEG5A1B13T9;:

C1>AeE(I01=36B c 5

BCCA

3T9;:AB

?BG1T5'Jn1 c EGF?1§/< ` � @;BE(I014BHLM>?BG1?A1B = A1I = A1C

B1C = B1I = B1A

369;:

C1A = C1I = C1B`

I01 c >?LMLM>?9Jn>AFQ9AE�>AeE(I01�E(I?BG161 c FOB c /O1T5FO5uE(IA@;5}E(I01CFO9 c 1T9AEHBG1�>Ae 4ABC `

..

..

.

..

.

..

..

.

..

..

.

..

..

..

..

..

..

..

.

..

..

..

..

..

.

..

.

..

.

..

.

..

.

..

.

..

.

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...................................................................................................................

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................

.

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

..

.

..

..

.

..

..

.

..

..

.

..

..

.

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................

.................................

.................................

.................................

.................................

.............................................................................................................................................................

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

..

.

..

..

..

..

..

..

..

..

..

..

..

..

..

..

...................................................................................................................................................

..

.................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................................................

A

A1

B

B1

C

C1I

Ð `�a J;34/O3 c 1�~�IAF c IPI?365N3M5*ÏA@;36BG1M57I?36J;1FO5N:?F ?FO:;1T:=FO9AEG>2003 × 2003 57Ï?@;3TBG1BG>0>?LM5�>AeuE(I01573TL815'F 0'1~�IAF c I[eq>?BHL 3=57Ï?@;3TBG1=¢§BF2: ` I01TBG1�L�F2¢4I"E�K01�3=:;>0>?BK;1XEð~�16149ªEð~�>SBG>0>?L5RFQeuE(I01*?LL8>?95(F2:01 `© I01�LM34FO9¢§34EG1�/O1T36:05�EG>E(I01NBG>0>?Lù34EE(I01N9;>?BðE(I6~�1T5HE c >?BH9;14B `� >?L81T>?9;1�I?365 1T9AEG1TBG1T:�E(I01NJ03T/Q3 c 1?6~N3T/ 16:36BG>A@;9;:Meq>?B3~�IAFQ/O1R3T9;:M@;J;>?9BG1§EG@;BG9?FO9;¢EG>�E(I01RBG>0>?L 34EuE(I01�9;>?BE(I6~�145(E c >?BH9;14Beq>?B�E(I01gnB5HE�EGFQL81A"FOLL81T:?FQ34EG14/ A@?E�E(I?3§E�E(IAFQ5}Jn14B5*>?9?FQ5'FEH1T:1T3 c I>AenE(I01N>"E(I01TB�BG>0>?L5

100EGFOLM1T5X0143 c 14J?EE(I01BG>0>?L 34E�E(I01N57>A@?E(I01T3T5(Ec >?BG9;1TB ` Ó>6~±L369"?9[?FQ5'FEE(I01=BG>0>?Lê34EE(I01=57>A@?E(I01T3T5(Ec >?BG9;1TBHÔ

� ��������� ��!����¡�� � � ��� $ ��� !�� � � � ��!��¤ ����� ��!�� �U¦���� � � � ����� � !���� !¨43�J?FG¡1 c 13¦1XE4¦1m?FQ5'FE4¦161

99eq>AFQ5 `.0/Q@;5�¢;¦1T9�¦14B3T/O1TLM1T9AE*�>?9 c >?905(F2:�¡1TBG1@;9ªJ;34/O3 c 18:;1

n × n J?FG¡1 c 145§ c I?3 c @;9;1:2- 1T9AEHBG1g14/D/21453§A@n1T5HE�Ï?@0F?3�¦1§E§¦1}?FQ5'FE4¦161�1'3T3 c EH14L8149AE:014@&3meq>AFO5X61XE c 1§/Q/O1�5'FEG@¦161�3T@ c >AFO9�5(@n:;1T5HE*0Ï?@0Fn3¦1XE4¦1gAFO5(FDE§¦1T1N143T3 c EG1TLM1T9AE

xeq>AFQ5 `0 9RA3�J0BG>A@A?1TB}ÏA@7- 34/2>?B5

x = p− 1 5(Fn145(EFOLJ;34FOB701§E

x = 2p− 1 5'F n 145(EuJ03TFQB ` 9 c >?LMLM1T9 c 1�J;3TB c >A/O>?BGFO1TB�/2145�J?FG¡1 c 145�34/DEG1TBG9034EGF ?1TLM1T9AE1T9M9;>AFOB1XE KA/O3T9 c c >?LL81R5'@;B@;9©¦1 c IAFOÏ?@0FO1TB7t:01�57>?BEG1�ÏA@n1R/O1T5g:;1§@&3 c 36571T5m5(FDEG@�¦1T1T5m34@ c >AFO905m9;>?BG:0>A@n1T5HE 1§E5(@n:;'1T5HE5*>AFO1T9AEuEG>A@?EH145�/O1T5m:014@&3�9;>AFQBG1T5 ` A9J036BðEGF c @0/DF214B*n>?98J0365'5*1EH>A@n�{>A@;B5�:2- @;9;1NJ0FG¡1 c 1�KA/O3T9 c I01¡3N@;9;1NJ?FG¡1 c 19;>AFOBG1A01XEuB§¦1 c FOJ0BG>0Ï?@n14L8149AE `p,� � | `

n1T5HE�FQLMJ03TFQB `?9gEG>A@?E*§FD/§?B5 1

2(n2+1)

J0FG¡1 c 1T5�9;>AFQBG1T5�1§E 12(n2−1) J?FG¡1 c 145KA/O3T9 c I01T5 `.0@0FO57Ï?@n1 c I?3"ÏA@n1RJ?FG¡1 c 1�K?/Q369 c I01�3�¦1§E§¦1g?FQ5'FE4¦161�143T3 c EG1TLM1T9AE

peq>AFQ5�1§EÏA@n1N/O3Jn14B5*>?909;1�1T5HEN57>?BEGFO1:2- @;9;1�J0FG¡1 c 1�K?/Q369 c I01Jn>A@;BN1T9AEHBG14B�:?36905�@;9;1J0FG¡1 c 19;>AFOBG1A

�6

/219;>?L8K?BG1�EG>"EG34/�:01MJ0365'5736¢41T5:2- @;9;18J?FG¡1 c 1MK?/Q369 c I01ª¡3@;9;1MJ?FG¡1 c 19;>AFOBG1B§¦143T/DFO56¦1T5Jn149;:03T9AEu/O3NJ0BG>?L814903":01�1T5HE 12(n2 − 1)p ` 3TFQ5§¥5(F4//- >?9¦14/DFOL�FO9;1�/Q3J0BG1TL�FG¡14BG1 149AEHB§¦1T1:03T905/Q3J?FG¡1 c 1:A@ c >AFO9g9;>?BG:;'>A@n1T5HEÑ J?@0FO57Ï?@n1�//- >?9g?FO1T9AE�:;1 //- 143TE§¦14BGFO14@;BÒ( c 14/Q3 c >?BGBG1T5'Jn>?9;:N3T@�9;>?L8K?BG1 EG>"EG34/?:01}?FQ5'FEH145:;145�J?FG¡1 c 145 9;>AFOBG145 34@ c >A@;B5�:01N/Q3�J;BG>?LM1T9036:;1 `n B*A/O3�J?FG¡1 c 1�:A@ c >AFO99;>?BG:;'>A@n1T5HE3=¦1XE4¦1AFO5(FDE§¦1T1N@;9;1R5714@0/O1Neq>AFO5 Ñ J0@0FQ5*ÏA@n1�/O3RJ;BG>?LM1T9036:;1R51- 3TBHB �1§EG1R3TJ;B§¡1T5 /21�J0BG1TL�F214BBG1§EG>A@;B�:?36905 c 1XEEH1gJ0FG¡1 c 1XÒ-1XE c 1§/Q/O1C:A@ c >AFQ9�5'@n:07145(E�3M¦1§E§¦1 AFO5(FDE§¦1T1

xeq>AFO5X63T/O>?B5}ÏA@n1/2145 1

2(n2 + 1) − 2 34@?EHBG1T5}J?FG¡1 c 145}9;>AFOBG145>?9AEg¦1XE4¦1�?FQ5'FE4¦16145 c I?3 c @;9;1N143T3 c EG1TLM1T9AE peq>AFO5 ` �- >¡@

1 + x+(

12(n2 + 1) − 2

)

p = 12(n2 − 1)p ` 98149M:�¦1T:?@0FEue23 c FQ/O1TLM1T9AEuÏ?@n1

x = p− 1 `p,� � (;`n1T5HEuJ03TFQB `?9�EG>A@?E*?FQ/0?B5 1

2n2

J0FG¡1 c 1T5 9;>AFOBG145�1§E 12n2

J?FG¡1 c 145mKA/O3T9 c I01T5 ` ¨T1NBG1T5HEH1:?@MB3TFQ5*>?909;14L8149AE145(E/21L �1TLM1NJ;>A@;Bu36BGBGF ?1TBC¡31 + x+

(

12n2 − 2

)

p = 12n2p

` 98149M:�¦1T:?@0FEue23 c FQ/O1TLM1T9AEuÏ?@n1

x = 2p− 1 `