wholeissue 34 2 - canadian mathematical societye 6 6 9@f @1:34a5+1 e?@9
TRANSCRIPT
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;�� A<9½CG1 e E�369<â�/H1ABCD
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E;<1P57?@C�EG1P�@An1
AFCE57?BF|EOA<9�/2?@5(369<â410c� 3Z�Nð4F
AB = 161ZE
BC = 12�<; �1ZEG1:CGMNFD9<1:C
EFc� L6�}ð4F
AB = x1ZE
BC = y�0; �1ZEG1:CGMNFD9<1:C
EF149�gq?@9 e EGF2?@9};01
x1�E
yc
c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c´ � ��è c #¯g°,±�°�² ´+Ó Â ¼ ¶/¸(°�¹¬¹ ´Cº µP¸�» ´�¼+½1¾ »�²�¸ ´ÃÀ µ�Á;±� ´ »�» ½ µ�¹ Ó À�Å µ+¯b» ´ ²GÆ�Ƕ/¸^Á�¸T¹�¹�¸ ´�½�È ¹�É ´ »!°;Ê Ä µ Ä!´�Ë ¹�¸(Ì ´ ¯0²�¸ Ä&¼,½ Ê+µ�¹ È ¹�É ´ »(° ½�Í6Î�½uË Ê È c
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ng2CG?@M
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(x, y)FD9}E)J01 e ?0?@CG;@FD9034EG1 K@/H3:9<1Pz{F|E)J~143 e J~?Bg
x369<;y369�FD9BEI1:â41:C¯c ð4A<K<K<?@5+1�E)J@34E
nFD5�3XKn?@5)F|EGF �@1QFD9BEI1:â41:C¯c��X1ZEG1:CGMNFD9<1QE)J01V9@A<M�L01:C?Bg�/H3ZE�EGF e 1{Kn?BFD9BEG5VFH905)F2;01�E)J01PCG1:â�F2?@9 |x| + |y| ≤ n
c´ � � � c·#¯�°,±�°�² ´�Ó Â ¼\Ä&Å�´aÔ µ ¼�Å�´ Á.Ê Ä µ � c
æX9~5)F2;01:5AB
3:9<;CD
?Bg CG1 e EG3:9<â�/21ABCD
z{FRE)JAD < AB
�nKn?BFD9BEG5F369<;
E36CG1 e J0?@5+149N5+?�E)J@34E
AFCEFD5X3�CIJ0?@M�LBA<5:c
� 3Z��àqgAB = 16
369<;BC = 12
�<;01ZEG1:CGMNFD9<1EF
c� L6�~àqg
AB = x369<;
BC = y�0;<1�EI14CIM�FH9<1
EFFD9 EI14CIMN5�?Bg
x369<;
yc
8 9�: <?>1@ �\DIH������,D��?J
´ è{è c #¯�°,±�°�² ´�Ó Â ¼�� Ç� XµP» Ä(´ ¯�� ¼ ¹�� Å�½�ÈsÄ&Å�´ ¹�² ½� È�½uË Ê È c
@?BA<C�CG1 e E�369<â�/H1:5�3:CG1X3:CIC�3:9<â416;�FD9>3X5+�BA<36CG1VK<3ZE)�EI14CI9�57? E)J@34EQE)J01+=�149 e /2?@571�3S57MN3:/|/214CV57�@A<3:CG1<c �:1�ESL<1>E)J01P3:CG1:3S?BgnE)J01�?BA@EG1:CQ57�@A<3:CG1P369<;
QE)J01P36CG143S?BgE)J01SFH909<14C"57�@A<3:CG1<cPàqg
S/Q = 9 + 4√
5��;<1�EI14CIM�FH9<1E)J01PC�34EGFH?N?Bg<E)J01P5(FH;<145"?BgnE)J01{CG1 e E�369<â�/H1:54c
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x > yc��J0149~E)J01}?BA@EG1:C�57�@A<3:CG1}J@3:5>5(FH;<1S/2149<â¯E)J
x + y3:9<;~E)J01 FD909<1:C�5+�BA<36CG1NJ@365�5(FH;<1/2149<â¯E)J
x − yc ��J01�â�F �@1:9NC�3ZEGF2?
S/Q = 9 + 4√
5e 369 E)J0149�L01PCG14K<CG145+149BEI1:;}3:5
(x + y)2
(x − y)2=(2 +
√5)2 c
ð4FH9 e 1x > y
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x
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x3:9<;
yF|g
x2 + y cos2 α = x sin α cos α3:9<;
x cos 2α + y sin 2α = 0c
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y3:CG1P9<?@9<���(1:CG?`c �
¾ ²�² ´ ¹ Ä ¸^µ,»e» ¼aÄ&Å�´ ²6µ+Á ´ ²�°+» Ò�Ä ¸!°�¹\ ¼�À µN° Ô ¸^¹ Å � Ò µ+¹�É ½�� É Ò�¼�´ ¹ º ¸^¹ Å �3Å ¸ ´ Á��s¸^É ÅÊ � Å °�°+» ½ ! ¸T¹ Å �N°�¹�É ½ ! ¸ ´�Ä ¹¬µ+Á$�3µ+¹ ÓIÍ ¸ Ä�Ò�> Ì+°�¹�µ�¯ Ò,½�À °�Á¿³µ+¹ ´(?² Ä ¸ ½ Æ�°�Á1µ+¹�¸^µ¬Ç�"5(FD9<â�E)J01PF2;01:9BEGFREGF2145
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x2 + 12y(1 + cos 2α) = 1
2x sin 2α
�369<;
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� �^ï �x cos 2α + y sin 2α = 0
c �q� �ð6?B/R�BFH9<â8E)JBFD5"5�=@5)EG1:M ?Bg`1:�@A<3ZEGF2?@905Vgq?@C
sin 2α369<;
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sin 2α =x(2x2 + y)
x2 + y2
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x2 + y2
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x3:9<;
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sin2 θ + cos2 θ = 1�B/2143�;@5QEI?
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y2(2x2 + y)2
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x 6= 0c
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�
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ð61ZE�EGFD9<âa = tan−1
(12
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(14
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(113
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2
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4
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tan(x + y) =tan x + tan y
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E�z{F e 1@�:z�1�?0L�EG34FH9
tan((a + b) + c
)=
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1 − tan a + tan b
1 − tan a tan btan c
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=12
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− 12
· 14
· 113
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· 14
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· 113
− 12
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4
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a+ b+ c ≥ 3abcc.<CG?:�@1�E)J@34E
a2 + b2 + c2 ≥ 2abcc
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(a + b + c)2 ≥ 9abc�05(FD9 e 1
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a2 + b2 + c2 − 13(a + b + c)2
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)
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= 23
(12
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))
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3(a + b + c)2 ≥ 1
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3:9<;1 + z2 = (z + x)(z + y)
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cð4FH9 e 1
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16 × 2 = 32MNFD9@A@EI145I�X369<;�J@3��@FD9<âN�VA0FH9 e =�M�?4�@1N3:/|/?Bg�E)J01P/D36CGâ41SL0?�ä61:5 � E�3 �:FH9<â
10 × 5 = 50M�FH9@A@EG1:5G�7cV��JBA<5Z�@E)J01+= z�?BA0/H;N�<9@FH57J}3ZE
� ç �6� 3ncRM�cO��369S=@?BA}5+1:18z�J6= E)JBFD5XMNFD9@FHM�F��(145QE)J01 Ä ° Ä µP» %n°�¯+'�¸^¹�É Ä ¸TÁ ´ Nà^9BEI14CG1:5IE)�FH9<â�/ =��:E)JBFD5V;<?0145Q9<?�E¥9<1 e 1:5(573:CGF|/R=�M�14369PE)J@34E F|EOâ�FR�@145OE)J01>1:3:CG/|F2145)E¥1:9<;PEGFHM}18A<5)FH9<âE)J01�â�FR�@149NCGA0/H1:54c��JBFD/H1NE)JBFD5�;@F �@FD5(FH?@9á?Bg�/D3�L0?BA<C�FH5{g23:5)E8FH9á?@9<1�571:90571@�`E)JBFH5�;<?0145P9<?�E>5+1:1:M?@K@EGFDM}34/`FD9�?�E)J01:C�54c>�`1Z/DFD9<1 M}?:�@145>L0?�ä61:5�gq?@C
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�03��61}3}gq1¯z MNFD9@A@EI145>EI?�57161�F|g =@?BA e 369½;<?�L01ZE�EG1:C�E)J@3:9 � ç ò0� 3ncRM�c ��JBFD/H1=@?BA`� CG1�;<?BFD9<âQE)JBFH5���àG� /|/:EIC^=�EI?{;@FD5)EIC�3 e E�=@?BAP38LBF|E�L:=PK<?BFH9BEGFD9<â"?BA@E�E)J@34E�E)J01:CG1"M�F2â4J�EL<1�57?@M�1SA<90CG1:34/DFD5)EGF e E)JBFD9<âZ5�FH9�E)JBFH5�K<CG?0LB/214M½c d EXE)J01�573:M�1SEGFDM�1B�nE)J0?BAnâ4J���E)JBFH5K<CG?0LB/214M ;<?0145O571614M%EG?�L01�M}?@CG18?Bg@3��+CG143:/�/|FDgq1��QK0CG?0L@/H1:M%E)J@369P5+?@M}1�?BgBE)J01�?@9<1:5E)J@34E¥z�1P1:9 e ?BA<9BEI14C�c � à�z{FD/|/n573�=�M�?@CG1{3�L0?BA@EOE)JBFH5X34E¥E)J01�149<;�c ��QF2; =@?BA�;0?�L<1�E�EI14CQE)J@3:9 � ç ò0� 3ncRM�c�½à^9}g23 e E+� � ç ò<ñ 3<cRM½c<FD5�Kn?@5(5(FHL@/H1<c�ð4A<K<�Kn?@571�E)J@34ES��14/|FH9<1�M�?4�@1:5
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15(M}34/D/<L<?�ä{369<;
8/D36CGâ41L<?�ä6145Z�6z�JBF e J�z{F|/D/0EG3��61
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(2x + 6y) + (98 − 3x − 5y) = 98 + y − xc
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2x + 6y ≤ 423:9<;
98 − 3x − 5y ≤ 42� ?@C
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�n = 0, 1, 2, . . .
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p > 2�OE)J01:CG1~FH5 3�5716�BAn1:9 e 1�573ZEGFH5)g|=@FD9<â}E)J01e ?@9<;BF|EGFH?@9�FD9NK<3:C�E � 3��¥3:9<;}J@3Z�BFH9<â
p3:5X57MN3:/|/2145)E�K<1:C�F2?0;`c
��ñ
� 1)ä:E+�nz�1SEGA<CI9~EI?�57?B/DA@EGFH?@905�g2CG?@M�?BA<C8CG143�;01:C�5�EG?�K0CG?0L@/H1:MN5Pâ�F �@1:9�FH9~E)J01d K<C�FD/ �6�6� � 9@A<M�L01:CS?BgVE)J01 À °�¯V¹ ´ ¯7�Q5)E�36C�EGFH9<âNz{FRE)J½E)J01���� æV/|FHMNK0FD3�;BF�à2EG34/DFD369<1�X14/|/H3 � 3ZEI14M}3ZEGF e 30�0��1:571:903ZEGF e ?�� � � 3�= �:���Zò �0â�FR�@149N34E ) �6�6� � ç ïZò�� � ï��:� �BcÑ c 7 143�;BFH9<â�E)J01�EI14M}K<1:C�3ZEGA<CG1:5QFH9���1:571:903ZEGF e ?�gq?@C E)J01>M}?@9BE)J@5X?Bg �X1 e 14M�L01:CO3:9<;�(369@A<3:C^=���ðZEI1Zg2369<?�9<?�EGF e 1458369�?0;0;�gq1434EGA<CG1@çX?@9~1:3 e J~;03�=�FD9}E)J@34E"K<1:C�F2?0;���1)ä e 14K@Egq?@C�E)J01P�nC�5IEX3:9<;�E)J01P/H3:5)E+�BE)J01P/2?6z�145)EOEI14M}K<1:C�3ZEGA<CG1{zP3:5XE)J01�5)A<M ?Bg�E)J01�/H?6z�1:5IEEI14M}K<1:C�3ZEGA<CG1:5�?@9 E)J01�;@3Z=NL<1Zgq?@CG1P3:9<; E)J01�;@3Z=�3:gDEG1:C¯c��J01V/2?6z�145)E<EI14M}K<1:C�3ZEGA<CG1QzP3:5
5◦ � ?@9 �X1 e 1:M}L<14C ñ 369<; 2◦ � ?@9��(369@A<3:C^= ñ�ï c�FD9<; E)J01{/H?6z�1:5IE¥EI14M}K<1:C�3ZEGA<CG1S?@9 �X1 e 14M�L01:C ��� cÊ�°+»(Ì ´�Ó Â ¼ ) °�µ�¹�¹�¸�² � µ Ä ²g¸ '�¸�² ½�ÈsÄ&Å�´ ¹�² ½ � ¯ ´+´ � ´ �nµ+¹ Ó©Í ¸ Ä�Ò�> Ì+°�¹¬µ+¯ Ò+½�À °�Á ³µ�¹ ´(?² Ä ¸ ½Æ�°�Á1µ+¹�¸^µ¬Ç� ´ É,¸(Ì ´�> Ì+°�¹�µ�¯ Ò Ð ²�%u¯b¸ Ä!´�Ï6Ò ±�Ç
�:1ZEt1�t2�. . .�t62
L<1 E)J01N/H?6z�1:5IE"EI14M}K<1:C�3ZEGA<CG1:5{gq?@C"E)J01 �B� ;03�=05>FH9�E)J01M�?@9BE)J@5"?Bg �X1 e 1:M}L<14CQ369<; �(369@A<3:C^=�c��1 ��9<?6z E)J@34E
t3 = 5369<;
t62 = 2c @?@C
i = 1�2�
. . .�59z�1�J@3��@1
ti+1 = ti + ti+23:9<;
ti+2 = ti+1 + ti+3�nJ01:9 e 1
ti = −ti+3c�à2EVgq?B/D/H?6zP5XE)J@3ZEgq?@C
i = 1�2�. . .�56��z�1PJ@3��@1
ti = ti+6c ��JBA<5Z�
t26 = t32 = t38 = t44 = t50 = t56 = t62 = 2�
t24 = t18 = t12 = t6 = −t3 = −5c
�P149 e 1B�BE)J01>/2?6z�145)E¥EG1:MNKn14C�34EGA<CG1�?@9 �X1 e 1:M}L<14C ��� FH5t25 = t24 + t26 = −3◦ �Oc
è c��:1ZEr3:9<;
sL<1QE�z�?�K<3:C�3:/|/21Z/:/|FH9<145�FD9�E)J01XK0/D369<1B�4369<;
P3:9<;
QE�z�?�Kn?BFD9BEG5�5(A e JE)J@34E
P ∈ r3:9<;
Q ∈ scQ�`?@905)F2;01:C e FHC e /H1:5
CP3:9<;
CQ5)A e J E)J@34E
CPFD5VEG3:9<â41:9BEEI?
r3ZE
P�CQ
FD5�EG3:9<â41:9BEOEG?s3ZE
Q�@3:9<;
CP369<;
CQ3:CG1�E�369<â4149BEV1(ä4EI14CI9034/D/ =SEG?1:3 e J�?�E)J014CV3ZEV5+?@M}1PKn?BFD9BE+�<5(3Z=
Tc 6FH9<; E)J01P/2? e A<5�?Bg
Tz�J0149
(CP , CQ)�B3:CGFH1:5?:�@14CQ34/D/nK03:FDC�5�?Bg e FDC e /2145�z{F|E)J�E)J01�â�FR�@149NK<CG?@K<1:C�EGF2145:c
Ê�°+» Ò�Ä ¸(°�¹I ¼ ) °�µ�¹�¹�¸�² � µ Ä ²�¸1'�¸^² ½�ÈsÄ&Å�´ ¹�² ½ � ¯ ´+´ � ´ Ç�:1ZE
KHL<1XE)J01 e ?@MNM�?@9{E�369<â4149BEO?BgE)J01 e FDC e /2145
CP3:9<;
CQc�à2E�FD5�?0L:�BF2?BA<5 E)J@3ZE�@A<36;0C�FD/D34EG1:C�34/H5
PO1TH3:9<;
QO2TK36CG1 e = e /|F e ��369<;�E)J01}g23 e E�E)J@34E∠K = ∠HM�1436905QE)J@34E
∠PO1T = ∠TO2Qc��JBA<5�� 4PO1T ∼ 4TO2Q� 5(FDMNF|/H3:CEICGFD369<â�/H1:5G�)��369<;�E)J01 g23 e E"E)J@3ZE
O1TO2FH5�35)EIC�34F2â4J�E�/|FH9<1SEG14/|/H58A<5�E)J@3ZE
PTQFH5�3:/D5+?~35)EIC�34F2â4J�E�/|FH9<10c
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H
K
O1
O2
P
Q
T
r
s
CP
CQ
�`?@905716�BAn1:9BEG/ =��TL014/H?@9<âZ58EI?�E)J01 e ?@905IEG3:9BE�/DFD9<1
PQc d /H57?��nz�1Nâ41ZE�g2CG?@M
E)J01P3�L0?:�@1{5(FDMNF|/H3:CGFRE�=�?BgnEICGFD369<â�/H1:5QE)J@3ZEPT = PQ · R
R + r
c� c � 3�� �X1�EI14CIM�FH9<1�z�J01�E)J01:CXE)J01�9@A<M}L<14C
20052004 e 369�L<1�zPC�F|E�EG1:9�365�E)J01�5(A<M?BgnE)J01P57�@A<3:CG1:5"?BgnE�z�?�K<?@5(FREGFR�@1>FD9BEI1:â41:C�54c� L6� �X1ZEG1:CGMNFD9<1�z�J01�E)J01:CVE)J01�9@A<M}L<14C
20042005 e 369�L01PzPCGFRE�EI149�365�E)J01 5(A<M?BgnE)J01P57�@A<3:CG1:5"?BgnE�z�?�K<?@5(FREGFR�@1>FD9BEI1:â41:C�54c
�4ò
À °�ÁIÂ+¸T¹ ´�Ó ²�°+» Ò�Ä ¸!°�¹ Ä ° � 3Z��µ+¹ Ó � L��E ¼ Æ�Ç �+µ Ò Á ´ ¹ ½ ¶ ´�Ò ¯V¹ ´�½�º�´ »^ÉP¸ Ò Á Çd 903ZEGA<C�3:/�9@A<M�L01:C
nFH5VE)J01S5)A<M ?Bg�E�z�?N57�@A<3:CG1:5"F|g�3:9<;~?@9@/ =NFDg�E)J01SK0CGFDM�1g23 e EG?@CGF��)34EGFH?@9�?Bg
n;0?01:589<?�E e ?@9BE�3:FD9~369�=~K0CGFDM�1 ?Bg`E)J01Sgq?@CGM
4k + 3EG?�369�?0;0;Kn?6z�14C ) 57161 ��c�ð4FH1:CGK0F)�905��:F�� ¾ » ´ Á ´ ¹ Ä µ+¯ ¼�Í+Å�´ °�¯ ¼ °�� ��Ò ÁI ´ ¯0²Z�@�>3:g29<14C+� � � ï+���Zò �K c ñ6��ï �@c
��1�J@3��@1"E)J018K0CGFDM�18K<?6z�1:C`g23 e EG?@CGF��)34EGFH?@90520052004 = 52004 ·4012004 3:9<;
20042005 = 24010 · 32005 · 1672005 �<57?Sz�1 e ?@9 e /DAn;01PE)J@3ZE 20052004 FH5VE)J01�5(A<M?BgnE�z�?�57�@A<3:CG1:5��@369<; E)J@3ZE20042005 FD5X9<?�EOE)J01P5)A<M ?Bg<E�z�?�57�@A<3:CG1:54c
� c �:1ZEPL<1S3�K<?BFH9BEXFH905)F2;01�E)J01�EIC�FH3:9<â�/21
ABCc{ð:3�=}E)J@34EVE)J01S/|FH9<145
AP�BP
�369<;CP
M}161�EOE)J01P5)F2;01:5"?BgABC
34EA′ � B′ �@3:9<; C′ �@CG1:5(Kn1 e EGFR�@1Z/R=`c �:1ZE
x =AP
PA′�
y =BP
PB′�
z =CP
PC′c
.<CG?:�@1�E)J@34Exyz = x + y + z + 2
cÊ�°+»(Ì ´�Ó Â ¼$�P´ µ�¹ Ï6À »^µ ÒNÓ¬´�È ¹ Ó ¯b¸ ´�Ò���½nº�´ µ Ò ¹ ´�½�� ¯bµ�¹/� ´ � Ô ¸�� Å�´ » º µ Ä µ,¸e»�» ´�½ Æ�° Ò�´ ¹ ½� ¯bµ�¹/� ´ � � ´ ° � ¯ ´�¼�È Ç � µ+¹ Ó µP»�» ½ �~µ�Á Ó¬´ ¹ ½�À3Í6½�Ë Ê È � ) °�µ+¹¬¹N¸^² � µ Ä ²g¸ '�¸�² ½EÈsÄ&Å�´ ¹�² ½� ¯ ´+´ � ´ �¿¶�Ç � Ç;Ê+Á ´�´ ¹(' ½ > µ,» Ä Â�°�Á1Á ´ » ½=Ä&Å�´ �X´�Ä&Å�´ ¯b»^µ�¹ Ó ²4�Cµ+¹ Ó¡Í ¸ Ä�Ò > Ì+°�¹¬µ+¯ Ò+½À °�Á ³µ�¹ ´(?² Ä ¸ ½ Æ�°�Á1µ+¹�¸^µ¬Ç ´ �#¯0² Ä ÉP¸!Ì ´aÄ&Å�´ ²�°+» Ò�Ä ¸!°�¹�°�� È ¹ Ó ¯b¸ ´NÒ�� Ç
�V36905�/D3�5)A0F|EG1@�8?@9 9<?�EG1:C�3[UV W ]/�� 3:FDCG1�;�� A<9�EIC�FH3:9<â�/21
UV Wc0æX9N3
PA
PA′ =[PAB]
[PA′B]=
[PAC]
[PA′C]
=[PAB] + [PAC]
[PA′B] + [PA′C]
=[PAB] + [PAC]
[PBC]
c
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A
B CA′
B′
C′P
�8� ?OíA`�01:9NK<?@573:9BESA = [PBC]
�Sb = [PCA]
1�ESc = [PAB]
çx =
PA
PA′ =SC + SB
SA
�y =
PB
PB′ =SA + SC
SB
�1ZE
z =PC
PC′ =SB + SA
SC
cæX9}3P34/2?@C�5>çxyz =
SC + SB
SA
× SA + SC
SB
× SB + SA
SC
=2SASBSC + S2
ASC + S2CSA + S2
CSB + S2BSC + S2
BSA + S2ASB
SASBSC
= 2 +SA + SC
SB
+SC + SB
SA
+SB + SA
SC= x + y + z + 2
c
���
�X´ �¬ÄT½ % ´ ÉP¸!Ì ´�Ä&Å�´ µ+¯bÉ Ò Á ´ ¹ Ä °�� º µ Ä µ,¸e»�» ´ Ç�X14�<9<1 K<?@5(FREGFR�@1½CG1:34/89@A<M�L01:C�5
α�
β�8369<;
γL:=
α + β + γ = 13:9<;
P = αA + βB + γCc{��J01:9
P − αA = βB + γC = (1 − α)A′ � J01:9 e 1@�α
−→PA + (1 − α)
−−→PA′ =
−→0cQà2E�gq?B/|/2?6zP5�E)J@34EAP
PA′ = x =1 − α
α
cð4FHM�FD/D36C�/R=��
y =1 − β
β
3:9<;z =
1 − γ
γ� J0149 e 1B�
xyz =(1 − α)(1 − β)(1 − γ)
αβγ369<;x + y + z + 2 =
1 − α
α+
1 − β
β+
1 − γ
γ+ 2
c��JBA<5��@FRE�FD5X5(A�� e F2149BE�EG?�57J0?6z�E)J@3ZE(1 − α)(1 − β)(1 − γ) = βγ(1 − α) + γα(1 − β) + αβ(1 − γ) + 2αβγ
c7 1 e 3:/|/DFD9<â�E)J@3ZE
α + β + γ = 1��E)JBFH5VF2;01:9BEGFRE�= FH5"14365)FD/ = e J01 e �61:;�c
� 1)ä:E+�Bz�1{EGA<CI9NEI?N5+?B/|A@EGF2?@905VEI?NK<CG?0LB/214M}58?Bg�E)J01 ï�� ��� à^C�FH57J � 34E)J014M}3ZEGF e 3:/æV/R=@M}K@FH36;�� 6FHC�5IE8.B3:Kn14C+�0â�FR�@149N34E ) �6�6� � ç ï��:� �BcÑ c � 3���B?@C"z�JBF e J�K<?@5(FREGFR�@1�FD9BEI1:â41:C�5
n;<?0145
2n;@F �@FH;<1 E)J01}5(A<M ?Bg¥E)J01N�<C�5)E
nKn?@5)F|EGF �@1>FH9BEG16â414C�5�� L6� �X1ZEG1:CGMNFD9<1@�@z{F|E)J}K<CG?0?Bg��0E)J0?@5+1SK<?@5(FREGFR�@1�FD9BEI1:â41:C�5
n� FDg 3:9�=�� z�JBF e J~J@3��@1E)J01PK<CG?@K<1:C�E�= E)J@3ZE
2n + 1;@F �@FH;<145VE)J01{5(A<M ?BgnE)J01{�<C�5)E
nKn?@5)F|EGF �@1>FH9BEG16â414C�5:c
Ê�°+»(Ì ´�Ó Â ¼ � ´ ° � ¯ ´�¼�È Ç � µ�¹ Ó µ,»e» ½ �~µ+Á Ó¬´ ¹ ½�À3ÍR½�Ë Ê È � ) °�µ+¹¬¹N¸^² � µ Ä ²�¸1'�¸^² ½#ÈsÄ&Å�´ ¹�² ½� ¯ ´+´ � ´ � ¾�Ó %uµ�¯ ÓÃÍ Ç �;Ç Xµ�¹�É ½ ~¸�» �&¯b¸ Ó �+µ Ò ¯b¸ ´ ¯ Ë ¹�¸(Ì ´ ¯0²�¸ Ä&¼,½ Xµ Ä!´ ¯b»!°�° ½ <�� ��µ�¹ ÓÍ ¸ Ä�Ò > Ì+°�¹�µ�¯ Ò,½�À °�Á ³µ�¹ ´(?² Ä ¸ ½ Æ�°�Á1µ+¹�¸^µ¬Ç ´ É,¸(Ì ´ � µ�¹ Ó µ,»e»VÐ ²E²P°+» Ò�Ä ¸(°�¹�Ç
�:1ZESn = 1 + 2 + · · · + n =
n(n + 1)
2
�@3:9<;�/H1ZEPL01�E)J01P571ZEQ?Bg�Kn?@5)F|EGF �@1FH9BEG16â414C�5:c
� 3�� ��18J@3Z�@12n | Sn
FDg03:9<;S?@9@/ =PF|g2n∣∣
n(n + 1)
2
�Zz�JBF e J�J0?B/H;05¥F|g0369<;�?@9@/R=FDg
n(n + 1) = 4npgq?@CO5+?@M}1
p ∈ P�B3:9<;SE)JBFH5QFD5�EICGAn1�FDg�3:9<;N?@9@/ =SFDg
n + 1 = 4pgq?@CQ57?@M�1p ∈ P
c� L6� ��1 e /H34FHM E)J@3ZE
(2n+1) - Sngq?@C 369�=
n ∈ Pc��0?S57161"E)JBFH5���5)A<K<K<?@5+1 � EG?
E)J01 e ?@9BEIC�36C^=6��E)J@3ZE(2n +1) | Sn
� E)J@3ZE¥FD5Z�65(A<K0Kn?@571�E)J@3ZE n(n + 1)
2= (2n +1)pgq?@C>57?@M�1
p ∈ Pc���J0149
n(n + 1) = 4np + 2p � z�J01:9 e 1@� 2p = np1z�J014CG1
p1 = n+1−4p ∈ P�45)FH9 e 1
4p = 2n(n+1)/(2n+1) < n+1c¥�`?@905716�BAn1:9BEG/ =��
n + 1 = 4p + p1 = 2np1 + p1 ≥ 2n + 1�@3 e ?@9BEIC�36;@F e EGF2?@9�c
���
è c�î@3 e JS?BgBE)J018K0/D3Z=@14C�5�FD9S3�EG1:909@FD5¥EI?BA<CG9036M}1:9BE¥K@/H3�=@16;S?@9<18M}3ZE e J�36âZ3:FD905)E¥1:3 e J?Bg E)J01�?�E)J01:C�54cNàqgO1+�@1:C^=�K@/H3�=@1:CVz�?@9�3ZE"/214365IE8?@9<1�MN34E e J��`5+J0?6zªE)J@34EXE)J01:CG1 FD5>3âZCG?BA<KA�B�C?Bg�E)J@CG161�K@/H3�=@1:C�5¥gq?@C�z�JBF e J
AL01:3ZE
B�BL<1434E
C�:369<;
CL<1434E
Ac
Ê�°+» Ò�Ä ¸(°�¹I ¼�Í ¸ Ä�Ò > Ì+°�¹¬µ+¯ Ò+½�À °�Á¿³µ+¹ ´(?² Ä ¸ ½ Æ�°�Á1µ�¹N¸Tµ�Ç�:1ZE
P1�P2�. . .�PnL01�E)J01{K0/D3Z=@14C�5Z�B369<; /H1ZE
biL<1�E)J01{9@A<M}L<14CV?Bg�K@/H3�=@1:C�5
PiL<1434E�c ��1 ��9<?6z E)J@3ZE
bi ≥ 1c ��1 e 3:9~365(5(A<M}1 � L:=~CG1Z/H36L<1Z/D/|FH9<â{FDg�9<1 e 14575(36C^=6�E)J@34E
b1FD5�M�FH9@FDMNA<M�3:9<;�E)J@34E
P1L<1434E
P2�P3�. . .�Pb1+1
c ð4FD9 e 1b1FH5`MNFD9@FHM�A<M��E)J01:CG1"FH5¥38K0/D3Z=@14C
Pk5)A e J>E)J@34E
P2L<1434E
Pk3:9<;
k > b1 +1� ?�E)J014C�z{FD5+1
b2 < b1�(c��J0149
PkL<1434E
P1�@3:9<;Sz�1 e 3:9 EG3��61
A = P1�B = P2
�@369<;C = Pk
c� c �:1�E
ABL01N3 e J0?@CG;�?Bg¥/2149<â¯E)J
6?BgO3 e FHC e /H1}?BgOC�36;@F|A<5
5e 1:9BEICG1:;�3ZE
Oc �:1�E
PQRS;<149<?�EI1�E)J01>57�@A<3:CG1�FD905 e C�F2L016;�FH9�E)J01�5+1 e EI?@C
OAB5)A e JPE)J@34E
PFH5V?@9�E)J01C�3�;BFDA<5
OA�SFH5"?@9SE)J01{C�3�;BFDA<5
OB�@3:9<;
Q369<;
R36CG1{K<?BFH9BE�5"?@9 E)J01{36C e ?BgnE)J01e FHC e /H1�L01ZE�z�1:1:9
A3:9<;
Bc �FD9<; E)J01P3:CG1:3�?Bg
PQRSc
Ê�°+» Ò�Ä ¸(°�¹I ¼ � ´ ° � ¯ ´�¼~È Ç � µ�¹ Ó µ,»e» ½ �~µ�Á Ó¬´ ¹ ½�À3ÍR½uË Ê È Ç�`?@905IEICGA e E"Kn?BFD9BEG5
F�G��3:9<;
H��365�FH9}E)J01�;@FD3�âZC�3:M½c{ð4FH9 e 1
OA = 53:9<;
AG = 3�Oz�1�57161�E)J@34E
OG = 4�Qg2CG?@M E)J01���J016?@CG14M�?Bg{.B=�E)J@3�â4?@C�3:5:cÇð4FH9 e 1
4OPF ∼ 4OAG�BF|EOgq?B/D/H?6zP5QE)J@3ZE
PF = 3t3:9<;
OF = 4tgq?@CQ57?@M�1
tc ��J0149
OH = OF + FH = 4t + 6t = 10t369<;QH = 3t
cµà2E}9<?6z gq?B/D/H?6zP5�g2CG?@M E)J01��J01:?@CG1:M ?Bgá.B=BE)J@36â4?@C�365 � FH9 4OQH��E)J@34E
(10t)2 + (3t)2 = 25cQ�P149 e 1B�
t2 = 25109
3:9<;[PQRS] = (6t)2 =
900
109
c..................
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.............................................................................................................................................................................................................................................................................................................................................
..................... . . . . . . . . ...........
..................... . . . . . . . . ...........
..................... . . . . . . . . ...........
A B
F
G
H
O
P
Q R
S
� cX.<CG?:�@1�E)J@34EOE)J014CG1P36CG1P?@9@/R=SE�z�? CG1:34/�9@A<M}L<14C�5x5(A e JSE)J@3ZE
(x − 1)(x − 2)(x − 3)(x − 4)(x − 5)(x − 6) = 720c
Ê�°+»(Ì ´�Ó Â ¼�È ¯+'�µ Ó�¼�È » ÄT½ Ê�µ+¹ � °�² ´�½~À�È�½aË Ê È � Ô ¸,� Å�´ » º µ Ä µ,¸e»�» ´�½ Æ�° ÒN´ ¹ ½�� ¯�µ+¹�� ´ �� °�²�³´ � Ò ¸�²©¶s³G µ8� Ï�º µ�¯V¯ ´ ¯g° ½ Ë ¹�¸(Ì ´ ¯0²�¸ Ä µ Ä #°+»�¸ ÄH´ ��¹N¸��0µ Ó¬´ À µ Ä µ,» Ò ¹ ¼ µ ½Gº µ+¯4� ´ »(°�¹¬µ ½ÊP±�µP¸T¹ � � ´ ° � ¯ ´�¼ È Ç � µ�¹ Ó µ,»e» ½ �~µ+Á Ó¬´ ¹ ½dÀ3Í6½yË Ê È � ) °�µ�¹�¹�¸�² � µ Ä ²g¸ '�¸�² ½~ÈsÄ&Å�´ ¹�² ½� ¯ ´+´ � ´ �y¶�Ç � Ç�Ê�Á ´+´ ¹ ' ½&> µ,» Ä Â�°�Á1Á ´ » ½\Ä&Å�´:�X´�Ä&Å�´ ¯b»^µ�¹ Ó ²4� ¾�Ó %uµ�¯ Ó Í Ç �;Ç Xµ�¹�É ½ ~¸�» �&¯b¸ Ó �+µ Ò ¯b¸ ´ ¯ Ë ¹�¸(Ì ´ ¯0²�¸ Ä&¼,½ Xµ Ä(´ ¯b»(°�° ½�<�� �aµ+¹ Ó¨Í ¸ Ä�Ò > Ì+°�¹�µ�¯ Ò,½ À °�Á ³µ�¹ ´(?² Ä ¸ ½Æ�°�Á1µ+¹�¸^µ¬Ç� ´ É,¸(Ì ´�Ä&Å�´ ²P°+» Ò�Ä ¸(°�¹y°��3Ê+Á ´�´ ¹('�Ç
ð61ZE�EGFD9<ây = x2 − 7x
��E)J01�â�FR�@149}16�BA<34EGFH?@9�FH5X5)A e)e 1:5(5(F �@14/ =N16�BA0FR�B34/2149BE�EG?(x2 − 7x + 6)(x2 − 7x + 10)(x2 − 7x + 12) = 720,
(y + 6)(y + 10)(y + 12) = 720,
y3 + 28y2 + 252y = 0,
y((y + 14)2 + 56
)= 0.��JBA<5��
y = 03:9<;
x ∈ {0�7} c
���
� c �:1ZEa�b ≥ 0
cX.<CG?:�@1�E)J@34E√
2(√
a(a + b)3 + b√
a2 + b2)
≤ 3(a2 + b2
) �z{F|E)JN1:�@A<34/DFRE�= FDg�3:9<;}?@9@/ =�F|g
a = bc
Ê�°+»(Ì ´�Ó Â ¼�È ¯+'�µ Ó�¼�È » ÄT½ Ê�µ+¹ � °�² ´�½~À�È�½aË Ê È � Ô ¸,� Å�´ » º µ Ä µ,¸e»�» ´�½ Æ�° ÒN´ ¹ ½�� ¯�µ+¹�� ´ �) °�µ+¹¬¹N¸^² � µ Ä ²g¸ '�¸�² ½�ÈsÄ&Å�´ ¹�² ½ � ¯ ´�´ � ´ �~µ�¹ Ó¿Í ¸ Ä�Ò�> Ì+°�¹�µ�¯ Ò,½aÀ °�Á ³µ�¹ ´(?² Ä ¸ ½ Æ�°�Á1µ+¹�¸^µ¬Ç ´ ±�¯ ´ ² ´ ¹ Ä�Ä&Å�´ µP±¬±�¯g°�µ � Å °�� º µ Ä µP¸�»e» ´ Ç
�"5(FD9<â8E)J01���3:A e J6=0�7ð e J6zP3:C ��à^9<16�BA<3:/|F|E^=√
a(a + b)3 + b√
a2 + b2 = (a + b) ·√
a2 + ab +√
a2 + b2 · b
≤√
2a2 + 2b2 + 2ab ·√
a2 + ab + b2 �
J01:9 e 1@�@5(FD9 e 12ab ≤ a2 + b2 �
√2(√
a(a + b)3 + b√
a2 + b2)
≤ 2(a2 + ab + b2)
= 2(a2 + b2) + 2ab ≤ 3(a2 + b2)c
î0�BA<3:/|F|E^= e 34/D/D5�gq?@Ca = b
� ?�E)J01:C�z{FH571@�2ab < a2 + b2 3:9<;}E)J01�FH9<1:�@A<34/DFRE�=FH585IEICGF e E���369<; e ?@9��@1:C�5714/ =���16�BA<3:/|F|E^=�J0?B/H;05�F|ga = b
�n365�FD58CG1:36;@F|/R= e J01 e �61:;�c8��J01K<CG?0?Bg�FD5 e ?@MNK0/H1ZEG1<c
� 1)ä:E+� z�1�EGA<CG9áEI?á57?B/DA@EGFH?@905�g2CG?@M�?BA<CPCG143�;01:C�5�EG?áK<CG?0LB/214M}5 ?BgQE)J01 ï�� ���à^CGFD5+J � 34E)J014M}3ZEGF e 3:/�æV/ =0MNK0FD3�; �6�6�4ò �<ð61 e ?@9<;�.B3:Kn14C+�0â�F �@1:9N34E ) �6�6� � ç ï��6ï �BcÑ c �X1�EI14CIM�FH9<1S34/D/�K03:FDC�5�?Bg�K0CGFDM�1�9@A<M�L01:C�5
(p, q)�@z{F|E)J
2 ≤ p�q < 100
�n5(A e JE)J@34Ep + 6
�p + 10
�q + 4
�q + 10
�03:9<;p + q + 1
3:CG1P3:/|/nK0CGFDM�1P9@A<M}L<14C�5:cÊ�°+»(Ì ´�Ó Â ¼ ) °�µ�¹�¹�¸�² � µ Ä ²g¸ '�¸�² ½�ÈsÄ&Å�´ ¹�² ½ � ¯ ´+´ � ´ �nµ+¹ Ó©Í ¸ Ä�Ò�> Ì+°�¹¬µ+¯ Ò+½�À °�Á ³µ�¹ ´(?² Ä ¸ ½Æ�°�Á1µ+¹�¸^µ¬Ç� ´ É,¸(Ì ´ � µ Ä ²g¸ '�¸�²�Ð %u¯b¸ Ä(´�ÏRÒ ±�Ç
��J01�?@9@/ =�K<34FHC�5X36CG1(p, q) ∈ {(7, 3)
�(13, 3)
�(37, 3)
�(97, 3)} cî6�@14C^=SK0CGFDM�1�;@F���14CG1:9BE g2CG?@M
2?@C
3FD5Q?Bg@E)J01�gq?@CGM
6k+1?@C
6k +5�Zz�J014CG1
kFD5V5+?@M}1>K<?@5(FREGFR�@18FH9BEG16â414C�cOæ"L:�BF2?BA<5)/R=
p 6= 2369<;
p 6= 3�B5(FD9 e 1
p + 6FH5VK0CGFDM�10cð4FH9 e 1
p + 10FH5{3}K<C�FHM}1@�
p 6= 6k + 5 � E)JBA<5Z� p = 6k + 1gq?@C�5+?@M}1NKn?@5)F|EGF �@1FH9BEG16â414C
k ≥ 1cQàqg
q > 6�:E)J018g23 e E�E)J@34E
q +10FH5QK0CGFDM�1�EG14/|/H5�A<5OE)J@3ZE
q = 6λ +1gq?@CQ57?@M�1{K<?@5(FREGFR�@1>FD9BEI1:â41:Cλ ≥ 1
cQ�P?6z�1+�@14C+�6z�18z�?BA0/2;SE)J0149�J@3��@1p + q + 1 = 6k + 1 + 6λ + 1 + 1 = 3(2k + 2λ + 1)
�z�JBF e JSFD5V9<?�E�K0CGFDM�10c ��JBA<5��
q < 6�B369<;�?0L:�@FH?BA<5(/ =
q 6= 2� 5(FD9 e 1
q + 4FD5VK0CGFDM�1¯�7c��J014CG14gq?@CG1B�¯z�1VM�A<5)E�J@3��@1
q = 3c � ?6z z�1VM�A<5)En�n9<;�K0CGFDM�1V9@A<M}L<14C�5
p = 6k+1gq?@C"57?@M�1PFH9BEG16â414Ck�1 ≤ k ≤ 16
��5)A e JNE)J@34Ep + 6
�p + 10
��3:9<;p + 4
36CG1S3:/|/K<C�FHM}1:54c5@=};BFHCG1 e E e J01 e �:FH9<�
k ∈ {1�2�6�16} � E)J@3ZE�FH5�� p ∈ {7
�13�37�97} c
���
è c �:1�EA3:9<;
BL<1N;@FD5)EGFD9 e E8K<?BFH9BE�5{?@9�3 e FDC e /21
Tc �:1ZE
CL01N3NK<?BFH9BE8;@FD5)EGFD9 e Eg2CG?@M
B5)A e J½E)J@3ZE |AB| = |AC| 3:9<; 5(A e J�E)J@34E BC
FH5PEG3:9<â41:9BE{EG?T3ZE
Bcð4A<K<K<?@5+1�E)J@34EOE)J01PLBFH571 e EG?@CQ?Bg
∠ABCM}161�EG5
AC34E�3PKn?BFD9BE
DFH905)F2;01
TcOð6J0?6zE)J@34E
∠ABC > 72◦ cÊ�°+» Ò�Ä ¸(°�¹I ¼�Í ¸ Ä�Ò > Ì+°�¹¬µ+¯ Ò+½�À °�Á¿³µ+¹ ´(?² Ä ¸ ½ Æ�°�Á1µ�¹N¸Tµ�Ç
�:1ZEOL<1�E)J01 e 149BEICG1 ?Bg
T�X/H1ZE
AB = AC = b369<;
BC = a�"3:9<;/21�E
∠ABC = αc ��1�E)J01:9jJ@3Z�@1
cos α =a
2b
�∠ABO = 90◦ − α
�83:9<;∠OBD = 90◦ − α
2
c"à^9�E)J01{FH57?@5 e 14/H1:5QEICGFD369<â�/H1AOB
��z�1PJ@3��@1OB =
AB
2 cos ∠ABO=
b
2 sin α
cà2E{FH5 �69<?6zP9½E)J@34E
BD =2ab cos α
2
a + b
3:9<;½E)J@3ZEsin 18◦ =
−1 +√
5
4
c ��J01 �43�z ?BgS��?@5(FD9<1:5â�FR�@145
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..
..
.................A
B
C
D
O
OD2 = OB2 + BD2 − 2 OB · BD · cos(
90◦ − α
2
) c��JBA<5��
OD2 < OB2 F|g�3:9<;~?@9@/R=NF|g BD2 < 2 · OB · BD · sinα
2� E)J@3ZEVFH5�� D FD5
FH905)F2;01TF|g�369<;N?@9@/R= FDg
2ab cos α2
a + b< 2 · b
2 sin α· sin
α
2
c
àqgnz�1 e 369 e 14/2b�6zPCGFREI1
sin α = 2 sinα
2cos
α
2
�@369<;a = 2b cos α
�6z�1Pâ41ZE
2 cos α cos α2
2 cos α + 1<
1
4 cos α2
c
��JBFD5{â�FR�@1454 cos α · 2 cos2
α
2− (2 cos α + 1) < 0
�nz�JBF e J���L:=�1:MNK0/H?:=BFH9<â�E)J01F2;01:9BEGFRE�=
2 cos2α
2= 1 + cos α
�<FH5816�BA0FR�B34/2149BEVEI?4 cos2 α + 2 cos α − 1 < 0
c��J014CG14gq?@CG1B�
cos α ∈(
−1 −√
5
4,
−1 +√
5
4
)
� E)JBA<5��
cos α <−1 +
√5
4= sin 18◦ c
�P?6z�1+�@14C+�sin 18◦ = cos 72◦ � J01:9 e 1@� α > 72◦ 5(FD9 e 1 α < 90◦ c
���
À °�Á1Á ´ ¹ Ä c d K<CG?0?BgnE)J@3ZEsin 18◦ =
−1 +√
5
4
gq?B/D/H?6zP5:c ��1PJ@3Z�@1sin 18◦ cos 36◦ =
2 sin 18◦ cos 18◦ cos 36◦
2 cos 18◦ =sin 36◦ cos 36◦
2 cos 18◦
=sin 72◦
4 cos 18◦ =cos 18◦
4 cos 18◦ =1
4
c �^ï ���1P34/H57? 5+1:1>E)J@3ZE
sin 18◦ − cos 36◦ = cos 72◦ − cos 36◦
= −2 sin72◦ + 36◦
2sin
72◦ − 36◦
2
= −2 sin 54◦ sin 18◦ = −2 sin 18◦ cos 36◦
= −1
2
c �^� �BCG?@M �^� �)�6z�1�â41�E
cos 36◦ = sin 18◦ + 12
c ��J0149��@A<5)FH9<â �qï �)�6z�1{57161�E)J@34Esin 18◦
(
sin 18◦ +1
2
)
=1
4
cð4FH9 e 1
sin 18◦ > 0�6z�1P?0LBE�3:FD9
sin 18◦ =−1 +
√5
4
c� c8ð4A<K<K<?@5+1
nFD5�369}FH9BEG16â414C ≥ 2
c �X1ZEG1:CGMNFD9<1PE)J01��nC�5IE";@FHâ�F|EX34gDEI14CQE)J01 ;<1 e FHMN3:/Kn?BFD9BE�FH9 E)J01P;<1 e FHMN3:/�1(ä6K<3:905(FH?@9}?BgnE)J01P9@A<M}L<14C 3√
n3 + 2n2 + nc
Ê�°+»(Ì ´�Ó Â ¼ ) °�µ�¹�¹�¸�² � µ Ä ²g¸ '�¸�² ½�ÈsÄ&Å�´ ¹�² ½ � ¯ ´+´ � ´ �nµ+¹ Ó©Í ¸ Ä�Ò�> Ì+°�¹¬µ+¯ Ò+½�À °�Á ³µ�¹ ´(?² Ä ¸ ½Æ�°�Á1µ+¹�¸^µ¬Ç� ´ É,¸(Ì ´ � µ Ä ²g¸ '�¸�²�ÐP²P°+» Ò�Ä ¸(°�¹�Ç��1PK0CG?:�@1�E)J@34EOE)J01{�<C�5)EQ;@FHâ�F|EQ34gDEI14COE)J01P;<1 e FHMN3:/nKn?BFD9BE�FH5
6c
�FDC�5)EQ?Bg�3:/|/��:z�1P?0L0571:C^�@1�E)J@34En =
3√
n3 <3√
n3 + 2n2 + n <3√
n3 + 3n2 + 3n + 1 = n + 1c
à2EQFD5X5(A�� e F2149BEOEI? 5+J0?6z�E)J@3ZEn +
6
10< 3
√
n3 + 2n2 + n < n +7
10
c �^ï ���J01{�<C�5)EOFH9<1:�@A<34/DFRE�=}?Bg �^ï �OJ0?B/2;@5Xgq?@C
n ≥ 2�@5(FD9 e 1
(5n + 3)3 < 125(n3 + 2n2 + n) ⇐=⇒ 5n(5n − 2) > 27c
��J01P571 e ?@9<;�FD9<16�BA<3:/|F|E^=}?Bg �qï �OJ0?B/H;05Vgq?@Cn ≥ 2
�@5(FD9 e 1(10n+7)3 > 1000(n3+2n2+n) ⇐=⇒ 100n2+470n+343 > 0
c��JBA<5�� �qï �OJ0?B/H;05Vgq?@C
n ≥ 2�@369<; E)J01>�nC�5IEV;<1 e FHMN3:/�;BF2â�FRE�FH5
6c
� c �X14�<9<1>E)J01>gHA<9 e EGF2?@9m?BgnE)J01�E)J@CG161PCG143:/@�B3:CGFD3�LB/2145
x�y�@369<;
zL:=
m(x, y, z) = max{x2 � y2 � z2} c�X1ZEG1:CGMNFD9<1@�¥z{FRE)JáK<CG?0?Bg��¥E)J01�M�FH9@FDMNA<M �B34/DAn1�?Bg
mFDg
x�
y��369<;
z�B36C^=áFD9 �5(AnL6ì^1 e EOEI?SE)J01{CG1:5IEICGF e EGF2?@905
x + y + z = 0369<;
x2 + y2 + z2 = 1c
���
Ê�°+»(Ì ´�Ó Â ¼�È ¯+'�µ Ó�¼�È » ÄT½ Ê�µ+¹ � °�² ´�½�À�È�½�Ë Ê È ��µ+¹ Ó~Ô ¸,� Å�´ » º µ Ä µ,¸e»�» ´�½ Æ�° ÒN´ ¹ ½ � ¯bµ�¹/� ´ Ç ´ É,¸(Ì ´�º µ Ä µ,¸e»�» ´ Ð ² %u¯b¸ Ä(´�ÏRÒ ±�Ç
��1P57J0?6z�E)J@3ZEOE)J01PCG1:�@A0FDCG16;NMNFD9@FHM�A<M ?BgmFH5 1
2
c�:1ZE
x�y�zL<1{5(A e J�E)J@34E
x + y + z = 03:9<;
x2 + y2 + z2 = 1cO�n/21436C�/R=��
x�y�@3:9<;
z3:CG1P9<1ZF|E)J014CQ34/D/<Kn?@5)F|EGF �@1{9<?@C�3:/|/n9<16âZ3ZEGFR�@1 � J0149 e 1B�BA<K�EI? 3 e J@369<â41P?Bg?@CG;<14CV?@C�5(FHâZ9��6z�1PMN3Z=�5(A<K0Kn?@571�E)J@34E
x ≤ 03:9<;
y�z ≥ 0
c¥��J01:92x2 = x2 + (−y − z)2 = 1 + 2yz ≥ 1
c��JBA<5��
m(x2, y2, z2) ≥ 12
c�<? e ?@M}K@/21�EI1�E)J01�K0CG?0?Bg��Oz�1}ì2A<5)ES?0L@5+14C^�@1~E)J@3ZEPgq?@C
x = −√
22
�y = 0
�z =
√2
2
�6z�1PJ@3Z�@1x + y + z = 0
�x2 + y2 + z2 = 1
369<;m(x2, y2, z2) = 1
2
c
� 1)ä:E+�:z�1{â�FR�@1>5+?B/|A@EGF2?@905OEI?SK<CG?0LB/214M}5X?Bg<E)J01 � 1�z��4143:/D369<; � 34E)J014M}3ZEGF e 3:/æV/R=@M}K@FH36;��nà � æjð6�@A<36;}ð614/H1 e EGFH?@9�.<CG?0LB/214M}5 �6�6�4ò â�F �@1:9N3ZE ) �:����� ç ï6��ï � ï6��� �@cÑ c �:1ZE
IL<1}E)J01}FH9 e 1:9BEICG1�?BgOEICGFD369<â�/H1
ABC�O3:9<;½/H1ZE
A′ � B′ �O3:9<; C′ L<1}E)J01CG1 � 1 e EGFH?@905S?BgIFH9
BC�CA
�¥3:9<;AB
�¥CG1457K<1 e EGF �@14/ =�c~��J01 e FHC e /H1NE)J@CG?BAnâ4JA′ �
B′ �@3:9<; C′ K<3:57571:5X34/H57?SE)J@CG?BAnâ4J Bc �FD9<; E)J01P3:9<â�/21
∠ABCc
Ê�°+»(Ì ´�Ó Â ¼dÔ ¸^É Ò�´ » È Á ´ ¹�É Ò µ,» À °NÌ�µ+² ½EÀ µ,»^µ � ¸�É ÒN´ ¯bµ ½�Ô µP»�»(°�¯4�0µ ½ Ê�±�µ,¸^¹ � � ´ ° � ¯ ´�¼È Ç � µ+¹ Ó µP»�» ½ �~µ+Á Ó¬´ ¹ ½�À3ÍR½�Ë Ê È � ) °�µ+¹¬¹N¸^² � µ Ä ²�¸1'�¸^² ½,ÈsÄ&Å�´ ¹�² ½ � ¯ ´+´ � ´ ��¶~Ç � Ç,Ê+Á ´�´ ¹(' ½> µP» Ä Â�°�Á1Á ´ » ½/Ä&Å�´$�X´�Ä&Å�´ ¯b»Tµ+¹ Ó ² ��µ�¹ Ó Í ¸ Ä�Ò > Ì+°�¹¬µ+¯ Ò+½EÀ °�Á ³µ�¹ ´(?² Ä ¸ ½ Æ�°�Á1µ+¹�¸^µ¬Ç� ´É,¸(Ì ´�Ä&Å�´ %u¯b¸ Ä(´�ÏRÒ ±?°�� È Á ´ ¹�É Ò µ,» À °�Ì�µ,²+Ç
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IEG?�E)J01�5(FH;<145
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�@CG1457K<1 e EGF �@14/ =�c...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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................................ . . . . . . . . . . . . ................
............................................................
◦◦
◦◦
A
A′
B
B′
C
C′
I
X
Y
Z
��ï
��J0149X�Y�Z36CG1SE)J01NM�F2;0�(K<?BFH9BE�5{?BgO5716âZM}1:9BE�5
IA′ � IB′ � IC′ CG1:5(Kn1 e EGFR�@1Z/R=`c��J014CG14gq?@CG1XY
FH5XK036C�34/D/H14/0EG?A′B′ 3:9<; Y Z
FD5XK<3:C�3:/|/21Z/<EG?B′C′ �@57?�z�1PJ@3��@1
∠A′B′C′ = ∠XY Z = 12∠XIZ = 1
2(180◦ − ∠ZBX)
= 12(180◦ − ∠ABC) = 90◦ − 1
2∠ABCæX9�E)J01P?�E)J01:CQJ@369<;��05)FH9 e 1
BIL@FD5+1 e EG5
∠ABC�:z�1�J@3��@1
∠C′BZ = ∠ZBI = 12∠ABC = ∠IBX = ∠XBA′ �
z�J01:9 e 1@�∠C′BA′ = ∠C′BZ + ∠ZBI + ∠IBX + ∠XBA′
= 4 · 12∠ABC = 2∠ABC
cð4FH9 e 1
A′ � B′ � C′ ��369<; B36CG1 e ?@9 e = e /|F e �4z�1>J@3Z�@1
∠C′BA′ + ∠A′B′C′ = 180◦ �z�JBF e J��6?@9�5)AnL05IEGF|EGA@EGFH?@9���â�F �@1:52 ·∠ABC +
(90◦ − 1
2∠ABC
)= 180◦ c���J01:CG1:�gq?@CG1@�
∠ABC = 60◦ c� c @?@C�Kn?@5)F|EGF �@1
x1�x2�y1�y2�@K<CG?4�@1>E)J01>FH9<1:�@A<34/DFRE�=
x21
y1
+x2
2
y2
≥ (x1 + x2)2
y1 + y2
cÊ�°+»(Ì ´�Ó Â ¼IÈ ¯+'�µ Ó�¼yÈ » ÄT½ Ê�µ+¹ � °�² ´�½\À�È�½dË Ê È � Ô ¸^É Ò�´ » È Á ´ ¹�É Ò µ,» À °�Ì�µ,² ½1À µP»Tµ� ¸�É ÒN´ ¯bµ ½�Ô µP»�»(°�¯4�0µ ½ ÊP±�µP¸T¹ � � °�²�³´ � Ò ¸�²X¶n³G µ � ÏRº µ+¯V¯ ´ ¯�° ½sË ¹N¸!Ì ´ ¯b²g¸ Ä µ Ä #°+»�¸ ÄH´ ��¹N¸��0µ Ó¬´À µ Ä µ,» Ò ¹ ¼ µ ½�º µ�¯ � ´ »!°�¹�µ ½ ÊP±�µP¸T¹ � Î µNÌ�¸ ´ ¯EÆ�°�² ½ ² Ä�ÒNÓ¬´ ¹ ÄT½EË ¹N¸!Ì ´ ¯b²g¸ Ä µ Ä #°+»�¸ ÄH´ ��¹N¸��0µ Ó¬´À µ Ä µ,» Ò ¹ ¼ µ ½�º µ�¯ � ´ »!°�¹�µ ½ ÊP±�µP¸T¹ �nµ+¹ Ó©Í ¸ Ä�Ò > Ì+°�¹¬µ+¯ Ò+½�À °�Á ³µ�¹ ´(?² Ä ¸ ½ Æ�°�Á1µ�¹N¸Tµ�Ç ´É,¸(Ì ´uÈ » Ä Ð ²E²�°+» Ò�Ä ¸!°�¹�²�µ+¹ Ó �V°�Á1Á ´ ¹ Ä ÇÊ�°+» Ò�Ä ¸(°�¹���c ��1PJ@3��@1
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≥ x21 + x2
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√
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x1
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y2
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a�b ∈ (0, 1)
c ��JBFD5XFD5X5(A eIe 14575)FR�@1Z/R=N1:�@A0F �B3:/H1:9BEOEG?a2(1 − b) + b(1 − a)2 ≥ b(1 − b)
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)= 1
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À °�Á1Á ´ ¹ Ä c���J01~?@CGFHâ�FH9034/OFD9<16�BA<3:/|F|E^=�FD5��@1:C^=½5)FHMNK0/H1}CG14/D34EGF �@1NEG?�E)J01~JBFHâ4J�/H1+�@14/?BgVMN34E)J½?B/R=@M}K@FH36;05��OLBA@E>F|E�FD5�3~â4?0?0;á? e)e 365)F2?@9�EI?�K<1:C�gq?@CIMµ;@F���14CG1:9BE{14/H1:M}1:9<�EG3:C^= EI1 e J@9@F2�BAn1:5"3ZEQ369�FD9BEICG?0;@A e EG?@C^=�/21+�@1Z/�3:9<;��@3:5Qz�1Z/D/^��gq?@CVâ4149<1:C�34/DF��)34EGFH?@905"?0L<�EG34FH9<1:;�L:=~36K0K0/ =@FD9<âPE)J01���34A e J6=<�(ð e J6zP36C �Pà^9<1:�@A<34/DFRE�=}EI?(√
y1,√
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yn≥ (x1 + x2 + · · · + xn)2
y1 + y2 + · · · + yn
c
� c �:1�EIL01"E)J01"FH9 e 1:9BEICG18?BgBEICGFD369<â�/H1
ABCc �:1�E Kn?BFD9BEG5
A1 6= A2/|F218?@9>E)J01�/|FH9<1
BC�@K<?BFH9BE�5
B1 6= B2/|F21P?@9 E)J01>/|FH9<1
AC�@3:9<;NK<?BFH9BE�5
C1 6= C2/|F21P?@9SE)J01{/|FH9<1
AB5+?PE)J@3ZE
AI = A1I = A2I�BI = B1I = B2I
�CI = C1I = C2I
cV.<CG?:�@1E)J@34EA1A2 + B1B2 + C1C2 = P
�:z�J01:CG1PFD5QE)J01PK<1:C�FHM}1ZEG1:CV?Bg 4ABC
cÊ�°+»(Ì ´�Ó Â ¼?Ô ¸^É Ò�´ » È Á ´ ¹�É Ò µ,» À °NÌ�µ+² ½XÀ µP»Tµ � ¸^É Ò�´ ¯�µ ½~Ô µ,»e»!°�¯ �bµ ½ ÊP±�µP¸T¹ � Ô ¸,� Å�´ »º µ Ä µ,¸e»�» ´�½ Æ�° ÒN´ ¹ ½ � ¯�µ+¹�� ´ � � ´ ° � ¯ ´�¼�È Ç � µ�¹ Ó µ,»e» ½ �~µ�Á Ó¬´ ¹ ½�À3ÍR½�Ë Ê È ��¶�Ç � ÇPÊ�Á ´+´ ¹ ' ½> µP» Ä Â�°�Á1Á ´ » ½/Ä&Å�´$�X´�Ä&Å�´ ¯b»Tµ+¹ Ó ² ��µ�¹ Ó Í ¸ Ä�Ò > Ì+°�¹¬µ+¯ Ò+½EÀ °�Á ³µ�¹ ´(?² Ä ¸ ½ Æ�°�Á1µ+¹�¸^µ¬Ç� ´É,¸(Ì ´�Ä&Å�´ %u¯b¸ Ä(´�ÏRÒ ±?°�� È Á ´ ¹�É Ò µ,» À °�Ì�µ,²+Ç�:1ZE
X�Y�ZL<18E)J01>gq1:1ZE�?Bg<E)J01>Kn14CIK<1:9<;BF e A0/D36C�5g2CG?@M
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c 7 FHâ4J�EVEICGFD369<â�/H1:5IAZ
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A1A2�¯z�1�J@3��@1
A1A2 = A1X + XA2 = AZ + Y Ac
ð4FHM�FD/D36C�/R=��6z�1P?0LBE�3:FD9
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..................................................................................................................................................................
..........................................................................................................................................................................................................................................................................
A
A1 A2B C
I
X
YZ
B1B2 = B1Y + Y B2 = BX + ZB369<;C1C2 = C1Z + ZC2 = CY + XC
c��J014CG14gq?@CG1B�A1A2 + B1B2 + C1C2 = (AZ + Y A) + (BX + ZB) + (CY + XC)
= (AZ + ZB) + (BX + XC) + (CY + Y A)
= AB + BC + CA = Pc
��ñ
� c d gHA<9 e EGF2?@9f(x)
FH5X;<1Z�n9<1:;}?@9SE)J01>FH9BEG1:C^�B34/[0, 1]
�@57?�E)J@34Ef(0) = f(1) = 0369<;
f
(a + b
2
)
≤ f(a) + f(b)c
gq?@CQ34/D/a369<;
bg2CG?@M
[0, 1]c
� 3Z�Nð6J0?6z E)J@34EnE)J01"16�BA<34EGFH?@9f(x) = 0
J@3:5�FH9@�<9@F|EG14/ =>MN369�=>57?B/DA@EGFH?@905 ?@9[0, 1]
c� L6� d CG1�E)J01:CG1>gHA<9 e EGF2?@905"?@9
[0, 1]z�JBF e J�5(34EGFD5(g|=�E)J01P36L<?4�@1 e ?@9<;@FREGF2?@905XL@A@EQ36CG19<?�E�FH;<149BEGF e 3:/|/R= �(1:CG?�
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� 3�� B?@Ca = b ∈ [0, 1]
�nz�1NJ@3��@1 E)J@3ZEf(a) ≤ 2f(a)
� J0149 e 1f(a) ≥ 0
�gq?@C83:/|/a ∈ [0, 1]
c �:1�Ean =
(12
)n �n ≥ 1
c ��1�z{FD/|/ 5+J0?6z L:=�FD9<;@A e EGFH?@9�E)J@3ZEf(an) = 0
gq?@CQ143 e Jnc
�03��6FD9<âa = 0
369<;b = 1
�Zz�1�J@3��@1f(
0 + 1
2
)
≤ f(0)+f(1)�Zz�JBF e JPâ�F �@1:5
f(
1
2
)
≤ 0 � z�J01:9 e 1@� f(a1) = f(
1
2
)
= 0c � 1(ä4E+�Bz�1P57J0?6z%E)J@34EQF|g
f(an) = 0�
E)J01:9f(an+1) = 0
cN�@3 �:FH9<âa = 0
369<;b = an
�`E)J01}L03:5(F e FD9<16�BA<3:/|F|E^=~=BF21Z/2;@5f(
0 + an
2
)
≤ f(0) + f(an) = 0cOð4FH9 e 1
f(
an
2
)
≤ 0�:z�1PJ@3Z�@1
f(an+1) = 0c
� L6� �:1ZEc > 0
c �X14�<9<1>E)J01{gHA<9 e EGFH?@9
f(x) =
{
0��F|g
x ∈ [0, 1] ∩ � �c��F|g
x ∈ [0, 1] ∩ (� \ � )
c��1�z{F|/D/<K<CG?4�@1>E)J@34E
f573ZEGFH5)��145�E)J01 e ?@9<;@FREGF2?@9}?BgnE)J01PJ6=0K<?�E)J01:5)FH54c
pnÕ:ë Ø �c �:1ZEx13:9<;
x2L01PC�3ZEGF2?@9034/n9@A<M�L01:C�54c��J0149 z�1PJ@3Z�@1
f
(x1 + x2
2
)
= 0 ≤ 0 + 0 = f(x1) + f(x2)c
pnÕ:ë Ø y<c �:1ZEx1L01P3PC�34EGFH?@903:/n9@A<M}L<14CV3:9<;
x2369�FHCGC�34EGFH?@903:/n9@A<M}L<14C�c��J0149 z�1PJ@3Z�@1
f
(x1 + x2
2
)
= c ≤ 0 + c = f(x1) + f(x2)c
pnÕ:ë Ø�� c �:1ZEx13:9<;
x2L01{FDCIC�3ZEGF2?@9034/n9@A<M�L01:C�54c��J0149 z�1PJ@3Z�@1
f
(x1 + x2
2
)
≤ c < c + c = f(x1) + f(x2)c
�4ò
��JBA<5���E)J01:CG1 1(ä:FH5IE FH9@�<9@F|EG14/ = M}3:9�= gHA<9 e EGFH?@905 ?@9[0, 1]
E)J@34E�36CG1ª9<?�EF2;01:9BEGF e 34/D/ = �(14CG?��@3:9<;N5(A e JSE)J@3ZEOE)J01 e ?@9<;@FREGF2?@905"?BgnE)J01PJ6=0K<?�E)J01:5)FH5"J0?B/2;`c cQ.<CG?4�@1�E)J@34E�369�=�K0CGFDM�1V9@A<M}L<14C
22n
+1e 3:909<?�E`L01VCG1:K0CG1:571:9BEG16;�365`3";@F���14CG1:9 e 1?BgnE�z�? �0gDE)J�Kn?6z�14C�5�?BgnFH9BEG16â414C�5:c
Ê�°+» Ò�Ä ¸(°�¹I ¼ ) °�µ�¹�¹�¸�² � µ Ä ²�¸1'�¸^² ½�ÈsÄ&Å�´ ¹�² ½ � ¯ ´+´ � ´ Ç@?@C
n = 1�6z�1PJ@3Z�@1
22n
+ 1 = 5c @?@COFH9BEG16â414C�5
x369<;
y5)A e JSE)J@34E
5 = x5 − y5 = (x − y)(x4 + x3y + x2y2 + xy3 + y4)�
z�1�J@3Z�@1x − y = 1
�¥?@Cx = y + 1
c ��149<1:C�34/D/ =���F|ga ≡ U (mod 5)
��z�J01:CG11 ≤ U ≤ 4
��E)J01:9a5 ≡ U (mod 5)
cj��JBA<5��Vg2CG?@M5 = x5 − y5 �Oz�1�â41�E
x ≡ y (mod 5)�0z�JBF e J e ?@9BEIC�36;@F e EG5�E)J01Sg23 e EVE)J@34E
x = y + 1cP�`?@905716�BAn1:9BEG/ =��
5FH5X9<?�EOE)J01P;@F���14CG1:9 e 1�?BgnE�z�?S�<gDE)J�K<?6z�1:C�5"?Bg�FD9BEI1:â41:C�54c
@?@Cn > 1
�0E)J01 1:�@A<3ZEGF2?@922n
+ 1 = x5 − y5 36âZ3:FD9}FHMNK0/|F2145 x = y + 1�5(FD9 e 1
22n
+ 1FD5XK<C�FHM}1<c ��JBA<5��
22n
+ 1 = (y + 1)4 + (y + 1)3y + (y + 1)2y2 + (y + 1)y3 + y4
= 5y4 + 10y3 + 10y2 + 5y + 1�
z�JBF e J FH5X3 e ?@9BEIC�3�;BF e EGFH?@9��@5(FD9 e 122n e 3:909<?�EVL<1P3PM�A0/|EGFDK0/H1�?Bg
5c
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���
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)5@= �~FD/|/DFD36M�.�c 5�14CG/|FH9<â4J0?���3:9<; @14CI903:9<;<? �{c �O?BAB�`ã1430�½K@AnL@/|FH57J016; L:=»E)J01� 34E)J014M}3ZEGF e 3:/ d 5757? e FH3ZEGF2?@9�?Bg d M�14CGF e 3P3:9<;}æOä:EG?@9~�{?BA<5+1�.0AnL@/|FH57J01:C�5�� �6�6�4òà�ð�5 � � � ����ñ ��� � �6ñ:� � � �<J@3:CG; e ?:�@14C+� � �6ñ�� ä:FDFnK03�â4145Z�0��ð�� ò� c �B�Æ ´ Ì*¸ ´ % ´�Ó Â ¼����������������� "!$#&%'�'(*)��,+.-/��½GË ¹�¸(Ì ´ ¯0²�¸ Ä&¼ °�� �X´ % º ¯ Ò ¹�²8%/¸,�9' ½� ¯ ´�Ó¬´ ¯b¸�� Ä °�¹ ½7�aº
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a2b2c2 ≥ 4S√
3c
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a2 + b2 + c2 ≥ 4S√
3 + (a − b)2 + (b − c)2 + (c − a)2c
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a2 + b2 + c2 ≥ 4S
√
3 +4(R − 2r)
4R + r+ (a − b)2 + (b − c)2 + (c − a)2
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m = 12(b + c − a)
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2(a + b − c)
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ab + bc + ca = s2 + r2 + 4Rr�^� �
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∑
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4R + r≥ 4s2r2 �
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4R + r
s
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4R + r
s=
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)
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a2 + b2 + c2 ≥ 4S
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ra�rb�rc�@K<CG?4�@1>E)J@34E
a
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b
rb+
c
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ra = S/(s − a)369<;}F|E�583:903:/H?0â�An1:5��E)J01{FH9<1:�@A<34/DFRE�= FH5"1:�@A0F �B3:/H1:9BEOEG?
a
ra+
b
rb+
c
rc=
2(ab + bc + ca) − (a2 + b2 + c2)
2S
≥ 2
√
3 +4(R − 2r)
4R + r
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c��K<CG?4�@18E)J@3ZE
tanA
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B
2+ tan
C
2≥√
3 +4(R − 2r)
4R + r
cÊ�°+» Ò�Ä ¸(°�¹ * BCG?@M E)J01 e ?@5(FD9<1�/H3�z¹z�1½â41ZE
a2 = b2 + c2 − 2bc cos Ac ð4FH9 e 1
S = 12bc sin A
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cEICG?BFD5O9<?@M}L0CG145�CZ�161Z/H5O9<?@9�9 �1:âZ34EGF|g25 EI1Z/H5O�@An1
a2 +b2 +c2 = 1c� ?@9BEICG1:CV�BAn1
3 − 5(ab + bc + ca) + 6abc(a + b + c) ≥ 0c
� ��è � c¡#¯g°,±�°�²�³´ ±�µ+¯ Ô µ�¹ Ò�´ » º�´ ¹�¸ Ä ° Ô¿Ò��¹�°�� ½ )b¾ Ê=�Ç Ô ÇEÊ�µ,É�µ,² Ä µ ½ �N°�É+¯�° �¹�° ½¾ ²V±�µ+É+¹ ´ c
æX9}; �145(FHâZ9<1PK036Cσ(n)
/D3�57?@M}M}1P;<145�;BFR�BFH5714A<C�5";BA�9<?@M}L0CG1P903ZEGA<CG14/nc
� 3Z� �<CG?BAB�@1:C�A<9N9<?@M�L@CG1P903ZEGA<CG14/nEG14/��@An1
σ(n) + 500 = σ(n + 2)c
� L6� ? �VAn14/�1:5IE�/H1{9<?@M�L@CG1�;<1{5+?B/|A@EGF2?@905";01{/D3PK<3:C�EGFH1 � 3Z� c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c
� � Ñ � c#¯�°,±�°�² ´�Ó Â ¼:� °�²�³´ � Ò ¸�²C¶n³G µ � ÏRº µ+¯V¯ ´ ¯�° ½CË ¹N¸!Ì ´ ¯b²g¸ Ä µ Ä #°+»�¸ ÄH´ ��¹N¸��0µ Ó¬´À µ Ä µ,» Ò ¹ ¼ µ ½dº µ+¯4� ´ »(°�¹¬µ ½ ÊP±�µP¸T¹¿µ+¹ Ó �µ�¹ Ä!´ »�¸^Áy°�¹ � ´ °�¯bÉ ´ #°,± ´ ² � Ò+½dº�Ò � Å µ+¯ ´ ² ÄT½Æ�°�Á1µ+¹�¸^µ<c
�:1ZEx1�x2�. . .�xnL01SCG1:34/`9@A<M�L01:C�5�5)A e JNE)J@3ZE
xk > 1gq?@C
1 ≤ k ≤ ncàqgnz�1{571ZE
xn+1 = x1�@K0CG?:�@1>E)J@3ZE
1
n
n∑
k=1
(logxk
xk+1 + logxk+1xk
)≤(
n∏
k=1
(1 + logn
xkxk+1
)
)1n c
� � Ñ � c·#¯g°,±�°�² ´+Ó Â ¼�Ô ¸ Å ³µP» ¼�º�´ ¹/��� ´�½�º ¯bµ+²�°�Ì ½ Æ�°�Á1µ�¹N¸Tµ0c�:1ZE
a�b�@369<;
cL<1{Kn?@5)F|EGF �@1{CG1:34/n9@A<M�L01:C�54c�ð6J0?6z�E)J@34E
∑
� ����� � �
a
b≥ 3
4+∑
� ����� � �
(a + c)2 + (a + b)c
(b + c)(2a + b + c)
c
ï6���
� � Ñ � c #¯g°,±�°�² ´+Ó Â ¼ Ê Ä µ+¹�» ´�¼ Æ�µ�Â+¸^¹�° %/¸ Ä � ½3Ô µ Ä&Å #¯g°;#¯ ´ ²R² ½�À�Å�´ »^Á\²3�!°�¯ Ó�½�Ô È�½Ë Ê È c
�:1ZED�
E�P369<;
FL01�E)J01 MNFH;<�)Kn?BFD9BEG5?BgO5)F2;01:5
BC�CA
��369<;AB
��CG1:5(Kn1 e EGFR�@1Z/R=���FH94ABC
c �:1ZEX�Y�n369<;
ZL<1SK<?BFH9BE�5�?@9NE)J015+1:âZM�149BEG5
BD�
CE�V3:9<;
AF�XCG1:5(Kn1 e EGFR�@1Z/R=`c��J018/DFD9<1:5
AX�BY
��3:9<;CZ
L<?BA<9<;S3 e 1:9BEIC�34/EICGFD369<â�/H1 � 5+J@36;<1:;�FH98E)J01X;@FD3�âZC�3:M �(c �:1�EX′ � Y ′ �369<;
Z′ L<1{E)J01SCG1 � 1 e EGF2?@9058?BgX�Y�<369<;
ZFH9E)J01~K<?BFH9BE�5
D�E�¥369<;
F�¥CG1:5(Kn1 e EGFR�@1Z/R=`c���J01Kn?BFD9BEG5
X′ � Y ′ �6369<; Z′ ;01ZEG1:CGMNFD9<18FH9S3�5(FDMNF|/H3:CM}3:909<1:CQ3:9<?�E)J01:C e 1:9BEIC�34/<EIC�FH3:9<â�/21P ′Q′R′ c
.
....
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......................................................................................................................................................................................................................................................................................................................
A
B Cq
D
q
EqF
P
Q
R
X
Y
Z
.<CG?:�@1�E)J@34E2 +
√3
4≤ [P QR]
[P ′Q′R′]≤ 8 − 4
√3�
z�J01:CG1[STU ]
CG14K<CG145+149BEG5QE)J01P36CG143S?Bg 4STUc
� � Ñ � c·#¯g°,±�°�² ´+Ó Â ¼�Ô ¸ Å ³µP» ¼�º�´ ¹/��� ´�½�º ¯bµ+²�°�Ì ½ Æ�°�Á1µ�¹N¸Tµ0c�:1ZE
a�b�@369<;
cL<1{Kn?@5)F|EGF �@1{CG1:34/n9@A<M�L01:C�54c�ð6J0?6z�E)J@34E
∑
� ����� � �
a
b+
(∑
� ����� � �a2
)12(∑
� ����� � �
1
a2
)12
≥ 2
3
(∑
� ����� � �a
)(∑
� ����� � �
1
a
) c
� � Ñ � c·#¯g°,±�°�² ´+Ó Â ¼�Ô ¸ Å ³µP» ¼�º�´ ¹/��� ´�½�º ¯bµ+²�°�Ì ½ Æ�°�Á1µ�¹N¸Tµ0c�:1ZE
a�b�@369<;
cL<1{Kn?@5)F|EGF �@1{CG1:34/n9@A<M�L01:C�54c�ð6J0?6z�E)J@34E
(∑
� ����� � �
a3
b2 − bc + c2
)(∑
� ����� � �
b2c2
a3(b2 − bc + c2)
)
≥(∑
� ����� � �a
)(∑
� ����� � �
1
a
)
≥ 16
(∑
� ����� � �
ab
a + b + 2c
)(∑
� ����� � �
c
2ab + bc + ca
) c
� � Ñ c #¯g°,±�°�² ´+Ó Â ¼ ¶~Ç ¾ Ç�#¯b¸ Ä&Å %/¸ 20¸ ÄT½~Ë ¹�¸(Ì ´ ¯0²�¸ Ä&¼�À °+»e» ´ É ´\À °�¯+' ½ Æ ´ ± Ò Â+»�¸,�s°��) ¯ ´ »^µ�¹ Ó c
��J01 34/|EGFREGAn;<145AD
�BE
�n369<;CF
?Bg 4ABC36CG1 K0CG?0;@A e 16;}EI?�M}161�EXE)J01e FHC e A<M e FHC e /H1P34E
X�Y�B369<;
Z�@CG1457K<1 e EGF �@14/ =�cX.<CG?4�@1>E)J@3ZE
AX
AD+
BY
BE+
CZ
CF= 4
c
ï6�6�
� � Ñ � c·#¯g°,±�°�² ´+Ó Â ¼nÈ ¯+'�µ Ó�¼sÈ » ÄT½ Ê+µ�¹ � °�² ´*½�À�È�½EË Ê È c�:1ZE
mL<1{3P9034EGA<C�34/�9@A<M}L<14C+�
m ≥ 2�B369<; /21�E
rL01{369�=�CG1:34/n9@A<M�L01:C�5(A e JE)J@34E
r ≥ 1/mc�àqg
a369<;
b3:CG18Kn?@5)F|EGF �@18CG143:/@9@A<M}L<14C�5Q5(34EGFD5(g|=BFH9<â
ab = r2 �6K<CG?4�@1E)J@34E1
(1 + a)m+
1
(1 + b)m≥ 2
(1 + r)m
c� ��è�� c·#¯g°,±�°�² ´+Ó Â ¼�Ô ¸�� Å�´ » º µ Ä µ,¸e»�» ´�½ Æ�° Ò�´ ¹ ½ � ¯�µ+¹�� ´ c
�:1ZE 4ABCL<1NC�F2â4J�E)�)369<â�/H16;�3ZE
A369<;�/H1ZE
OL01 E)J01NMNFH;<�)Kn?BFD9BE>?Bg
BCc
�:1ZEML01P3�Kn?BFD9BEQFD9 E)J01�K@/H3:9<1�?Bg 4ABC
�@369<;�/21�EM ′ � M ′′ � N � N ′ �03:9<; N ′′;<149<?�EI1�E)J01½?@C�E)J0? e 1:9BEICG145�?Bg 4MAB
� 4MAC� 4AM ′M ′′ � 4NAB
�X3:9<;4NAC
�<CG1457K<1 e EGF �@14/ =�c8àqgOFD5XE)J01�MNFH;<�)Kn?BFD9BE"?Bg
M ′M ′′ �<5+J0?6z%E)J@34E OFH5"3:/D5+?E)J01PMNFH;<�)Kn?BFD9BEV?Bg
N ′N ′′ c� ��è Ñ c·#¯g°,±�°�² ´+Ó Â ¼�Ô ¸�� Å�´ » º µ Ä µ,¸e»�» ´�½ Æ�° Ò�´ ¹ ½ � ¯�µ+¹�� ´ c
�:1ZE�E)J01NFD9 e FHC e /H1�?Bg 4ABCJ@3Z�@1 e 1:9BEICG1
I3:9<;�M�1:1ZE�E)J01}5)F2;01:5
AC3:9<;
AB34E
E369<;
F�BCG1:5(Kn1 e EGFR�@1Z/R=`c B?@C�3{K<?BFH9BE
M?@9SE)J01�/DFD9<1{5716âZM}1:9BE
EF�B5+J0?6zE)J@34E 4MAB
369<; 4MCAJ@3Z�@1>E)J01{573:M�1{3:CG1:3{FDg�3:9<;}?@9@/ =�F|g
MI ⊥ BCc
� ��è�è c·#¯g°,±�°�² ´+Ó Â ¼ �µ+¹�°�² ¾ Ç Í ²6µN° Ò ²�²P°�ÉP»!° Ò,½¬ÈsÄ&Å�´ ¹�² ½ � ¯ ´�´ � ´ c�:1ZE
a�b�@3:9<;
cL<1{9<?@9<�)9<16âZ3ZEGFR�@1>CG1:34/n9@A<M}L<14C�5X5(A e J�E)J@34E
a ≤ b ≤ c�B3:9<;/21�E
nL<1{3�K<?@5(FREGFR�@1>FD9BEI1:â41:C¯c".<CG?4�@1>E)J@3ZE
(a + (n + 1)b
)(b + (n + 2)c
)(c + na) ≥ (n + 1)(n + 2)(n + 3)abc
c� ��è � c·#¯g°,±�°�² ´+Ó Â ¼ �µ+¹�°�² ¾ Ç Í ²6µN° Ò ²�²P°�ÉP»!° Ò,½¬ÈsÄ&Å�´ ¹�² ½ � ¯ ´�´ � ´ c
�:1ZEa�b��3:9<;
cL01 9<?@9<�(9<1:âZ34EGF �@1 CG1:34/�9@A<M}L<14C�5�z{FRE)J
a2 + b2 + c2 = 1c.<CG?:�@1�E)J@34E
∑
� ����� � �(1 − 2a2)(b − c)2 ≥ 0
c� ��è � c·#¯g°,±�°�² ´+Ó Â ¼ �µ+¹�°�² ¾ Ç Í ²6µN° Ò ²�²P°�ÉP»!° Ò,½¬ÈsÄ&Å�´ ¹�² ½ � ¯ ´�´ � ´ c
�:1ZEa�b��3:9<;
cL01 9<?@9<�(9<1:âZ34EGF �@1 CG1:34/�9@A<M}L<14C�5�z{FRE)J
a2 + b2 + c2 = 1c.<CG?:�@1�E)J@34E
3 − 5(ab + bc + ca) + 6abc(a + b + c) ≥ 0c
� ��è � c #¯�°,±�°�² ´�Ó Â ¼yÔ µ+¹ ÒN´ » º�´ ¹N¸ Ä ° Ô¿Ò��¹�°�� ½;)�¾ ÊI�Ç Ô Ç�Ê+µ+É+µ+² Ä µ ½ �N°�É+¯�° �¹�° ½ÊP±�µP¸T¹`c�:1ZE
σ(n);01:9<?�EG1>E)J01P5)A<M ?BgnE)J01P;@F �@FD5+?@C�5"?Bg<E)J01P903ZEGA<C�3:/�9@A<M}L<14C
nc
� 3Z� �FD9<;N3P903ZEGA<C�3:/�9@A<M}L<14Cn5)A e JSE)J@34E
σ(n) + 500 = σ(n + 2)c
� L6� ? �P?6z%M}3:9�=N57?B/DA@EGFH?@905X36CG1�E)J01:CG1�EI?�K036C�E � 3��
ï6���
� � ��� ��� � �
� °�±�¯g°¬Â+» ´ Á¡¸^² ´ Ì ´ ¯�± ´ ¯VÁ1µ+¹ ´ ¹ Ä » ¼ �0»!°�² ´�Ó Ç Í,Å�´;´+Ó ¸ Ä °�¯/¸�²nµP»�%uµ ¼ ²u±#» ´ µ+² ´�ÓÄ ° �g°�¹�²�¸ Ó¬´ ¯ �!°�¯�± Ò Â+»�¸,�bµ Ä ¸!°�¹\¹ ´ % ²�°+» Ò�Ä ¸!°�¹�²n°�¯�¹ ´ %©¸^¹�²g¸^É Å3Ä ²n°�¹;±�µ,² Ä ±�¯g°¬Â+» ´ Á\²+Ç
��è Ñ � c ) �6�6� � ç ï�ï:� � ï6ï�ñ �£#¯�°,±�°�² ´�Ó Â ¼�Ô ¸ Å ³µP» ¼\º�´ ¹���� ´*½�º ¯bµ+²�°�Ì ½ Æ�°�Á1µ�¹N¸Tµ0c�:1ZE
ABCL<1{369N3 e A@EI1:�(3:9<â�/21:; EICGFD369<â�/H1<c
� 3Z��.<CG?:�@1�E)J@3ZE tan A
A+
tan B
B+
tan C
C>(
6
π
)2 c� L6��.<CG?:�@1�E)J@3ZE
A cot A + B cot B + C cot C <(
π
2
)2 c�qe � ? �X1�EI14CIM�FH9<1SE)J01NL<145)E e ?@905IEG3:9BEG5
c1 ≥(
6
π
)2 3:9<;0 < c2 < c3 ≤
(π
2
)2
5(A e JSE)J@3ZE∑
� ����� � �
tan A
A≥ c1
369<;c2 ≤
∑
� ����� � �A cot A ≤ c3
c
À °�Á;±�°�²�¸ Ä!´ °��C²�°+» Ò�Ä ¸!°�¹�². ¼ Æ�° ¼ º µ+¯6Â�µ�¯bµ ½ � ´ Â�µ�¹ ´ ² ´ Ë ¹N¸!Ì ´ ¯b²g¸ ÄT¼+½ � µ�¹�µ�¯ ½� ´ Â�µ+¹�°�¹ ��Ê�µP» ´ Á Ô µ,»e¸ '�¸R³� ½ ² Ä�ÒNÓ¬´ ¹ ÄT½ Ê+µ�¯bµ32 ´ Ì+° À °+»e» ´ É ´*½ Ê�µ+¯�µ42 ´ Ì+° ½�º °�²6¹N¸Tµ µ�¹ Ó� ´ ¯5� ´ É�°NÌ�¸^¹¬µ8�3µ+¹ Ó ¶E¯�µ,É�°+» 2 Ò Â Ô ¸�»(°76² ´ Ì�¸R³� ½ #¯�µ+¹�2bµ�¹N¸ ½ Ê ´ ¯6Â+¸^µ<c
� 3��O369<; �2e � d K<K@/R=BFH9<â�E)J01 d�� � � � à^9<16�BA<3:/|F|E^=��Bz�1�?0LBE�3:FD9 � gq?@CV3:9�=}3 e A@EG16�369<â�/H16; EIC�FH3:9<â�/21ABC
�tan A
A+
tan B
B+
tan C
C≥ 3
(tan A tan B tan C
ABC
)13 � �^ï �
369<;tan A + tan B + tan C ≥ 3(tan A tan B tan C)
13c �^� �
�"5(FD9<â �^� �¥369<; E)J018z�14/|/2� ��9<?6zP9 F2;01:9BEGFRE�=tan A tan B tan C = tan A + tan B + tan C
�z�1�â41�E
tan A tan B tan C ≥ 3(tan A tan B tan C)13�
z�J01:9 e 1@�tan A tan B tan C ≥ 3
√3c �^ñ �
ð4FH9 e 1ABC ≤
(13(A + B + C)
)3=(
π3
)3 �BFD9<16�BA<3:/|F|EGFH1:5 �qñ �¥3:9<; �qï � FDM}K@/R=SE)J@34Etan A
A+
tan B
B+
tan C
C≥ 9
√3
π
�
ï6� �
z{F|E)J�1:�@A<34/DFRE�=�FDg¥3:9<;�?@9@/ =~F|gA = B = C = π
3
c{��JBA<5Z�c1 = 9
√3
π
FD9�K<3:C�E �2e �7cð4FH9 e 1 9
√3
π>(
6π
)2 �@K<3:C�E � 3Z� FD5X3:/D5+? K0CG?:�@1:;�c� L6��369<; �2e � ð4FH9 e 1
limx→0
x cot x = 1�4z�1>MN3Z= 1(ä4EI149<;
h(x) = x cot xEG?PE)J01
e ?@9BEGFD9@An?BA<5XgHA<9 e EGF2?@9f(x)
;<1Z�n9<1:;N365
f(x) =
{
x cot xFDg
x 6= 0,
1FDg
x = 0.
��J0149f(x)
FH5 e ?@9BEGFH9@An?BA<5"?@9 [0, π
2
] c � 1)ä:E e ?@905(FH;<14CQ3{/H1:MNM}30çr ØZÙSÙ Õ �c ��J01{gHA<9 e EGF2?@9
f(x) = x cot xFH5 e ?@9 e 3��@1�?@9 [
0, π2
] c#¯�°�°�� * ��J01P571 e ?@9<;};01:C�FR�B3ZEGFR�@1P?Bg
f(x)FH5
f ′′(x) =2(x cos x − sin x)
sin3 x< 0
�5(FD9 e 1
tan x > x?@9 (
0, π2
) c ��JBA<5��f(x)
FD5 e ?@9 e 3Z�@1P?@9 [0, π
2
] c��J014CG14gq?@CG1B�
f(A) + f(B) + f(C) ≤ 3f
(A + B + C
3
)
�
E)J@34E�FD5Z�A cot A + B cot B + C cot C ≤ π
√3
3
�z{F|E)J~16�BA<3:/|F|E^=~F|g¥3:9<;�?@9@/ =�F|g
A = B = C = π3
c{��JBA<5Z�c3 = π
√3
3
FH9�K036C�E �2e �7cð4FH9 e 1 π
√3
3<(
π2
)2 �@K<3:C�E � L�� FD5X3:/D5+? K<CG?4�@16;`c�qe ��à2E8?@9@/R=~CG1:MN3:FD9058EI?��<9<;
c2c @?@CVE)J@34E�K0A<CGKn?@571�z�1 FD9BEICG?0;@A e 1�369<?�E)J014C/214M}MN3<ç
r ØZÙSÙ Õ�y<c �:1�EfL01�3 e ?@9 e 3��@1}gHA<9 e EGF2?@9á?@9á57?@M�1}CG143:/¥FH9BEG1:C^�B34/
I�¥369<;�/H1ZE
a�
b�
u�
v ∈ IL<1�5)A e J E)J@3ZE
a ≤ u ≤ v ≤ b369<;
u + v = a + bc ��J0149
f(u) + f(v) ≥ f(a) + f(b)c
#¯�°�°�� * àqga = b
�BE)J01PCG1:5)A0/|E�FH5"?0L:�@FH?BA<5:c � ?6z 365(5(A<M}1>E)J@34Ea < b
c��:1ZE`L01{E)J01/DFD9<1 e ?@9BE�3:FD9@FH9<âPE)J01�Kn?BFD9BEG5 (
a, f(a)) 369<; (
b, f(b)) c ð4FD9 e 1
`FH5�9<?�EX�@14C�EGF e 34/���FREJ@365��n9@FREI1S5)/2?@K<1<c �:1�E
y = Ax + BL01�E)J01 16�BA<34EGFH?@9�?Bg
`c�ð4FH9 e 1
fFD5 e ?@9 e 3��@1@�z�1�J@3��@1
f(x) ≥ Ax + Bgq?@CQ1+�@1:C^=
x ∈ [a, b]cQà^9NK<3:C�EGF e A0/H3:C+�6z�1PJ@3Z�@1
f(u) ≥ Au + B3:9<;
f(v) ≥ Av + Bc
�P149 e 1B�f(u) + f(v) ≥ (Au + B) + (Av + B) = A(u + v) + 2B
= A(a + b) + 2B = (Aa + B) + (Ab + B)
= f(a) + f(b)c
ï6���
�~F|E)J0?BA@E¥/H?@575V?Bg�â4149<1:C�34/DFRE�=��:z�1�M}3�= 365(5(A<M}1�E)J@34E0 < A ≤ B ≤ C < π
2
c� ?�EG1>E)J@3ZEA + B > π
2� J01:9 e 1@� 0 < π
2− A < B
c ��J01:9 π2
− A < B ≤ C < π23:/|/@L014/H?@9<âVEG?>E)J01"FD9BEI14C^�B3:/ [
0, π2
] 3:9<;�5(34EGFD5(g|=B+C =
(π2
−A)+ π
2
c�5@= �:1:MNM}3� 369<;SE)J01 e ?@9 e 3��@FRE�=N?Bgf�6z�1�J@3��@1
f(B) + f(C) ≥ f(
π2
− A)+ f
(π2
) cð4FH9 e 1
f(
π2
)= 0
�6z�1P?0LBE�3:FD9f(B) + f(C) = f
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λ = f(A) + f(B) + f(C) ≥ f(A) + f(
π2
− A) c
�:1ZEu = min
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} cQ��J01:90 < u ≤ v < π
2z{F|E)Ju + v = 0 + π
2
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π2
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≥ 1c ��J014CG14gq?@CG1B�
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ABCz{F|E)J
AB = AC = k� 3 e ?@905IEG3:9BE��)�0E)J0149~365∠A → 0
z�1S57161{E)J@34EA cot A → 1
�B cot B → 0
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cQ�P149 e 1B�λ → 1
c ��JBA<5��c2 = 1
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a1
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ai = 1W�V4e
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RZ^V k [ �~} RZV4e>[#`2U k ]?e4U%]?U�e>[c]�RZo�d�U(k, `, m)
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� /H1ZEf(a)
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i=1
(
∑̀
j=1
ai+j−1
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c%��J0149fFH5 e ?@9BEGFD9@An?BA<5Z�"3:9<;
f(a)FD5e /21436C�/R=j1)ä�K0CG1:5(5(FHL@/H1½gq?@C
(k, `, m)gq?@C�34/D/
a ≥ 1c � ?6z��
f(1) = k`m �83:9<;f(a) > am ≥ a � J0149 e 1B� f(a) → ∞ 365
a → ∞ c£5@=áE)J01�à^9BEG1:CGM�1:;@FD34EG1�¥3:/|An1{��J01:?@CG1:M��@gq?@C�369�=�FD9BEI1:â41:Cn ≥ k`m ��E)J01:CG1�1)ä6FD5)E�5X5+?@M}1 a ≥ 1
5)A e J E)J@3ZEf(a) = n
�@3:9<; E)JBA<5Z�nFH5"1)ä�K0CG1:5(5(FHL@/H1>gq?@C
(k, `, m)c
��J014CG14gq?@CG1B�2005
FH5Q1)ä�K0CG1:5(5(FHL@/H1�gq?@C(k, `, m)
��3:9<;1987
FH5�9<?�E¥1(ä6K<CG14575)F2LB/21gq?@C(k, `, m)
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c à^9 E)JBFD5 e 3:5+1B�3`3 ≤ k`m ≤ 2005
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c B?@CP�Bä61:;`E)J01SFH9<1:�@A<34/DFRE�=
1987 < k`m ≤ 2005e 369�J0?B/2;�gq?@C"?@9@/R=��<9@F|EG14/ =~MN369�=
m��3:9<;z�J01:9
`3:9<;
m3:CG1NL0?�E)J~��ä616;}E)J01 FD9<16�BA<3:/|F|E^= e 369�J0?B/2;�gq?@C8?@9@/ =~�<9@F|EG14/ =~M}3:9�=
kc ��1}9<1:16;á?@9@/ = e ?@905)F2;01:C
` = 3�4�5� 369<;
7� gq?@C8FDg
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nFH5�34/H57?
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k�`�@3:9<;
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45
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1
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��J01�57?B/R�@14C{K<CG?4�@16;½E)JBFH5�FD9<16�BA<3:/|F|E^=ágq?@CP3:9�= 9<?@9<�(9<1:âZ34EGF �@1�CG143:/Q9@A<M�L01:C�5a�b�c�¥369<;
dc½ð61�E�EGFH9<â
x = 14(a + b + c + d)
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(a + 1)(b + 1)(c + 1)(d + 1)≥ 256
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1
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∑
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= 16x5 + 39x4 + 84x(1 − x) + 2(1 − x2) + 10 > 0c
àqgx > 1
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> 55x3 − 86x2 + 84x + 12
= 43x(x − 1)2 + 12x3 + 41x + 12 > 0c
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�L1 = 1
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gq?@Cn ≥ 1
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bn/2c∑
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k
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22k
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Kn?B/ =09<?@M�FH34/H5Pn(x) =
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k
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4
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c��1 e /D3:FDM�E)J@34E
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Pn+1(x) = Pn(x) +x
4Pn−1(x)
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n ∈ �c �0?}145)E�3�LB/DFD5+J �^ï �)�@9<?�EI1>�nC�5IEOE)J@3ZEx
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k=1
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4
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Pn(x) +x
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k=0
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k
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4
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k=1
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k − 1
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4
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k
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4
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k
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4
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k
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4
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ï�ï6ñ
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2
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1 + c2qn+12
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c236CG1{EG?}L<1S;<1�EI14C)�MNFD9<16;`c�ð6?B/ �@FD9<â
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k
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(−z)k
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k
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1
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k
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4k
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V (z) = zn/n!c���J01
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(z∗k, 0, 0, . . . , 0)
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S =s∑
k=1
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)
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365∆ → 0
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0
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z
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0
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z
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(n − 1)!
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j=1
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1j+1
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0
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j=1
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1j+1
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n − 1− 1
n
(1
1n− 1
2n+
2
2n− 2
3n+
3
3n− 3
4n+ · · ·
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n
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1
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vb
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C0V0
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vb
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k ]8 | n + 1
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4
7∑
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4
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n = 7W�V4e
n = 8W*^~o k ^g^gRZu�RpdLR�[?R�U%^ �A� `2U*]?U jlk ]?U �
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n = 8k − 1jlk ]S^ k t_U
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8�
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369<; E)J0149�;014�<9<1Ai = {(x + a)2 | a ∈ Bi}
gq?@Ci = 1
�2�3�4c��J0149}L:=SEG16;BF2?BA<5XL@A@E�5)EIC�34F2â4J�EGgq?@C�zP36CG; e 3:/ e A0/H3ZEGF2?@905��:z�1��n9<;SE)J@3ZE¥E)J01{5(A<M?Bg�E)J01N14/H1:M}1:9BE�5�FD9
Ak1:�@A<34/H5
8x2 + 248x + 2604gq?@C
k = 1�2�3�4c�ð4FH9 e 1
Sx = A1 ∪ A2 ∪ A3 ∪ A4�0?BA<C e /D3:FDMFH5"145)E�3�LB/DFD5+J01:;�c�<? e ?@MNK0/H1ZEG1QE)J01XK<CG?0?Bg��ZFRE`9<?6zÇ5)A�� e 1:5`EG?>57J0?6z E)J@34EnE)J01"5IEG3ZEI14M�149BEG5
P (n)FH5QEIC�An1{gq?@Cn ∈ {15
�16�23�24�31�32�39�40} cð61ZE�EGFD9<â
x = 03:9<;
x = 1FD9½E)J01�K<3:C�EGFREGF2?@9�â�F �@1:9 3�L0?:�@1B�¥z�1~5+1:1�E)J@3ZE
P (31)369<;
P (32)3:CG1>EICGAn10c B?@COE)J01�5)F ä�CG14M}34FH9@FD9<â��B34/DAn145�?Bg
n�@5+�BA<36C�FH9<âP1:3 e J14/H1:M}1:9BEVL014/H?6zªâ�FR�@145QE)J01PCG1:�@A0FDCG16;NK<3:C�EGFREGF2?@9054c
{1�2�. . .�15}
= {1�7�8�14} ∪ {2
�9�15} ∪ {3
�6�11�12} ∪ {4
�5�10�13} �
{1�2�. . .�16}
= {1�6�9�16} ∪ {2
�4�8�11�13} ∪ {3
�5�12�14} ∪ {7
�10�15} �
{1�2�. . .�23}
= {1�3�10�13�19�21} ∪ {2
�8�22�23} ∪ {4
�5�9�11�15�17�18}
∪ {6�7�12�14�16�20} �
{1�2�. . .�24}
= {1�5�7�10�13�15�16�20} ∪ {4
�14�22�23}
∪ {2�3�11�17�19�21} ∪ {6
�8�9�12�18�24} �
{1�2�. . .�39}
= {1�8�11�15�16�19�27�28�35�37} ∪ {2
�5�20�29�30�38�39}
∪ {3�6�7�12�13�14�17�21�22�23�26�32�33}
∪ {4�9�10�18�24�25�31�34�36} �
{1�2�. . .�40}
= {1�6�18�20�32�34�35�37} ∪ {2
�7�14�15�16�28�30�39�40}
∪ {3�4�5�9�13�21�22�27�29�36�38}
∪ {8�10�11�12�17�19�23�24�25�26�31�33} c��JBFD5 e ?@M}K@/21�EI145VE)J01{K<CG?0?BgGc
� � �� ����J����' "�� D ! � !%� � (4� .�. ��? �&��� �@��� ����� � �����E� � � * � ��'$�;� � � �#��� � ( �!�����%����� ������ ��� ��# 6 ���������������� ����� �!��"$#%�&������'�(
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����' {12�22�
. . .�
n2} F���n ∈ {15
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{12�22�. . .�232} = {12
�52�82�92�152�182�192} ∪ {22
�72�122�202�222}
∪ {32�42�62�112�132�172�212} ∪ {102
�142�162�232}
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��è�è Ñ c ��?@CICG1 e EGF2?@9�c ) �6�6� � ç ï6ï�ï � ï6ïZò � #¯�°,±�°�² ´�Ó Â ¼0�+Ò µ+¹ Ï�º °�²8�V°ÃÆ�°�Á ´ ¯�°Ô ³µ�¯ Ñ�Ò�´ � ½�Ë ¹�¸(Ì ´ ¯0²�¸ Ó µ Ó1Ó¬´ ! µ,»e»Tµ Ó °+»�¸ Ó�½ ! µP»�»^µ Ó °+»e¸ Ó�½ ÊP±�µP¸T¹`c
�:1ZEABC
L01>3�EICGFD369<â�/H1�z{F|E)JS5(FH;<145a ≥ b ≥ c
?@K<K<?@5(FREI18E)J01>3:9<â�/2145A�B�
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AHL01 Kn14CIK<1:9<;BF e A0/D36CXEG?}E)J01 5)F2;01
BCz{F|E)J
H?@9
BCcð61ZE
m = BH369<;
n = CHcV.<CG?:�@1�E)J@3ZE
a(bm + cn) − bc(b + c)FH5VKn?@5)F|EGF �@1@�9<16âZ3ZEGFR�@1B�<?@C �(14CG?�3 eIe ?@CG;BFH9<â>365
∠AFD5"?0LBEGA<571@�@3 e A@EI1B�0?@C�CGFHâ4J�E)�(3:9<â�/21:;�c
¾ ²�² ´ ¹ Ä ¸^µ,»e» ¼�Ä&Å�´ ²6µ+Á ´ ²P°+» Ò�Ä ¸(°�¹  ¼ À�Å ¸e± À3Ò ¯ Ä ¸�² ½ÃÔ ¸^²�²P° Ò ¯b¸yÊ�° Ò�Ä&Å�´ ¯V¹¡Ê Ä µ Ä(´Ë ¹N¸!Ì ´ ¯b²g¸ ÄT¼+½�� °,±#»�¸^¹ ½�Ô <�½dË Ê È � À�Å µ�¯b» ´ ²?Æ�Ç~¶/¸^Á�¸T¹�¹�¸ ´�½sÈ ¹�É ´ »(°ÃÊ Ä µ Ä(´ Ë ¹�¸(Ì ´ ¯ ϲ�¸ Ä&¼,½ Ê�µ+¹ È ¹�É ´ »(° ½dÍRÎ�½dË Ê È � �+Ò µ�¹ ÏRº °�² �g°©Æ�°�Á ´ ¯�° Ô ³µ�¯ Ñ�Ò�´ � ½IË ¹N¸!Ì ´ ¯b²g¸ Ó µ Ó Ó¬´! µ,»e»Tµ Ó °+»�¸ Ó�½ ! µP»�»^µ Ó °+»e¸ Ó�½ Ê�±�µ,¸^¹ � � ¯�µ+¹��0¸�²8�V° � µNÌ�¸ ´ ¯ � µ�¯ ��³G µ À µP±#¸ Ä ³µ�¹ ½ )�¾ Ê ³È »(Ì�µ�¯ ´ �À3Ò Â ´ ¯�° ½ #¯b¸ ´ ÉN° Ó¬´uÀ ³°�¯ Ó °¬Â�µ ½ Ê�±�µ,¸^¹ �#Æ�¸�� Å µ�¯ Ó ) Ç�� ´ ²�² ½ Æ3µ+¹�� Å °��µP»!°�² !#´ ¯ Ó¬´ ² ½�À�È�½Ë Ê È � � ´ ° � ¯ ´�¼XÈ Ç � µ�¹ Ó µ,»e» ½ �~µ�Á Ó¬´ ¹ ½�À3Í6½sË Ê È � !#´+Ó�Ò »^µ � Ç Ô¿Ò ¯ ÄT¼+½ ¶s°�Ì ´ ¯ ½ È�½Ë Ê È � Î µNÌ�¸ ´ ¯�Æ�°�² ½ ² Ä�ÒNÓ¬´ ¹ ÄT½�Ë ¹N¸!Ì ´ ¯b²g¸ Ä µ Ä #°+»�¸ ÄH´ ��¹N¸��0µ Ó¬´1À µ Ä µ,» Ò ¹ ¼ µ ½uº µ+¯4� ´ »(°�¹¬µ ½ÊP±�µP¸T¹ ��¶�Ç � Ç#Ê+Á ´�´ ¹(' ½ > µP» Ä Â�°�Á1Á ´ » ½�Ä&Å�´ �X´�Ä&Å�´ ¯b»^µ�¹ Ó ²4��µ+¹ Ó ´�Ä!´ ¯ ��Ç. ;°�° ½�º ¸(°+»TµË ¹N¸!Ì ´ ¯b²g¸ ÄT¼+½ �+µ Ô ¸^¯�µ Ó µ ½�À�È�½uË Ê È c
ð4FH9 e 1m = c cos B
369<;n = b cos C
�6z�1PJ@3Z�@1a(bm + cn) = abc(cos B + cos C)
c�"5(FD9<â8E)J01 �43Zz ?Bg`��?@5(FD9<1:5XCG14Kn1434EG16;B/R=���z�1P?0LBE�3:FD9a(bm + cn) − bc(b + c) = abc(cos B + cos C) − bc(b + c)
= b(ac cos B) + c(ab cos C) − bc(b + c)
= ba2 + c2 − b2
2+ c
a2 + b2 − c2
2− 2bc(b + c)
2
=a2b − bc2 − b3 + a2c − b2c − c3
2
=b + c
2(a2 − b2 − c2)
= −bc(b + c) cos Ac
��J014CG14gq?@CG1B��E)J01�1(ä6K<CG14575)F2?@9a(bm + cn) − bc(b + c)
FD5�Kn?@5)F|EGF �@1@��9<16âZ3ZEGFR�@1B�`?@C�(1:CG? F|g�3:9<;}?@9@/R= FDg
cos AFH5X9<1:âZ34EGF �@1@�@K<?@5(FREGFR�@1B�0?@C �(1:CG?��BCG1:5(Kn1 e EGFR�@1Z/R=`c
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(# ����&� � �&��� �
)��� ./� ? ��� ? ���%� � � ��"$�������8��� �����h�
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)� � .�!
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��è�è�è c ) �6�6� � ç ï�ï6ï � ï�ï6� ��#¯�°,±�°�² ´�Ó Â ¼ Å µ+Á ! µ�¹ Í,Å�Ò µ�¹ ½ �~µ+¹�°+¸ Ë ¹N¸!Ì ´ ¯b²g¸ ÄT¼°���Ê �0¸ ´ ¹�� ´�½ �~µ�¹�°+¸ ½ ! ¸ ´�Ä ¹�µ�Á�c�¥F �@1:9�K<?@5(FREGFR�@1�CG1:34/<9@A<M}L<14C�5
a�b�c5)A e J�E)J@3ZE
a + b + c = 1�BK<CG?4�@1�E)J@3ZE
(1 − a)(1 − b)(1 − c)(1 − a2
)2+(1 − b2
)2+(1 − c2
)2 ≤ 1
8
c
Ê�°+» Ò�Ä ¸(°�¹I ¼�Ô ¸�� Å�´ » º µ Ä µP¸�»e» ´*½ Æ�° Ò�´ ¹ ½ � ¯bµ�¹/� ´ c��1PK0CG?:�@1�E)J@34E
L ≥ 8��z�J014CG1
L =(1 − a2)2 + (1 − b2)2 + (1 − c2)2
(1 − a)(1 − b)(1 − c)=
∑
� ����� � �
(1 − a)(1 + a)2
(1 − b)(1 − c)
cBCG?@M(1 + a)2 = (1 + 1 − b − c)2 =
((1 − b) + (1 − c)
)2 ≥ 4(1 − b)(1 − c)�
369<;N5)FHM�FD/D36COFD9<16�BA<3:/|F|EGFH1:5Vgq?@C(1 + b)2
3:9<;(1 + c)2
�BFRE�gq?B/D/H?6zP5�E)J@34EL ≥ 4(1 − a) + 4(1 − b) + 4(1 − c) = 8
c� � ������������' "�� ������� � ��� ���#����� �!�������� � � ��� �� �� ! � �$! � ��� � �)����� + &-�4�>��� ����� � �����
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������� � ��������� �����������������/�! #"%$&"�'�(*)H+�-~� D?0�9pKLKi(�1%'��1�=K;"�'�!"� �i-* �9�1I)�9Z'g<�'�:?3�)4(@5A:�<� �C(� �9iD�1*5 ; "*KLKi(@�I(�"�� �*0E(@'g'y1�K;"�5I9�B�9��i�M541�:;"*Kp9�1��g< �%0J(='G'G1%Ki"�5�92B�9Z<�A g(*(�Dc(I�
� U�[a�b�cu2U8o k ^gR�[?R�T�Ux]?U%W*d!V�s4tvu2U*]�^ w `IRZag`q^GW�[?RZ^ j X
1
a+
1
b+
1
c=
1
abc
�
8 ] k T�U�[#`�W%[a
a2 + 1+
b
b2 + 1+
c
c2 + 1≤ 3
√3
4
�
�:�
Ê�°+» Ò�Ä ¸(°�¹1 ¼ ! ° � Ò ° � º µ À µ+¹ ½�À µ�¹ Í,Å ° Ë ¹N¸!Ì ´ ¯b²g¸ ÄT¼ °�� Ô ´+Ó ¸��0¸^¹ ´ µ�¹ Ó Å µ�¯VÁ1µ � ¼,½À µ�¹ Í+Å ° ½ ! ¸ ´�Ä ¹�µ�Á�c
��J01�â�F �@1:9 e ?@9<;@FREGF2?@9�FD5"16�BA0FR�B34/2149BEOEI?ab + bc + ca = 1
c ��JBA<5��∑
� ����� � �
a
a2 + 1=
∑
� ����� � �
a
a2 + ab + bc + ca
=∑
� ����� � �
a
(a + b)(a + c)=
2
(a + b)(b + c)(c + a)
cæX9�E)J01P?�E)J01:CQJ@369<;��<L:=SE)J01 d�� � � � à^9<16�BA<3:/|F|E^=��6z�1PJ@3Z�@1(a + b)(b + c)(c + a) = (a + b + c)(ab + bc + ca) − abc
≥ 89(a + b + c)(ab + bc + ca) = 8
9(a + b + c)
≥ 89
√
3(ab + bc + ca) = 89
√3c
��J014CG14gq?@CG1B�∑
� ����� � �
a
a2 + 1≤ 3
√3
4
cî0�BA<3:/|F|E^=}J0?B/H;05VFDg�3:9<;}?@9@/ =�F|g
a = b = c = 1√3
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A0RL^V k [
120◦�
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RZV w `IR�ag` 4Ai+1Bi+1Ci+1RL^k uI[�W*RLV4U�e j ] k t 4AiBiCi
u*X_]?U��!U�a?[?RLV4\�U%W�ag`HT�U%]�[?UgnHRLVq[#`2U k o4o k ^gR�[cU8^#R�e2U��p[#`�W%[RZ^ �
BiCiRL^�[#`2U>o!U%]co4U*V4eIR�acs2dLW�] u�RL^yU*a?[ k ] kIj
AiAi+1
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60◦ W*^ n → ∞�
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ï��6ï
Ê�°+» Ò�Ä ¸(°�¹  ¼ È ±�°�² Ä °+»e¸^² � Ç�¶ ´ Á�¸^² ½�! µ�¯gÌ�µ�' ´ ¸(° �s¸^É Å Ê � Å °�°+» ½nÈsÄ&Å�´ ¹�² ½ � ¯ ´�´ � ´*½Áy° Ó ¸�� ´�Ó Â ¼�Ä&Å�´X´�Ó ¸ Ä °�¯¯c��1>5IEG3:C�E�z{FRE)JPE)J01{?0L0571:C^�B3ZEGF2?@9�E)J@3ZE¥E)J01>K0CG?0L@/H1:M��B3:5V5IEG3ZEI1:;��6FH5�FH9 e ?@CICG1 e E�c��J01 e ?@9<;BF|EGFH?@9
∠A0 6= 2π/3;<?0145 9<?�E�â�A<3:C�369BEG161QE)J@34E�E)J01X3:9<â�/2145¥?Bg 4AnBnCn36K0K<CG?@3 e J
π/33:5
n → ∞ cVà^9Sz�J@34E�gq?B/D/H?6zP5Z�:z�18z{FD/|/n365(5(A<M}1>E)J@3ZE∠A0
FD5X5(A e JE)J@34E∠An 6= 2π/3
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A01)ä6FD5)E�c½.@1:CIJ@36K05Z�¥?BA<C>CG143�;01:C�5�z{FD/|/O57J@36CG15+?@M}1SFH905)F2â4J�E�5>?@9�E)J01�571ZE�?Bg �+K0CG?0JBF2LBF|EG16;N�B3:/|An1:5 �{gq?@C
∠A0� z�1Pz{F|/D/`3:/D5+?�5(3Z=~3L@FREVM}?@CG1�?@9 E)J@3ZEQ/D34EG1:C¯c d E¥E)JBFH5XKn?BFD9BE+��z�18z{F|/D/@ì2A<5IEV3:575)A<M�1�E)J@34EOE)JBFD5"571ZE�FD5X9<?�EE)J01�1:9BEGFDCG1{FD9BEI14C^�B3:/
(0, π)�@c
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.............................
��
��
ωn
.........................
.....................
.............................
.........................
.....................
AnAn+1
Bn
Bn+1
Cn
Cn+1
�:1ZEωn
;01:9<?�EG1�E)J01�3:9<â�/21CnBnAn
�jn = cos 2ωn + i sin 2ωn
� E)J01:9jn = cos 2ωn − i sin 2ωn
�I�xn =
BnCn
AnBn
�03:9<;an�bn�@369<;
cnL01>E)J01 e ?@MNK0/H1(ä
9@A<M�L01:C�5NCG1:K0CG1:571:9BEGFD9<â~E)J01�Kn?BFD9BEG5An�
Bn�V369<;
Cn�XCG1:5(Kn1 e EGFR�@1Z/R=`c d 9�14365�=FH9<;BA e EGF2?@9�5+J0?6zP5�E)J@34E 4AnBnCn
FH5�FD5+?@5 e 1Z/2145>3:9<;~E)J@3ZE0 < ωn < π
2
cS��J01:9∠CnBnCn+1 = ∠An+1BnAn = ∠Bn+1CnBn = 2ωn
�@369<;(an − cn)jn = bn − an
� �^ï �(cn − bn)jn = cn+1 − bn
� �^� �(an+1 − bn)jn = an − bn
� �^ñ �369<;
(bn+1 − cn)jn = bn − cnc �Dò �
BCG?@M 16�BA<34EGFH?@905 �qï �)� �|ò �)�@3:9<; �^� �)�@CG1:5(Kn1 e EGFR�@1Z/R=��6z�1P?0LBE�3:FD9an =
1
jn + 1bn +
jn
jn + 1cn� �^� �
bn+1 =1
jn
bn +jn − 1
jn
cn� �q� �
cn+1 = (1 − jn)bn + jncn� � � �
369<;�g2CG?@M �^ñ �¥369<; �q� �)�6z�1Pâ41�Ean+1 =
jn
jn + 1bn +
1
jn + 1cnc � � �
��1�J@3��@1cos ωn =
BnCn
2AnBn
?@Cxn = 2 cos ωn
c �n/21436C�/R=��0 < xn < 2
c
�6�
�"5(FD9<âjnjn = 1
3:9<;½16�BA<34EGFH?@905 �q� �)� � � �)��3:9<; � � �)��z�1}?0LBE�3:FD9�3}CG1 e A<CGCG1:9 e 1}gq?@CE)J01P5+1:�@An149 e 1 {xn} ç
xn+1 =Bn+1Cn+1
An+1Bn+1
=|cn+1 − bn+1||bn+1 − an+1|
=
∣∣∣∣
j2n − jn + 1
jn(cn − bn)
∣∣∣∣
∣∣∣∣
j2n − jn − 1
jn(jn + 1)(cn − bn)
∣∣∣∣
=|(jn + 1)(j2
n − jn + 1)||j2
n − jn − 1|=
|jn + 1| · |(j2n − jn + 1)jn|
|(j2n − jn − 1)jn|
=|jn + 1| · |(jn + jn − 1)|
|(jn − jn − 1)|
=| cos 2ωn + 1 + i sin 2ωn| · |2 cos 2ωn − 1|
| − 1 + 2i sin 2ωn|
=
√2 + 2 cos 2ωn · |2 cos 2ωn − 1|
√
1 + 4 sin2 2ωn
=
√
2(1 + cos 2ωn) · |2(2 cos2 ωn − 1) − 1|√
1 + 16 sin2 ωn cos2 ωn
=2 cos ωn · |4 cos2 ωn − 3|
√1 + 16 cos2 ωn − 16 cos4 ωn
=xn · |x2
n − 3|√
1 + 4x2n − x4
n
c�X14�<9<1SE)J01SgHA<9 e EGFH?@905
f : [0, 2] → [0, 2]365
f(x) =x · |x2 − 3|√1 + 4x2 − x4
3:9<;g : [0, 2] → [0, 4]
3:5g(x) =
(f(x)
)2=
x2(x2 − 3)2
1 + 4x2 − x4
cV��J01:9xn+1 = f(xn)
cæ"L0571:C^�@1PE)J@3ZEVFDg�E)J01S5716�BAn1:9 e 1 {xn} J@3:583�/DFDMNFRE L
�0E)J01:9L =
L · |L2 − 3|√1 + 4L2 − L4
�z�JBF e JNâ�FR�@145
0�1�03:9<;
23:5XKn?@5(5(FHL@/H18�B34/DAn145Xgq?@C
Lc
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f(x) =√
3J@3:583�57?B/DA@EGFH?@9
x0FD9
(0, 2)� L:=}E)J01Nà^9BEI14CIM}16;BFH3ZEI1P�¥34/DAn1��J01:?@CG1:M��¥5(FD9 e 1
f(0) = 0�f(2) = 2
� 369<;fFD5 e ?@9BEGFD9@An?BA<5�?@9
[0, 2]�(c~��J01:9
x1 = f(x0) =√
3��369<;�E)J01�â416?@M}1ZEIC^=~K<CG?0LB/214M�;<?0145�9<?�E�M}3��61�571:90571<c�ð4FDM��FD/D36C�/R=��VF|E ;<?0145N9<?�ESM}3��61�5+14905+1�F|g
x0FD5N3½57?B/DA@EGFH?@9�?Bg
fn(x) =√
3��z�J014CG1
fn(x);<149<?�EI145
f(f(. . . f
︸ ︷︷ ︸
n� � �� ����
n ≥ 1
(x))) c��{1:9 e 1@�:E)J01�5+1�EO?Bg �+K<CG?0JBFHL@FREI1:;P�B34/DAn145 �Vgq?@C
x0
FH5 {x | fn(x) =
√3}
∩ (0, 2)c
�A<C�E)J01:C7��z�1PJ@3Z�@1
g′(x) =−2x(x2 − 3)(x2 + 1)
(
x2 −(3 −
√6)) (
x2 −(3 +
√6))
(1 + 4x2 − x4)2�
5+?}E)J@34E8L<?�E)J~gHA<9 e EGFH?@905g369<;
f36CG1�5)EIC�F e EG/ =~FD9 e CG1:3:5(FD9<�@9 [
0,√
3 −√
6] 3:9<;
[√3, 2
] �@369<;N5IEICGF e EG/R=�;<1 e CG1:3:5(FD9<âP?@9 [√3 −
√6,
√3] c
ï��6ñ
� ?6z e ?@905)F2;01:CQE)J01 ;@F���14CG1:9 e 1g(x) − x2 =
2x2(x2 − 1
) (x2 − 4
)
1 + 4x2 − x4
c�ð4FH9 e 11 + 4x2 − x4 = 5 − (x2 − 2)2 > 0
gq?@Cx ∈ [0, 2]
�6z�1{J@3Z�@1g(x) − x2 > 0
gq?@C1+�@1:C^=x ∈ (0, 1)
369<;g(x) − x2 < 0
gq?@C"1+�@14C^=x ∈ (1, 2)
c>�P149 e 1B�f(x) > xgq?@CO1+�@1:C^=
x ∈ (0, 1)�f(x) < x
gq?@CO1+�@1:C^=x ∈ (1, 2)
��3:9<;f(x) = x
FDg<369<; ?@9@/ =FDgx = 0
�x = 1
�0?@Cx = 2
c�:1ZE
a =√
3 −√
6cOð4FD9 e 1
a < 1�:z�1{J@3��@1
a < f(a)c �:1�E
I =[a, f(a)
] cà2E8FH5{1:3:5�=~EI? e J01 e �}E)J@3ZE1 < f(a) <
√3c���JBA<5���z�1}J@3��@1
f(f(a)
)< f(a)
�5(FD9 e 1f(a) > 1
c»ð4FH9 e 1fFH5}5)EIC�F e EG/ =Ç;<1 e CG1:3:5(FD9<â½369<; e ?@9BEGFH9@An?BA<5~?@9
I3:9<;
I ⊆[√
3 −√
6,√
3] ��z�1~5+1:1}E)J@3ZE
f(I) =[f(f(a)
), f(a)
] � z�JBF e J�FHMNK0/|F2145E)J@34E
f(I) ⊆ I��5(FD9 e 1
f(f(a)
)=
√53461345
− 16741345
√6 > a
c ��J01:CG1Zgq?@CG1@�a < f
(f(a)
)< 1 < f(a) <
√3c
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z�JBF e J FH5}9<?�E�+K<CG?0JBFHL@FREI1:; �:� E)J01:CG1~1(ä:FH5IEG5
n0 ∈ �5(A e J�E)J@3ZE
xn0∈ I
c���J0149½F|E�z{FD/|/Ogq?B/D/H?6zE)J@34Exn ∈ I
gq?@CV1+�@14C^=n ∈ �
z{FRE)Jn ≥ n0
�@5)FH9 e 1f(I) ⊆ I
cð4A<K<K<?@5+1B�Bgq?@C¥E)J01{K0A<CGKn?@571�?Bg e ?@9BEIC�36;@F e EGF2?@9��6E)J@34E¥E)J01P5716�BAn1:9 e 1 {xn} J@365E)J01{gq?B/D/H?6z{FH9<â�K0CG?@Kn14C�E^=`çxn /∈ I
gq?@CV1+�@14C^=n ∈ �
c�:1ZE�A<5";<149<?�EI1�E)JBFH5XK<CG?@K<1:C�E�=}L:= P c ��1 e ?@905(FH;<14CQ571+�@1:C�34/ e 36571:54cpnÕ:ë Ø �c
x0 < ac��J0149 � L:=�FD9<;@A e EGFH?@9���z�1�?0L�EG34FH9
xn < agq?@CS1+�@14C^=
n ∈ ��V5(FD9 e 1
fFD55)EIC�F e EG/ =�FH9 e CG14365)FH9<âP?@9
[0, a]c � àqg
xk < agq?@CQ57?@M�1
k ∈ ��BE)J0149
f(xk) < f(a)�?@C
xk+1 < f(a)c�5@= K<CG?@K<1:C�E�= P �Oz�1�â41�E xk+1 < a
c �¡5<A@E>E)J01:9½z�1�J@3��@1xn < xn+1
gq?@CV1+�@1:C^=n ∈ �
�<5)FH9 e 1x < f(x)
gq?@CX1+�@14C^=x ∈ (0, 1)
c���JBA<5��@E)J015+1:�@An149 e 1 {xn} FH5¥FH9 e CG14365)FH9<â"369<;SL0?BA<9<;<1:;S3�L0?:�@1B��3:9<;{E)J01:CG1Zgq?@CG18FRE¥J@365�38/|FHM�F|E+�Lc 7 1 e 34/D/ E)J@3ZE�E)J01}Kn?@5(5(FHL@/H1S�B3:/|An1:5P?Bg
L36CG1
0�1� 369<;
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