* (x, y) (3, -4) (5, 0) (2, 4.58) x 2 + y 2 = 25 S = 2 2 - 1 2 2 × 3 2 - 1 3 2 × 4 2 - 1 4 2 ×···× 2004 2 - 1 2004 2 S 8338 50705 A B C A + B = C C x 2 + y 2 =1 y =7x +5 A B O ∠AOB x (x 2 - 3x + 1) x+1 =1 f (a, b) a b f (1, 5) = 1 + 2 + 3 + 4 + 5 = 15 f (3, 6) = 3 + 4 + 5 + 6 = 18 f (133333, 533333)
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2L?D0)8<CN4GBDCN7 d B�4�=nI<4,0�BD4�;>gSb!��F>=94 � 2k ≤ 2n �BGF<23BäCN4 � k ≤ 2n−1
b� 0~79;�} H9?D;3�<06BGF/0�89034�CN?D0�8R?D0L4G=/.AB�IL:OCJ798>= d BDC1;<7Rgq;<?
p ≥ 3b+�{ãqg
p = 1;<?
p = 2 � BGF/0L7fBGF/0 d ;<798<CQBDC1;<7fBGF<2�B�2L7�:fBGF<?D0L0O4�=nI<4,0�BD4�F<2��<0O2 d ;<KMKR;<7T0�.103KR037>BCJ4��>2 d =n;>=94G.Q:�BE?[=n0 � 4GCJ7 d 0+}60 d 2L7 d F/;/;<4,0);<7<. : 2p−1 < 34G=nI/450�B[4Lb@��F/0)?D0L4G=/.AB�CN4g12L.N4,0�gq;<?_BGF/0L450 d 2�450L43b �ãqg
p = 3 � .J0�B S = {x1 � x2 � x3} b � 0 d F/;/;<450)gq;>=9?�450�B[4 A1 � A2 � A3 � A42�KO;<79� ∅ � {x1} � {x2} � {x3} � {x1 � x2} � {x1 � x3} � {x2 � x3} � {x1 � x2 � x3} �4�= d F~BGF<23BäBGF/0�CN7>BE03?[4,0 d BDC1;<7R;>gY2�7�:�BGF<?D0L0);>g�BGF/03K#CJ4s79;<79��0LKMH<B{:�b .1032�?[.Q: � 79;<790;>guBGF/06gq;>=9? d 2�7fI/0 ∅ b c .N4,; � BGF/0)450�B[4 {x1 � x2} � {x1 � x3} � {x2 � x3} d 2L7/79;�Bs2L.A.I90 d F/;<450L7 � I90 d 23=94,0OBGF/0�CJ?�CN7>BE03?[4,0 d BDC1;<7TCJ4)0LKMH<B{:�b ã1B+gq;>.A.1;�}�4+BGF<2�B�;<790f;>g!BGF/01� 03.J0LKO0L7>B�4,0�BD4WK�=94GB_I/0 d F/;<4,037 � 452�: {x1} b��YF/037~032 d F6;>g>BGF/0 d F/;<4,037)450�B[4WKM=94EBd ;<7>B[2LCN7
x1
b��YF>=94 � BGF/0 d ;<7 d .A=94�CJ;<7OF/;>.J8/4�gq;<? p = 3b�L0�B
p ≥ 4I90M2O�� �0�8fCJ7>BD0��303?�bR�3=9H9H9;<4,0�BGF<2�B�BGF/0O?D034�=/.QB�F/;>.18<4+gq;<?�2O4,0�B}+CABGF
p − 103.J0LKO0L7>B[4Lb@�L0�B
S = {x1 � x2 � . . . � xp} � 2�798~.10�B A1 � A2 � . . . � A2p−1I9068>CJ4EBDCJ7 d Bä4�=nI<4,0�BD4s;>gS4G= d F)BGF<23B
Ai ∩ Aj ∩ Ak 6= 0 � gq;<?ä2L.A.i < j < k
b�L0�B
npI/06BGF/0�7<=9KOI903?�;>g�4,0�BD4
AiBGF<23B d ;<7>B[2LCN7
xpb6ãqg
np < 2p−2 � BGF/0L7BGF/0�7<=9KOI903?_;>g>4,0�BD4Ai79;�B d ;<7>BD23CJ7<CN79�
xpCJ4
2p−1−np > 2p−1−2p−2 = 2p−2b�YF/034,0 4,0�BD4�gq;<?DK 2 d ;>.A.10 d BDC1;<7 ;>geKR;<?D0´BGF<2�7
2p−28<CN4GBDCN7 d B 4G=nI/450�B[4#;>g
{x1 � x2 � . . . � xp−1} 4�= d F�BGF<23B Ai ∩ Aj 6= ∅ gq;<?+2L.A. i2�798
j � d ;<7>BE?[2�8<C d BDCJ79�BGF/0+.103KOKM2nb���F>=94np ≥ 2p−2
i = Ai \ {xp} gq;<? i = 1 � 2 � . . . � npb��YF/037
A′1 � A′
2 � . . . � A′np
gq;<?EK2 d ;>.A.10 d BDC1;<7);>g9KR;<?D0�BGF<2L72p−2
8<CN4GBDCN7 d B_4�=nI<4,0�BD4�;>g {x1 � x2 � . . . � xp−1} b_¦s4�CN79�BGF/0�.103KOKM2 � }60M890L8<= d 0�BGF<2�BsBGF/0L?D0O0G �CN4GB r2L798
s4G= d FRBGF<23B
A′r ∩ A′
s = ∅ bRã1Bgq;>.N.J;�}�4WBGF<23BAr ∩ As = {xp} b!Ü>;<?ä032 d F k 6∈ {r � s} � 4�CN7 d 0 Ar ∩ As ∩ Ak 6= ∅ �}60�F<2��<0
Ar ∩ As ∩ Ak = {xp} � g1?D;<K }6F>C d F6}60�890L8<= d 0�BGF<2�B xpCJ4@2 d ;<KOKO;<703.J0LKO0L7>B�;>gY23.N./BGF/0
Ai� 4Lb
pE�"6�� �9bnp = 2p−2
b�L0�BkI/0�4�= d FsBGF<23B
xp 6∈ Ak2�798�8/03�9790
Bk = Ak∪{xp} bu�YF/037 Bk 6= Akb�3=9H9H9;<4,0�BGF<23B
Bk 6= Ajgq;<?!2L.A.
jbW�YF/037�BGF/0�4,0�BD4
A1 � A2 � . . . � A2p−1 � Bkgq;<?EK2 d ;>.N.J0 d BDCJ;<7);>g
2p−1 +18<CN4GBDCN7 d B_4G=nI/450�B[4!;>g
S � 2�798+BGF/0�CN7>BE03?[4,0 d BDC1;<7~;>g/2�7�:6H923CJ?;>gu8<CN4GBDCN7 d B!450�B[4�CJ7~BGF>CN4 d ;>.N.J0 d BDCJ;<7~CJ4@79;<79��0LKMH<B{: � d ;<7>BE?[2�8<C d BDCJ79��BGF/0+.J0LKMKO29b�YF/03?D03gq;<?D0 � gq;<?O032 d F k4�= d FáBGF<23B
xp 6∈ Ak � BGF/0L?D0 0G �CN4GB[4 j4G= d FâBGF<2�B
Ak ∪ {xp} = Ajb��3CN7 d 06BGF/0)7<=9KRI/0L?�;>g
Ai� 4s}6F>C d F�89;R79;�B d ;<7>BD23CJ7
xpCN4�BGF/0452LKR0+2L4!BGF/0+7<=9KRI/0L?�;>g
Ai� 4!}6F>C d F d ;<7>BD23CJ7
xp � }60�890L8<= d 0�BGF<2�BWBGF/0 d ;>.A.10 d BDC1;<7;>gnBGF/0Ai� 4�CJ4
i � 2L798MBGF/0L?D0�gq;<?D0 � a CN4�2 d ;<KOKO;<7�03.J0LKO0L7>Bs;>g�23.N.nBGF/0 Ai� 43b@�YF>CN4P03798/4sBGF/0CJ798>= d BDC1;<7M4GBD0LHM2�798)BGF/0�H9?D;/;>gDb
bãqg � gq;<? i > i1 � 79;<7906;>g9BGF/0+.ACJ79034 `iKR0L0�B[4!BGF/0�CJ7>BD0L?[C1;<?!;>g
Tk1� BGF/037 Tk1
CN4;<790);>g�BGF/06?D0���CJ;<7/4Lbsã{7MBGF/0);�BGF/03? d 2L4,0 � 4�=nI<4GBDCQBD=<BDCJ79� Tk1
gq;<?Tk � }60+gq;>.N.J;�} BGF/0452LKR0�?D032�45;<7<CJ79��2�4!2�I/;L�<0/bW�3CJ7 d 0 � 2�Bä0L2 d F~4EBE03H � BGF/0�7<=9KOI903?!;>g9.NCN790L4!?D0LKM2LCN7<CJ79�BE;RI90 d ;<7/4�CJ8903?D0�8OCN4P8/0 d ?D032�4GCJ79� � BGF>CN4�H/?D; d 03454@}+CA.N.u0,�<037>BD=92L.A.Q:R4EBE;<H � ��C �<CN79��=94�2?D0���CJ;<7~}6F>C d F~CJ4@2�BE?DCN2�79��.J09b� CABGFRBGF>CJ4�H9?D; d 0L4�4 � 032 d F�;>g�BGF/0 Ti
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b ��F/0�gq;>.N.J;�}+CJ79�4GB[23BD0LKO0L7>B[4s2L?D060L�<=/C �>2L.J0L7>B ��JC � f(X2)
CN4@?D0�8>= d C1I>.10+CN7Z[X] ��NCNC � f(X)
CN4!?D0L8<= d CJI<.J0�CJ7Z[X] � ;<?�BGF/0L?D0�0G �CN4GB_H9;>.Q:<79;<KMCN2L.N4 G(X)
2�798H(X)CN7
Z[X]2�798�2�=97<CQB
u;>g
Z�ABGF<23BuCJ4 � 2�7+CJ7��<03?[BDCJI<.J0P0�.103KR037>B_;>g Z \{0} � 4�= d FBGF<2�B
uf(X) = G2(X) − XH2(X)b
p�x:#�x�yQy-�'#o* � �L0�Bf(X)
I90Ý2L7�:âHn;>. :/79;<K�CJ23.P;>gZ[X]
}6F>C d FáCN4OCN?E?D0L8<= d CJI<.J0ÝCN7Z[X]
b c 454G=9KR0OBGF<2�Bf(X)
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CJ4!79;�Bä2�4,�>=92�?D0PCN7Zgq;<?ä032 d F6=97<CAB
u;>g
Z;<?WBGF<2�BAC
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f(X2)CJ4�CN?E?D0L8<= d CJI<.J0+CJ7
Z[X]b
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1 + · · · + a2n
gq;<?ä2�7�:a1 � . . . � an ∈ Z
b2
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Zb c�<=92�8/?[2�BDC d Hn;>. :/79;<K�CJ23.
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Z[X]CNg/2L798);<7<. :+CNgf(X)
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CJ4ä2�4,�>=92�?D0�CJ7Z �;�BGF/0L?�}+CJ450 � CABWF<2�4!79;6?D;/;�BD43b ��gn4�Hn0 d CJ23.<CNKOH9;<?[B[2�7 d 0CJ4@BGF/0 d 2L4,0+}6F/037
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Z � 4�CN7 d 0 f(X2)BGF/0L7�F<2�4sB�}6;CJ?D?D0�8>= d C1I>.106�>=92�8<?[23BDC d g12 d BE;<?[4 � I>=<B�79;�?D;/;�B[4Lb�P;�}á}60sH9?D;3�<0 � }+CABGF/;>=<Bu=94GCJ79� d ;<KOH<.10G P7<=9KOI903?[4 � BGF<2�B�2�7�: � �303790L?[23.NCQ��0�8��I<CJ�<=92�8/?[2�BDC d Hn;>. :/79;<K�CJ23.�;>g / [X]
F (X) = AX4m + BX2m + CI90�2�H9;>.Q:<79;<KMCN2L.9CN7 / [X]}+CABGF
A 6= 0b_��F/0L7
F (X)CN4@?D0�8>= d C1I>.10+CN7 / [X]
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m = 1b � 0�KM2�:�2L.N4,;R2�4�4�=9KO0
A > 0b��P;�}�4G=9H9H9;<4,06BGF<23B
F (X)CN4CJ?D?D0�8>= d C1I>.10~CN7 / [X]
b�ã1Bsgq;>.A.1;�}�4�g1?D;<K �JCAC � ;>g��YF/0L;<?D0LK � BGF<23BB2 − 4AC < 02�798�BGF<2�B � gq;<?�2L7�:T?D032L.W7<=9KRI/0L? S � }60RF<2��<0 AC < S2
;<?A(2S − B) < 0
b�3CJ7 d 06BGF>CJ4�CN4 d .1032�?[.Q:�0L�<=/C �>2L.J0L7>B�BD;MBGF/0 d ;<7>BE?[2�8<C d BDC1;<7B2 < 4AC
2�798C < 0 �BGF/0�H9?D;/;>g�CN4 d ;<KMH/.J0�BD09b
2
c 4)2�7Ý2�H/H/.AC d 23BDCJ;<7';>g!�YF/0L;<?D0LK � � }60 d ;<7/4GC18/0L?��Ngq;<?�2L7�: a � b ∈ Z � BGF/0Hn;>. :/79;<K�CJ23.Fa,b(X) = X4 − 2(a + b)X2 + (a − b)2
b� 0��9?[4GB�890�BE03?EK�CJ790f790 d 03454�2�?{:Ý2�798�4G=.� d CJ0L7>B d ;<798<CQBDC1;<7/4~;<7
= 16ab � 2(a − b) + 2(a + b) = 4a �2�798 −2(a − b)+2(a+ b) = 4bCJ4ä2�45�<=92L?D0�CJ7
Z �BGF<23B�CJ4 � CNg92L798~;<7<.Q:�CNg92�B�.1032�4EB;<7906;>gnBGF/0�03.J0LKO0L7>B[4
a � b � 2�798 abCJ4@2�4,�>=92�?D0�CJ7
Zb� 0�79;�BD0�BGF/0�gq;>.N.J;�}+CJ79� d ;<7/4,0L�<=n037 d 034 �
� b���F/0~H9;>.Q:<79;<KMCN2L.Fa,b(X)
CJ4�?D0L8<= d CJI<.J0)CJ7��p[X]
gq;<?s2L7�:�H/?DCNKR0p � I/0 d 2L=9450 �IL:�BGF/0M.N2�}#;>g!�<=92�8/?[2�BDC d ?D0 d CNH9?D; d CQB�: � BGF/0MH9?D;/8>= d B+;>gä79;<79�G4,�>=92�?D034+CN7�� p
CN423.Q}�2�:/4+2R4,�>=92�?D0MCJ7��pb��N�YF>=94 � CAB�CJ4�0L2L4[:�BD;��9798�CN?E?D0L8<= d CJI<.J0RH9;>.Q:<79;<KMCN2L.N4;>g��
[X]}6F>C d F�2�?D0O?D0�8>= d C1I>.10MCJ7��
p[X]gq;<?�2L7�:TH/?DCNKR0MCJ7>BD0��303?
p �BGF/0 d 2L4,0
a = 5 � b = 41CN4 d ;<7/4GC18/0L?D0L8�CJ7� £ � H/H�b �������>����£ � ���L��� � b
CAg_2L798�;<7<.Q:RCAg79;<790�;>g�BGF/0�0�.103KR037>BD4a � b � ;<? ab
CN4P2�4,�>=92�?D0)CJ7Z�A}6F>C d F d ;<7/4EBDCABD=<BD0L4 � gq;<?4�Hn0 d CN� d �>2L.A=n0L4+;>g
a � b ∈ � � 2)�<0L?{: d ;<KMKR;<7�0� �03? d CJ450�CN7�2L7�:�CN7>BE?D;/8<= d BD;<?{:d ;>=9?[4,0�;<7O0G LBD0L7/4GC1;<7��n03.J8/4 � b� CABGF)KR;<?D0+�30L7903?[2L.ACAB{: � IL:)B[2��LCJ79� A = 1}60+�30�BWg1?D;<K���F/0�;<?D03K � BGF/0�gq;>.J�.1;�}+CN79� d F<2L?[2 d BD0L?[CA�G2�BDC1;<7�;>g9BGF/0+03.J0LKO0L7>B[4@;>g�45;<KR0�23.1�30LI/?[23C d�d .J;<4�=9?D0+;>g
�JC � F (X) = X4 + BX2 + CCJ4@26I<CJ�<=92�8/?[2�BDC d Hn;>. :/79;<K�CJ23.�;>g
Z[X]b
�NCNC � � 0 d 2�7)}�?DCQBE0 ξCJ7M45;<KR0�;>gnBGF/0�gq;>.N.J;�}+CJ79��gq;>=9?_}�2�:/4 �
±√
B ±√
B2 − 4C
2
b}6F/03?D0
B2�798
C2L?D0+03.J0LKO0L7>B[4@;>g
Z4G= d F6BGF<23B
B2 − 4CCN4�79;�BW2�45�<=92L?D0�CN7
Z � 2�798 B ± 2√
C2�?D0+79;�B!4,�>=92�?D034@CN7
ZCNg
CCN4@2645�<=92L?D0+CN7
Zb
����&P��#���$ ���'6 ��� ��2�BD23.NCJ;á��b!¨_=n03?[4,037>�[�>2LCJ� � U = Z1Ò~Vn\{Ë Ç{XEXGË�Õ�Ç>×�Z�ÑLZA\NZ�ÌqØ[Â/X[Z�Ì{Ë�X[Z{Æ�Ï'Î�Æ�X
f(X2) � �L����� �<�������>������� b ��� �L0�;<7/2L?D8 ÜYb T .1;<4GCJ7/45��C � ¨ä03?[2L.J8 �/b c .10G L2�798/0L?[45;<7 � 2�798 �L;<?D037 b)�32�?[45;<7 �
�LÍ�Ë = Z �9ÌqØ �[É�Ë�×EÆ�ÏYÕ ��ZN\A\NZJÓ�Ò �>Æ/ÈPË¡\A\q¼�Ç/ÌqÏ/Ó�Ò�Ë Ó<ÌqÍYË�ÒOÓ/ÌJZ{×[Ó�\¡Â�Æ�Ò~V�Ë�ÌJZ�ÌJZ�Æ�Ï �c KR03?DC d 2L7f��23BGF/03KO2�BDC d 2L.��';<7>BGF>.Q: � $������ �����>� �G� �>� � ���>�" b ���Ý� b T 0�CABGF���C d F/;>.J45;<7 � � ã{7>BE?D;/8>= d BDC1;<7~BE; c I<4GBE?[2 d B c .J�30�I<?[2 � � � CA.10,: � �" : � b £�� ¨ä0L;<?D�30�- @;>. :>2�2L7986¨W2�I9;<?�����0����; � � -9?D;/I>.103KO4_2L798+�YF/0L;<?D0LKM4_CN7 c 7/23.Q:<4�CN4!ã[ã � ��LH9?[CJ79�303? � �" � � b ¢�� ��BE0,�<037�¤Y;<KM2�7 � � Ü�CJ03.J8~�YF/0L;<?{:� � �LH9?[CJ79�303? � �� � >¢ b
m ≥ n � 0�B�4�C a � b � c > 0 �KR;<7>BE?D03?��<=n0am
bm + cm+
bm
cm + am+
cm
am + bm≥ an
bn + cn+
bn
cn + an+
cn
an + bn
b
� �� Þ � � Û b¤¼nX�Æ3V�Æ�É�½Ë�VYÓLX�Ë�Z{×DÍYË,\�Æ�Ó<Ì1Ó�ZN\A\�Ë¡]8Å_ÆnÇ>Ë�Ï<] ��XDÓLÏY×EË>b-<;>=9?
a � b � c ∈ (0, 1) � BE?D;>=>�<03?!.N26H/.A=94sHn0�BDCABD06I9;<?D79064G=9H�@03?DCJ03=9?D060�Bä.J26H<.N=94��?[2�798/0+I/;<?E790�CJ7<g @03?DCJ03=9?D0�8/0a+ b+ c+abc
0L7�BD0L7/2L7>B d ;<KOH>BE0�8/0�.N2 d ;<7>BE?[23CJ7>BD0ab + bc + ca = 1
�@7�89;<7/790~=97T�>=92�8<?DCA.J2�B �0L?D0~CJ7/4 d ?DCNH<BDCJI<.J0 d ;<7��<0G �0O8/;<7>B�.1034P.1;<79��=n0�=9?[4�8/0L4d'È;�B @0L464,;<7>Bm � n � p
0�Bq � H9?[CJ4)8/2L7/4 d 0�B�;<?D8/?D0/bä�3CW.1034~8>CJ2��3;<7/2L.J0L46KO0L4G=9?D0L7>B d0�B
d′ � KR;<7>BE?D03?��>=n0 √mp + nq ≤ 1
2(d + d′)
b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b bÞ �Pæ+à b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø�ËÝZ1Í8½Ó�\qØ�Æ�Ë�Ï�× �¡Ë�]8Æ�X[ÓLÉ�Æ<Ê�]8Å�Æ�ÒOÓ�Ï>Z1Ó/b
�L0�BABC
I/0s2L7�:+2 d =<BD0��G2�79��.J0�8PBE?[CJ2L79��.10/b!�L0�Br2L798
RI/0�BGF/0@CN7/?[2�8>CN=94_2L798d CJ? d =9KM?[2�8>CN=94 � ?D0L4�Hn0 d BDCQ�<0�.Q: � 2L798·.10�B
a1a2 · · · an ≥ 1 � b�12 � -9?D;3�<0�BGF<2�B En ≥ n(1 + r2)
1 + r
gq;<?n = 3
2L798n = 4
b�1I � ? -9?D;3�<06;<?!8>CJ4�H9?D;3�<0�BGF<23B
En ≥ n(1 + r2)
1 + r
gq;<?n = 5
bÞ �Pæ�� b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø �<Ó3É�ZA\�Ë¡Â��� X,Ì�Æ�Ó���Ë¡]���Ï>Z�ÊLË�X[ÉGZ�Ì1Ø�Æ�Î�¼/\�ÆLZ{Ë�É�ÌJZJ]�Å�Æ�ÒOÓ�Ï>Z1Ó/b
�L0�Ba � b � c � d � 2L798 r
I/0sH9;<4�CQBDCQ�<0@?D0L23.�7<=9KOI903?[4_4�= d F�BGF<23Br =
4√
abcd ≥ 1b-9?D;L�<0PBGF<23B
1
(1 + a)2+
1
(1 + b)2+
1
(1 + c)2+
1
(1 + d)2≥ 4
(1 + r)2b
Þ � � � b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø ��Z�ÌDǧÁ>ÊLÆ�Ï/Ó�X¡Ç3]8Æ_Ç>×DÍ9ÓLX�Ë¡É3Ì1]�Å_Æ�ÒOÓLÏ<ZJÓ9bãqg
m2L798
n2�?D0~H9;<4�CQBDCQ�<06CN7>BE0L�30L?[4P4G= d FOBGF<2�B
m ≥ n � 2L798RCNg a � b � c > 0 �H9?D;3�<0�BGF<23Bam
bm + cm+
bm
cm + am+
cm
am + bm≥ an
bn + cn+
bn
cn + an+
cn
an + bn
bÞ � � Û b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø�ËÝZ�×DÍ�Ë¡\�Æ�Ó/ÌJÓ3ZA\N\{Ë�]8Å�ÆnÇ<Ë�Ï/] ��XDÓLÏY×EË>b
Ü<;<?a � b � c ∈ (0, 1) � �9798�BGF/0�.1032�4EB!=9H/Hn03?sI/;>=9798R2L798�BGF/0)��?D0L2�BE034GB!.1;�}603?I9;>=9798O;>ga + b + c + abc � 4G=nI�;{0 d BWBE;)BGF/0 d ;<7/4EBE?[2LCN7>B ab + bc + ca = 1
b
� � �Þ � � Þ b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø �<Ó3É�ZA\�Ë¡Â��� X,Ì�Æ�Ó���Ë¡]���Ï>Z�ÊLË�X[ÉGZ�Ì1Ø�Æ�Î�¼/\�ÆLZ{Ë�É�ÌJZJ]�Å�Æ�ÒOÓ�Ï>Z1Ó/b�12 � -9?D;3�<0�BGF<2�BäCNg 0 ≤ λ ≤ 4 � BGF/0L7 � gq;<?ä2L.A.nHn;<4GCABDC �<0+?D0L23.�7<=9KOI903?[4 x � y � z � t �
GI90�BGF/06H9;>CJ7>B@;>gYCN7>BE03?[4,0 d BDC1;<7O;>g�BGF/0�.ACJ790
AD�q0G LBD0L798/0�8 � }+CQBGFOBGF/0~.ACJ7906BGF<?D;>=n�3F C}6F>C d FRCJ4PH/2�?E�2L.A.10�.<BE;
ABb��L0�B
EI/0PBGF/0�Hn;>CN7>Bä4�= d F6BGF<2�B
ACGECJ4~2�?D0 d B[2�79��.J0 � 2�798�.J0�B FI/0�BGF/0�H9;>CJ7>B64G= d FÝBGF<23B
BFGECN4P2�?D0 d BD2L79��.10/bM�L0�B
HI/0)BGF/0)CN7>BE03?[4,0 d BDC1;<7;>g
AG2�798
BFbs�L0�B
O1
I90PBGF/0+CJ7>BD0L?[450 d BDCJ;<7O;>gnBGF/08<CN2��3;<7/23.J4ä;>g<BGF/0P�>=92�8<?DCA.J2�BE03?[2L.CDHF � 2L798�.10�B O2I90�BGF/0�CJ7>BD0L?[450 d BDCJ;<7~;>g<BGF/0�8>CJ2��3;<7/2L.N4�;>g<BGF/0P�<=92�8/?[C1�.J2�BE03?[2L.
) CJ4sBGF/0�;<7<. : d ?DCQBDC d 2L.uHn;>CN7>B�;>gfb6�P;�BD06BGF<2�B
f(
1
2, 1
2, 1
2
)
= 3
2
bc .J45; � 2�4PBGF/0 d ;<7 d =9?D?D0L7>B�H9;>CJ7>BP2�H/H9?D;<2 d F/0L4�2O4�CJ890O;>g 4ABC � BGF/0MgJ=97 d BDCJ;<7 fd .1032�?[.Q:T2�H/H9?D;<2 d F/0L4�CAB[4)KM2� LCJK�=9K �>23.N=n0�;>g2b ��037 d 0 � 1
bã1B�gq;>.A.1;�}�4�BGF<2�BMP · MC = MQ · MD � 4,;fBGF<2�B CDQP
CJ4 d : d .AC d�F/0L7 d 0 �
∠MPQ = ∠CDQb@�L0�B
O1
2L798O2
890379;�BE0PBGF/0 d 037>BE?D0L4s;>gΓ1
2�798Γ2 � ?D0345H90 d �BDCQ�<0�.Q:ub��YF/037�BGF<?D;>=n�3FM45;<KR0+2L79��.10 d F<2�4GCJ79� � }60�F<2��<0
∠APQ + ∠QBA = ∠APM + ∠MPQ + ∠QBA
= 180◦ − ∠CPA + ∠CDQ + ∠QBA
= 1
2∠AO1C + ∠DBQ + ∠QBA
= ∠AO1O2 + ∠DBA
= ∠AO1O2 + ∠O1O2B = 180◦ �2�798~BGF/0+?D0L4G=/.ABägq;>.A.1;�}�43bo�oEb = ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �@Ó�Ï>Z�Ë,\�Å�Ë¡ZNÉ �G] �/ZJÏY×EË¡\A\�Ë¡É5] ��X[Ó�Ï�×GË�b��/Õ�Z�Ì{Æ�X�É É�×EÆ�ÒOÒfË�ÏYÌDÔW��F/0+gq;>.N.J;�}+CJ79�PH9?D;/;>g�=9450L4ä�<0L?{:MK�= d F~BGF/0�4�2�KO0�2L?D��=9KR037>B2�4�CJ7���;>.A=<BDC1;<7'ã � 0G d 03H<BPBGF<2�B�F/0L?D0OBGF/0�2L79��.103462�?D0f8>CJ?D0 d BE0L8�b'ã1B�CJ4�CN7>BE03?D0L4EBDCJ79�BE; d ;<KOH/2�?D0OBGF/0O=94,0�;>g!=9798<CN?D0 d BD0�8Ý2�798�8>CJ?D0 d BE0L8Ý2�79��.J0L43b Ü>;<?PBGF/0O=9798<CN?D0 d BD0�82�79��.J0L4
∠XY Z2�798
∠XY ′ZCN7/4 d ?[C1I/0�8jCN7 2 d CJ? d .J0 � ∠XY Z = ∠XY ′Z
;<?∠XY Z = 180◦ − ∠XY ′Z
2 dGd ;<?D8<CN79�R2�4Y2L798
Y ′ .NCJ0f;<7TBGF/0�452LKR0�2L? d ;<?;<7OBGF/0~;<H/Hn;<4GCABD0)2L? d 4P8/0�BD0L?DKMCN790�8�IL:X2�798
Zb�Ü<;<?äBGF/0~8<CN?D0 d BD0�8R2�79��.J0L4�g1?D;<K
Y XBD;
Y Z2L798fg1?D;<K
Y ′XBD;
Y ′Z}60MF<2��<0
(Y X, Y Z) = (Y ′X, Y ′Z)� CN7¤Y03CN4G�¡� 4s79;�B[23BDCJ;<7 � gq;<?ä2�7�:~gq;>=9?!Hn;>CN7>BD4
X � Y � Y ′ � Z ;<7M2 d CN? d .10/b �Å_Ë�ÒOÓ�X%Ê�Ç<Ë¡ÉLÔ� b
MP · MC = MA2 = MB2 = MQ · MD � 89;<7 d .1034)�>=923BE?D0RHn;>CN7>BD4C � P � Q � D 4,;<7>B d ; d : d .NCJ�<=n034Lb
� b)-<;>=9?_8/0L4W?[2LCN4,;<7/4ä890�4[:/K4@0�BE?[C10��J.N2�8/?D;>CQBE0�89034 d 0L7>BE?D034O1O2
034GB�K4@0�8>CJ2�BE?DC d 0d ;<KMKM=9790 � .1034 d ;<?D8/0L4 AC0�B
& P Q�R S�JLKMHoi$H�PBS�JWYB Z S\[ C ](' S\[oJ_^DRnX9HDTUh\Y�H�q J_^DRMi$H�PBS�J
AY_h$T&^�J_^ Z J J_^DRT_R�S�JUIWRMH�q$J_^DR�S�PBS\R�p_i$H�PBS�JsTUPVIWTUXVRMH�q 4ABC
XaP9R�Y@H�S6J_^DRnXaPVS\RBC
�
) �$5�%� ��(�V3u%�Eg,�.�Cr%D*% F5*�W1( 4%�)�,D*�#�+�; ),�-��+\*rE.�9;0/�*�?1�D%�#UA2�FE�3�� *r#L)&3�� .�;0/3�D;$�(5%���44S�JUIWHD[�h$TUR Z T_HDH�IW[�PVS Z JURoY�j�Y_JWR d KnP�J_^
H�S6J_^DR�i\R IWi$R�S\[�PVTUhDX Z I86�PBY7R TWJUH�I H�q EF ] KM^�P9T&^6PBY J_^DRbXBPBS\Rx = 1
2X
� 4 qN
PVY Z XBY7HH�SoJ_^DRMXaPVS\RBC 9 KM^�P9T&^-^ Z Y�R�:�h Z JWP9H�S y = 0; ] J_^DR�S N
d�h\Y_J@^ Z Q�R6TUHDH�IW[�PBS Z JWR Y(
1
2X, 0
)
�(f ^DR�S ] Y_PVS\TUR ND = NE Z IWR�I Z [�PaP>H�q\J_^DR�S�PVS\R p&i\H�PVS�JsTUPBIWTUX9R ] KMR�^ Z Q�R
ND2 =(
1
2X)2
= NE2 =(
1
4a)2
+(
1
2Y)2
�
<�R�S\T_R ] X Z S\[ YY Z JWPVY_qaj
x2 − y2 = 1
4a2
�
= H�S�Q�R�I�Y7RrX�j ] q H�I Z S�j�i\H�Y&P�JWP9H�S�H�q\Q�R�I�JWR&cA
KM^DH�Y2R�TUHDH�IW[�PBS Z JWR YsY Z JWPVY_qajMJ_^�PVYR>:�h Z JWPVH�S �@?D+�� � R_c�T_R�i�J
A = BH�I
A = C ] KM^DR IWRnJ_^DRnJUI�P Z S\N�X9R�PBY`[DR�N�R�S\R I Z JUR�]
N =(
1
2X, 0
) PVYnJ_^DR TUR S�JUIWRgH�q J_^DR0S�PVS\R p&i\H�PVS�J�TUPVIWTUXVR ] Y&PBS\T_R-P�JnPBY6R�:�hDPV[�PBY_J Z S�Jq9IWH�dD ] E ]mZ S\[ F
� f ^DR IWRrq H�IWR ] J_^DR X9HDTUh\YeH�qmJ_^DRgQ�R�I�JWR&cA
PBYeN�P�Q�R�SA6 j J_^DRR>:�h Z JWPVH�S Z 6\H�Q�R ] KM^�P9T&^n[DR Y2T_IWP�6\R�Y Z IWR�TWJ Z S\N�hDX Z I ^�jDi\R IB6\H�X Z KnPaJ_^bd Z�C H�I Z c PVY BC�@?D+�� � KnPaJ_^OJ_^DRni$H�PBS�JWY
bW�3=9H9H9;<4,0�BGF<23BB = (−1, 0) � C = (1, 0) � 2L798 A = (x, y) � }+CABGF
x > 1b��L0�B
P = (x, 0)2L798
Q = (2x − 1, 0)bs��F/0L7
B = ∠CAP = ∠PAQ �BGF>=94 �4GCJ7 d 0A + 2B = π/2
� � AB ⊥ AQb!��037 d 0 � y
x + 1=
x − 1
y�BGF<23BäCN4 �
x2 − y2 = 1b
� 0@�30�B9BGF/0@;�BGF/0L?�I/?[2L7 d F(x < −1)
;>gLBGF/0@F�:<Hn03?GI/;>.J2ä}6F/037B = C+π/2
bã{7 d ;<7 d .A=94�CJ;<7 � BGF/0�.J; d =94ä;>g A CN4�BGF/0�?D0 d BD2L79��=/.J2L?_F�:<Hn03?GI/;>.J2 d 037>BE?D0�8�23BYBGF/0�KMCJ89�Hn;>CN7>B�;>gBC � }+CQBGF)BGF/0P�<03?[BDC d 034 B � C 0G d .N=n8/0�8ub
�YF>CN4 d ;<KOH<.10�BE034�BGF/0+H9?D;/;>gu;>gY;>=9? d .N2LCNK#2�I9;3�<09b�P;�} }60)}+CN.A._4,F/;�}�BGF<2�B � gq;<?P0L2 d F��� �0�8 n � BGF/0MgJ=97 d BDCJ;<7 g(λ, n)
F<2L4+CAB[4KO2¡ �CNKM=9K }6F/0L7λ = 1
b!Ü>;<?n ≥ m � }60�F<2��<0
(
g(λ, n) − g(1, n))
−(
g(λ, m) − g(1, m))
=1
2λ
(
1 + λm)1m − (1 + λn)
1n
)
≥ 0 �
�����
=94�CN79�PBGF/0�.103KOKM2nb���F/0L7g(λ, m) − g(1, m) ≤ lim
n→∞
(
g(λ, n) − g(1, n))
≤ 0 �2�798~BGF>=94 � g(λ, m) ≤ g(1, m) � gq;<?!23.N. m bÜ�CN7/2L.A.Q: � 79;�BD0äBGF<23B lim