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d*�- y�-IE<-7a�(70>?�6@A

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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�:

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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

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e ∈ {1K2K3} K�*)$�9�: e = 3

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b ∈ {1K2} R��f$�9�: b

y'6@&�22K):H0V°�&)-�(768-7E<Eb0

a = 1Rs�f$�9�:�K

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a = 1R

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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

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3 + 6 + 4=

6

13

R

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42

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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

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62

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4

R

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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

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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

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13 +

√13

2+

√13

22+ · · · +

√13

2n+ · · ·

=√

13(

1 +1

2+ · · · +

1

2i+ · · ·

)

=√

13

(

1

1 − 1

2

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= 2√

13R

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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

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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

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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�*

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n(70@E<E

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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

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= 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

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2(lw + lh + wh) = 2(1629) = 3258J 2 K619�*52H;)0�y�$'(I&�J>0�-IE

lwh = 12600J 3 R

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MNK

"�-.2H;OM = r − PM = r −

√3

2

619�*ON =

r2 − (

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r −√

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N

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ST

r

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r2 − 14

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1 +√

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+ r2 K

r(

2 +√

3)

= 1 +√

3 + 34

+ 14

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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

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

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

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)

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

A

B CD E

20 21

8 12 9

α β γ

β+γ α+β

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AKBKs6@9�*

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A;�6@E�9�$X{7&�A�2H;)0@A

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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

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B6@9�*

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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

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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

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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. . .

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Ka2 = 1

Ka3 = 1

K|6@9�*an+1an−2 − anan−1 = 2

K{l$�Av68(I(

n ≥ 3Rf»�AC$@y�032H;�682

an

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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

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f(x) = x/(1−x)-IEv:4$�9Dy�0<®z$�9V2H;)0 -79'2C0@A�y'68(

(0, 1)Rv!�9z"%;�682

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1 − a+

b

1 − b+

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3

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3

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abc

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abc

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a = b = cR

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s R ¹3 × n

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K. . .

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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

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k + 1a�0�2H;)0z9�&�J>a�08Av$'{�2H;)0

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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

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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

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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^-.{

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ai

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/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

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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:<;

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aKbKcKdEH&L:<;�2H;�682

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r]la1`

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R�f0G2C0@ACJ=-I9�0

nR

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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)

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RF@0G2N&�EB61E<E<&�J>0d 6= 0

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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

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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

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Z10G2F2C-I9�Ox = 0

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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

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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

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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. . .

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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

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a'-7E�01:C2FE∠PEQ

619�*x2H;�6G2PH

a�-IEb0@:C2CE∠APB

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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. . .

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ai ∈ IRf�f08��9�0

a =1

n

n∑

i=1

ai

Rf»�AC$@y�032H;�682

n(n − 2)

2f(a) +

n∑

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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

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ΓR}�1&�AF2H;)08A

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BDK)"�-.2H;

E( M�-I9�O5$�9>2H;)0,E<61J>05EH-+*)0,$'{

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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. . .

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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

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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

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(+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

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m = n = 5R `

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tK�*)0=?�6@AF230G23*#� 6@&�24AC0>*�0

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AB = mr0[2

AC = nrKm0G2

n*)08&D®=9�$�J>a)AC08EzA�E010G(7E�?L$�EH-.2C-.{+E@R�W_$�9'24AC0@A}°�&L0

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Eb0@OGJQ089'2CEAC

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KLK�J>$�9'24AC0@AB°'&L0

ML = LK = KNR

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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

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

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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

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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

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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. . .

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[

1√2,√

2

] K$�9=6

n∑

i=1

3

ai + 2ai+1

≥n∑

i=1

2

ai + ai+1

K

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p&\ n&n R �0�s���o����2#c����h� �y���s���o���$#5��m�������� >@# RZ1$'-.2

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:H$'&�?�0EF

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Z1$'-.2a1Ka2K. . .

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?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�$# +

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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

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:H$�9�{7&�EH-+$�952H;'-IE J=6GM=;�6Gy�0VO80@9�08AF682C01*sR

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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

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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

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∠BACK)6@9�*>E<-I9�:H0

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QK�"%0XE�010=2H;�682

AP = AQRxS�;)0@AC0G{l$�AC0�K

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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

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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

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AB! fgQ.ck\@Q�igRgX lbQ.m n � P:Q'R

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��p U�]�q7RVU e R ! XIa AD =√

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x =BF

F A

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CE

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K1"%0V;�6Gy�0BF

F A· AE

EC· CS

SB= −1

r2H;�682�-IEGK

xCS

BS= y

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K1"%0V$)a'2F6@-I9

xBC

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R ] � `� $1"³"%0�6@?�?�(�MXW_0@9�0G(768&�EG�DS�;)01$�AC08J -79 4ABD

2C$=O80G2BF

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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=

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1

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�X-.2H;%2H;)0z&�EH&�6@(�9�$D2C6G2C-+$�95{l$�ANE<-7*�08EB619�*,619�O[(70@Ev-79�24AF-76@9�O[(+0ABC

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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

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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

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-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

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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��/;�� �

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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

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d d

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b − d

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A

B CH

r1

r2

bc

c′ b′F@0G2da�0f2H;)0 (70@9�O�2H;%$'{D2H;)0f2C6@9�O80@9'2�{+AC$�J

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b + c − a

2

RZ8-7J,-I(I61AF(�M�K'{l$�A^24AC-I619�O[(70@E

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2

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2= d

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+ (b2 − 2c2)(2b2 − c2)

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