wholeissue 34 2 - canadian mathematical societye 6 6 9@f @1:34a5+1 e?@9

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ð4A<C8/2145 e ã?�EZ�145AB

1ZECD

;�� A<9½CG1 e E�369<â�/H1ABCD

3��@1 eAD < AB

��?@9e J0?BFH5)F|EO/2145XKn?BFD9BEG5F1ZE

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

d­e /2?@5716; CGFHâ4J�E e FDC e A0/H3:C e =B/DFD9<;<14C�J@365N369 FH9BEG16â414C C�3�;BFDA<5}3:9<;�369 FH9BEG16â414CJ014FHâ4J�E�c¥��J01{9@A<M�14CGF e 34/0�B34/DAn1P?Bg<E)J018�@?B/|A<M�1>FD5Qgq?BA<COEGFDM�145�E)J01{9@A<M�14CGF e 34/0�B34/DAn1?Bg<FREG5OEG?�EG34/05)A<CGg23 e 1�36CG143 � FD9 e /DAn;BFH9<â�F|E�5OEI?@K 3:9<;�L0?�E�EI?@MS�7c �X1ZEG1:CGMNFD9<18E)J01�5(M}34/D/H�1:5IEQKn?@5(5(FHL@/H18�@?B/|A<M�1>gq?@COE)J01 e =B/DFD9<;<14C�c´ � � � c·­#¯�°,±�°�² ´�Ó Â ¼\Ä&Å�´aÔ µ ¼�Å�´ Á.Ê Ä µ � c

d 909<1�369<;d5nCG1:9<;@3>K0/D3Z=S3>âZ36M}1"z�JBF e JSL<1:â�FH905¥z{FRE)JS3�K0F|/21�?BgnEI?0?�E)J@K@F e �65:c��J01+=�34/|EG1:CG9034EG1PEGA<CG905Xz{FRE)J d 909<1 â4?BFD9<â{�nC�5IE�c�æX9~143 e J}K@/H3�=@1:C7� 5XEGA<CI9��<5+J01�MNA<5IECG1:M}?:�@1

1�3�@?@C

6EG?0?�E)J@K0F e ��5Qg2CG?@M�E)J01{K0F|/210c ��J01{K0/D3Z=@14C z�J0? CG1:M}?:�@145QE)J01>/H3:5)EEI?0?�E)J@K@F e �Xz{FH905�E)J01�âZ3:M�10c B?@C�z�JBF e JP?Bg�E)J01Q�B34/DAn145�?Bg

ng2CG?@M

36EG?

40FD9 e /DA<5)FR�@1;<?0145�5nCG149<;03�J@3Z�@1P3�z{FH909@FD9<â>5)EIC�3ZEI1:â+=

´ � � � c·­#¯�°,±�°�² ´�Ó Â ¼\Ä&Å�´aÔ µ ¼�Å�´ Á.Ê Ä µ � c� 3Z� �X1ZEG1:CGMNFD9<1P34/D/<FD9BEI1:â41:C�5

xgq?@C z�JBF e J x − 3

3x − 2

FD5X369�FH9BEG16â414C�c� L6� �X1ZEG1:CGMNFD9<1P34/D/<FD9BEI1:â41:C�5

ygq?@C¥z�JBF e J 3y3 + 3

3y2 + y − 2

FD5X369�FD9BEI1:â41:C¯c´ � � � c·­#¯�°,±�°�² ´�Ó Â ¼\Ä&Å�´aÔ µ ¼�Å�´ Á ² Ä µ � c

à^9�3�5716�BAn1:9 e 1�?BgVgq?BA<CP9@A<M�L01:C�5���E)J01�571 e ?@9<; 9@A<M�L01:CPFD5�E�z{F e 1}E)J01~�nC�5IE9@A<M�L01:C¯c d /D5+?��0E)J01S5(A<M¹?Bg`E)J01��nC�5IE"3:9<;�gq?BA<C�E)J~9@A<M}L<14C�58FH59�<E)J01S5(A<M¹?Bg`E)J015+1 e ?@9<;N3:9<; E)JBFDCG;�FH5

7�@369<; E)J01{5(A<M ?BgnE)J01P57�@A<3:CG1:5�?Bg<E)J01{gq?BA<CQ9@A<M}L<14C�5XFH5

78c

�X1ZEG1:CGMNFD9<1P3:/|/n5(A e J�5716�BAn1:9 e 1:54c

�:ï

´ � ��� c·­#¯�°,±�°�² ´�Ó Â ¼\Ä&Å�´aÔ µ ¼�Å�´ Á.Ê Ä µ � cd /H3ZE�EGF e 1SKn?BFD9BEXFH583 Kn?BFD9BE

(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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�:1ZEx369<;

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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

�6z�1{5(A eIe 14575)FR�@1Z/R=N?0L�EG34FH9 E)J01�1:�@A0F �B3:/H1:9BEV1:�@A<3ZEGF2?@905x + y

x − y= 2 +

√5�

x + y =(2 +

√5)x −

(2 +

√5)y�

�6�

369<;x+

√5x = 3y +

√5yc���J01�C�3ZEGF2?�?Bg@E)J0185(FH;<145Q?Bg@E)J018CG1 e E�369<â�/H1�E)J0149P=BF21Z/2;@5

x

y=

(3 +

√5)(

1 −√

5)

(1 +

√5)(

1 −√

5) =

√5 + 1

2

�z�JBF e J FH5QE)J01���?B/H;<149 7 34EGFH?�����1 e 369N34/H57? e ?@M}K@A@EI1 y

x=

√5 − 1

2

c������� ������������� ���������������������� ����� �!��"$#%�&������'�(

´ è � c·­#¯�°,±�°�² ´�Ó Â ¼$�X´ Ì ´ ¹ �+Ò ¯b¸36� ½ > µ+É+¯ ´  ½�À ¯g°�µ Ä ¸^µ<c�X1ZEG1:CGMNFD9<1>E)J01PCG1Z/H3ZEGF2?@9057JBFHK}L01ZE�z�1:1:9

x3:9<;

yF|g

x2 + y cos2 α = x sin α cos α3:9<;

x cos 2α + y sin 2α = 0c

�|d 5(5(A<M}1>E)J@3ZEVL<?�E)Jx369<;

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

2 cos2 α = 1 + cos 2α369<;

2 sin α cos α = sin 2α�E)J01{�nC�5IEV?BgnE)J01�E�z�?Nâ�F �@1:9}16�BA<34EGFH?@905VFH5X5)A e)e 1:5(5(F �@14/ =N16�BA0FR�B34/2149BE�EG?

x2 + 12y(1 + cos 2α) = 1

2x sin 2α

�369<;

x sin 2α − y cos 2α = 2x2 + yc

��JBA<5��4E)J01XE�z�?{â�FR�@149S1:�@A<3ZEGF2?@905`=@FH14/H;>E)J01Xgq?B/D/H?6z{FH9<âX16�BA0FR�B34/2149BE 5�=@5)EG1:Mª?Bg01:�@A<36�EGF2?@905Zçx sin 2α − y cos 2α = 2x2 + y

� �^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<;

cos 2α�6z�1P?0LBE�3:FD9

sin 2α =x(2x2 + y)

x2 + y2

3:9<;cos 2α = −y(2x2 + y)

x2 + y2

�5(FD9 e 1

x3:9<;

y3:CG1�9<?@9<� �(14CG?`c�ð6�@A<3:CGFD9<â�L<?�E)J 16�BA<34EGFH?@905�369<; 3:K<K@/R=BFH9<â E)J01.B=BE)J@36â4?@CG1:3:9�à�;01:9BEGFRE�=��

sin2 θ + cos2 θ = 1�B/2143�;@5QEI?

x2(2x2 + y)2

(x2 + y2)2+

y2(2x2 + y)2

(x2 + y2)2= 1

�z�JBF e J�5(FDM}K@/DF|��145QEG?

(2x2 + y)2

x2 + y2= 1

c�P149 e 1B�

4x2 + 4y − 1 = 0�6z�J014CG1�z�1PJ@3Z�@1P36âZ3:FD9�A<5+1:; E)J01>g23 e EOE)J@3ZE

x 6= 0c

) ¾�Ó�* ��1 e ?BA0/H;}J@3Z�@1{5+�BA<36CG1:;}16�BA<34EGFH?@905 �qï �¥3:9<; �q� ��EI?�â41ZEx2 sin2 2α − 2xy sin 2α cos 2α + y2 cos2 2α = (2x2 + y)2

�x2 cos2 2α + 2xy cos 2α sin 2α + y2 sin2 2α = 0

�:ñ

369<; E)J0149}36;<;SEI?Nâ41�Ex2(sin2 2α + cos2 2α) + y2(sin2 2α + cos2 2α) = (2x2 + y)2

�?@C

x2 + y2 = (2x2 + y)2�

z�JBF e JNâ�FR�@1454x2 + 4y − 1 = 0

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(12

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(14

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(113

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1 − tan a + tan b

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=12

+ 14

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(a + b + c)2 ≥ 9abc�05(FD9 e 1

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�FDC�5)E+�057?@M�1�âZCG?BA<9<;}CGA0/H1:54c � ? E)J@CG?6z{FD9<âPL<?�ä6145:c � ?�5)E�3 e �:FH9<âPL<?�ä6145:c � ?BA36CG1{9<?�E�34/D/H?6z�16;�EI? K@A@EV?@9<1{L<?�ä{FH905)F2;01{3:9<?�E)J01:C¯c � @?@CO5+?@M}1{CG143657?@9��:z�J01:9�à ;0?E)JBFH5�E^=0K<1P?BgnK<CG?0LB/214M z{FRE)J 3PâZCG?BA<K}?Bgn5)EGAn;01:9BE�5Z�6E)J01+=�34/RzP3�=05¥zP369BE EI?PEIC^=SE)J0145+1579<143 �Z= E)JBFD9<âZ5 � ���JBFD/H1�FRE�M}3��61:5 5+14905+1�EI?�â41�E ?BA@E�3�K0FH1 e 1�?Bg"5 e C�3:K K<3:Kn14C�369<; ;0?�3�L@FRE?Bg�FH9��@145)EGFHâZ34EGFH?@9��n/H1ZE+� 58/H?0? �NL<1Zgq?@CG1Pz�1�/21436K�c{ð4FD9 e 1���14/|FH9<1�FH5�g23:5)EG1:C"?@9~57MN3:/|/214CL<?�ä6145"3:9<;}�VA0FH9 e =�FH5Vg23:5)EG1:CQ?@9�/H3:CGâ41�L<?�ä6145Z��/21�E+� 5QEIC^=}J@3Z�BFH9<â{�`1Z/DFD9<1PM}?:�@1{3:/|/�?BgE)J01N57MN3:/|/OL0?�ä61:5 � E�3 �:FH9<â

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

32MNFD9@A@EI145�3:9<;}E)J01:9~5(FREG583:CG?BA<9<;

���

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1/H3:CGâ41�L<?�ä ��VA0FD9 e =�� 5>EGFHM}1NFD5PCG1:;@A e 16;�EG?

9 × 5 = 45M�FH9@A@EG1:5P3:9<;�`1Z/DFD9<1@� 5OEGFDM�1�FH9 e CG1436571:5¥EG?

16×2+6 = 38MNFD9@A@EI145:c ��1836CG189<?6z 34E � ç ò0� 3ncRM�c365QE)J01{�<9@FH57JBFH9<â8EGFDM�10c ��JBFD5XFD5";<1Z�n9@FREI1Z/R=NL<1�E�EI14C�cE5<A@E�FH5VFREOE)J01PL<145)EQK<?@575)F2LB/21

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1557MN3:/|/XL0?�ä61:5 3:9<;

2/D36CGâ41�L0?�ä61:54c���JBFH5�EG3��61:5�J014C

15 × 2 + 2 × 6 = 42MNFD9@A@EI145:c¥�VA0FH9 e =Pz{FD/|/BE)JBA<5VM�?4�@1

15(M}34/D/<L<?�ä{369<;

8/D36CGâ41L<?�ä6145Z�6z�JBF e J�z{F|/D/0EG3��61

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43M�FH9@A@EG1:54cNð6?~F|E�5+1:1:MN5�E)J@3ZE

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x57MN3:/|/<369<;

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16 − x369<;

10 − y57MN3:/|/XL<?�ä6145:cfà^9áE)JBFH5 e 36571@�V��14/|FH9<1�EG3��61:5

2x + 6yM�FH9@A@EG1:5�369<;��VA0FH9 e =NE�3 �6145

3(16 − x) + 5(10 − y) = 98 − 3x − 5yMNFD9@A@EI145:cð4A<K<K<?@5+1�9<?6z E)J@34E>E)J01+= �<9@FH57J½E)J01�ì^?0L 3ZE � ç ò<� 3ncRM�cQ?@C�1:3:CG/|F214C�c � ��1@� /D/5+J0?6z E)J@34EOE)JBFD5 e 36909<?�EQ3 e EGA<3:/|/R=NJ@3:K<K<1:9�c �{à^9�E)JBFD5 e 3:5+1B�01:3 e J�z�?@C �65Vgq?@C�34EQM}?@5)E

42MNFD9@A@EI145Z��57?�E)J01�EG?�EG34/�EGFDM�1 E)J@3ZE�E)J01+=�z�?@C ��z�?BA0/2;�J@3��@1 EI?�L<1N3ZE�M}?@5)E

84MNFD9@A@EI145:c BCG?@M�E)J01{FD9@gq?@CIMN34EGFH?@9N3�L0?:�@1B��E)J014FDCOEG?�EG34/0EGFHM}1>FH5

(2x + 6y) + (98 − 3x − 5y) = 98 + y − xc

àqg�E)J014FDCVEI?�E�3:/�EGFHM}1SFH5834E"M�?@5IE84MNFD9@A@EI145Z�<E)J01:9

98 + y − x ≤ 84 � E)J@3ZEFH5��x − y ≥ 14

cu5<A@Ex369<;

y3:CG1P9<?@9<�(9<1:âZ34EGF �@1{FD9BEI1:â41:C�5Qz{FRE)J

0 ≤ x ≤ 163:9<;

0 ≤ y ≤ 10c>��J014CG14gq?@CG1B�

x − y ≤ 16� L01 e 3:A<571

xFD5�3ZE�M}?@5)E

16�(cS�P149 e 1B�<z�1J@3Z�@1

14 ≤ x − y ≤ 16c

��J01jK<?@575)F2LB/21ÇK03:FDC�5(x, y)

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2x + 6y ≤ 423:9<;

98 − 3x − 5y ≤ 42� ?@C

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n

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P = x + y�6z�J014CG1

x369<;

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x − 3√

x + 1 = 3√

y + 2 − yc

è c}à^9�3NK0/D369<1B��/21�EΓL<1N3 e FDC e /21�z{F|E)J e 149BEICG1

O369<;�C�36;@F|A<5

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s(i, k)�

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Ai369<;

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n5(A e J8E)J@3ZE ?@9<1 e 3:9 e ?B/2?BA<C

ne CG?@5(5+145 5+?�E)J@34E�E)J01Q�B3:/|An1:5

s(i, k)36CG1{3:/|/�16�BA<3:/�c� Ä � Ïá¿���Á�ÇÃ� ��� � ÅOÊZÆ6Ë

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f(f(x − y)

)= f(x)f(y) − f(x) + f(y) − xy

c� c �FD9<;�34/D/@EIC�FHK@/2145"?Bg�9<?@9<�)9<16âZ3ZEGFR�@1>FD9BEI1:â41:C�5

(x, y, m)5)A e JSE)J@3ZE x! + y!

n!= 3n� z{FRE)JSE)J01 e ?@9��@149BEGF2?@9

0! = 1�7c

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x1 = a�:z�J01:CG1

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xn+1 = 3x3n − 7x2

n + 5xngq?@C

n ≥ 1c

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Ñ c �X1ZEG1:CGMNFD9<1P34/D/nK03:FDC�5(x, y)

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�y3 + 1 − x2y − x2 = 0

cè c �:1�E

A�B�<3:9<;

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kc �:1ZE�E)J01�/DFD9<1:5

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B369<;

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ABM�1:1ZE�5

hFH9

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gFH9

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A��z�J0149<1+�@1:C

B3:9<;

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FD9�E)J01�K0/D369<1 � E)J@3ZE�FD5Z�x3:9<;

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t = 01(ä:3 e EG/R= ?@9<1�/H3:M}KáFD5S5Iz{FRE e J01:;�?@9�c d E369�=�FH9BEG16â414CVEGFHM}1

t ≥ 1�`1)ä:3 e EG/ =}E)J0?@571S/D36MNK<5�3:CG1 5Iz{FRE e J01:;�?@9Nz�JBF e J�36CG1 34E"3;@FD5)E�369 e 1N?Bg

2005g2CG?@M¹5+?@M}1S/H3:M}K}z�JBF e J�FD5�34/HCG143�;6=~5Iz{FRE e J01:;�?@9�c�.<CG?:�@1�E)J@3ZE1+�@1:C^=�/D36MNK z{F|/D/�L01P5Gz{F|E e J016;N?@9N3ZEQ5+?@M}1�EGFHM}1<c

� c �:1�EQ(n)

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E)J@34EQ(

Q(Q(20052005)

))

= 7c

� c �:1ZErL01{E)J01SC�36;@F|A<58?Bg�E)J01PFH905 e CGFHL<1:;}57K<J01:CG1S?Bg�3PEG1ZEIC�36J016;@CG?@9

ABCD3:9<;/21�E

r1�

r2�

r3�O369<;

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ABCD369<;�34/H57?}EG3:9<â41:9BE"EI?�E)J01K0/D369<145 e ?@9BEG34FH9@FD9<â�E)J01P?�E)J01:C¥E)J@CG161{g23 e 145:cX.<CG?4�@1>E)J@3ZE

2

r=

1

r1

+1

r2

+1

r3

+1

r4

c� c d 5716�BAn1:9 e 1

x0�x1�x2�. . .��?Bg�CG143:/`9@A<M}L<14C�58FD58K<1:C�F2?0;BF e z{F|E)J}K<1:C�F2?0;

p�<F|g

xn+p = xngq?@CQ34/D/n9<?@9<�)9<16âZ3ZEGFR�@1>FH9BEG16â414C�5

n369<;

p > 0c

� 3Z��.<CG?:�@1�E)J@3ZEOE)J01:CG1�1)ä6FD5)E�5X3P5+1:�@An149 e 1�z{FRE)J�Kn14CGFH?0;2�6z�JBF e J�5(34EGFD5(�n1:5

xn+1 = xn − 1

xn

�n = 0, 1, 2, . . .

c� L6��.<CG?:�@1�E)J@3ZEPgq?@CP369�=áFH9BEG16â414C

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

�:1ZEX�Y�<369<;

ZL01PE)J01Pgq161�E"?Bg�E)J01�Kn14CIK<1:9<;BF e A0/D36C�5�g2CG?@M

IEG?�E)J01�5(FH;<145

BC�CA

�@3:9<;AB

�@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

(y1 + y2)

(x2

1

y1

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= x21 + x2

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y1

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2y1

y2

≥ x21 + x2

2 + 2

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y1

· x22y1

y2

= x21 + x2

2 + 2x1x2 = (x1 + x2)2 c

Ê�°+» Ò�Ä ¸(°�¹��<c �:1�Ea =

x1

x1 + x2

369<;b =

y1

y1 + y2

c>��J01:91 − a =

x2

x1 + x2

3:9<;1 − b =

y2

y1 + y2

�@3:9<; E)J01�?@C�F2â�FD903:/<FD9<16�BA<3:/|F|E^= e 369}L01PCG1�zPC�F|E�EG1:9�FD9 E)J01{gq?@CGMa2

b+

(1 − a)2

1 − b≥ 1

�z�J01:CG1

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)

�a2 − a2b + b − 2ab + a2b ≥ b − b2 �

?@Ca2 + b2 ≥ 2ab

�6z�JBF e J FH5QEIC�An1<c

�B�

Ê�°+» Ò�Ä ¸(°�¹��@c�ð4FD9 e 1 x2

y≥ 2x − y

gq?@Cy > 0

�6z�1P3:K<K@/R=SE)JBFH5QE�z{F e 1�EI?�?0LBE�3:FD9

a2

b+

(1 − a)2

1 − b≥ (2a − b) +

(2(1 − a) − (1 − b)

)= 1

�z�J01:CG1

a369<;

b3:CG1�;014�<9<16;N3:5VFH9}ð6?B/|A@EGF2?@9 � c

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,√

y2, . . . ,√

yn)369<; ( x1√y1

,x2√y2

, . . . ,xn√yn

) �6z�J014CG1�1:3 e Jxi369<;

yiFD5X3�K<?@5(FREGFR�@1>9@A<M�L01:C+ç

x21

y1

+x2

2

y2

+ · · · +x2

n

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

IEG?�E)J01X5(FH;<145

BC�CA

�AB

CG1:5(Kn1 e EGFR�@1Z/R=`c��J0149IX = IY = IZ

c 7 FHâ4J�EVEICGFD369<â�/H1:5IAZ

�IA1X

�IA2X

�IYA

3:CG1 e ?@9<âZCGAn149BE+��369<;�5(FD9 e 1XFD5�E)J01XMNFH;<�)Kn?BFD9BE ?BgB5716âZM}1:9BE

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?�

Ê�°+»(Ì ´�Ó Â ¼ ) °�µ�¹�¹�¸�² � µ Ä ²g¸ '�¸�² ½NÈsÄ&Å�´ ¹�² ½ � ¯ ´+´ � ´ � Î µNÌ�¸ ´ ¯EÆ�°�² ½ ² Ä�Ò�Ó¬´ ¹ ÄT½/Ë ¹N¸!Ì ´ ¯b²g¸ Ä µ Ä­#°+»�¸ ÄH´ ��¹N¸��0µ Ó¬´IÀ µ Ä µ,» Ò ¹ ¼ µ ½�º µ+¯4� ´ »(°�¹¬µ ½ ÊP±�µP¸T¹ �sµ�¹ Ó �P¸ >,Å ° Ò,½ ­#°+» ' À °�Á1Á Ò ¹�¸ Ä&¼À °+»�» ´ É ´�½ ~¸T¹ Ä!´ ¯�~µ�Ì ´ ¹ ½ � � ½uË Ê È Ç. ´ ÉP¸!Ì ´�Ä&Å�´ ²�°+» Ò�Ä ¸!°�¹�°�� � µ Ä ²g¸ '�¸�²+Ç

� 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

��J@3ZE e ?@M}K@/21�EI145 E)J01�MN34EG1:C�FH34/@?@9{�0/21Xgq?@C�E)JBFH5¥9@A<M�L01:C ?Bg�E)J01 À °�¯V¹ ´ ¯¯c¥ð6149<;M�18=@?BA<CQ9@F e 1P57?B/DA@EGFH?@905X3:9<;}â41:9<14C�3:/|F��)3ZEGF2?@9054c

���

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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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a2 + b2 + c2 ≥ 4S√

3c

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√bc + b

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≥ 33√

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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cð4FH9 e 1

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cÏ � ÄnÁqÀÇÊ4Ã�ÆBÅ � � �

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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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2(a + b − c)

c¥��JBFH5�=@FH14/H;05∑

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(b + c − a)(c + a − b)

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ab + bc + ca = s2 + r2 + 4Rr�^� �

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� ����� � �(b + c − a)(c + a − b) = 4r(4R + r)

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4R + r

a2 + b2 + c2 ≥ 4S

3 +4(R − 2r)

4R + r+ (a − b)2 + (b − c)2 + (c − a)2

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ra�rb�rc�@K<CG?4�@1>E)J@34E

a

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b

rb+

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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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ï6�6�

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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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ð6?BF|Ea�b1ZE

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

.

....

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

..

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

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

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

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 ≤

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>(

)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

(π2

− A) c ��J01:9

λ = f(A) + f(B) + f(C) ≥ f(A) + f(

π2

− A) c

�:1ZEu = min

{A� π

2− A

} 3:9<;v = max

{A� π

2− A

} cQ��J01:90 < u ≤ v < π

2z{F|E)Ju + v = 0 + π

2

cE5@= �:14M}MN3 � 3�âZ34FH9��6z�1Pâ41�Ef(u) + f(v) ≥ f(0) + f

(π2

)= 1 + 0

c��J@3ZEQFD5Z�

f(A) + f(

π2

− A)

≥ 1c ��J014CG14gq?@CG1B�

c2 ≥ 1càqg�z�1 e ?@905(FH;<14C�369�3 e A@EI1 FH57?@5 e 14/H1:5�EICGFD369<â�/H1

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

�n369<;C cot C → 0

cQ�P149 e 1B�λ → 1

c ��JBA<5��c2 = 1

c� � ������������' "�� � ��.@* & �#� � � ����+ � � = �1���� ? � � � � � ���# � � =�� ��� ��� ��� � M � � � � ���4=E� M �������

� ��# � � D . ��� ./� &�� � ( � !"* &��;� M �����;' � ��� � � =�� ��� ��� ��� � � � ���� � � + �������� �5M�!"� � ���<� ( * � � � � �� ����L��D � ��� � �

(6��� � � � � ����' � ��������� �

)��MC. � � . �>? � � ��� ��� ��� �!��� ��� ����� � ����� �;F * ��'������ ���

����' D � ���# � �B�;� � ��� �����;� M ������� ��# � D ! � !%�@� (�� .�. �? �&��� ��� ���J��� � �&� �;� � � * � ��'$�;�� � ����� �� ����'������6���E6 ����

(6��� � � � � ����' � ��� ����� �

)( � � ����� ���������� ���;F%6��� � � � � � " � ��� ����' �5��!������F���

c2

(

����� �������������������������������! #"%$&"�'�(*),+�-/.�02143*57680�9;:;(=<>'�:?3�)4(@5A:�<CB29�5ED#(@5A:F 1%'G'@(@-H.�(*D#"�5E)I1� ?->.ID?0J"2"*KL<M6 9�5E)�'�"� ?<ONMPH�

QSR T�U*V,W*V�XYRZVI[?U�\%U%]�^k�

`�

m� \�]?U%W%[?U*]_[#`�W�V

2� W�VYRZVI[?U�\%U%]

nRZ^bacW%dLdZU�e(?f%$J g(='G'#9h+*Ki(8jlk ]

(k, `, m)R j [#`2U*]?UmU#n�RL^#[�o k ^#Rp[?R T�Uq]?U%W*drV�s4tvu2U*]�^

a1

�a2

�. . .

�ak

^gs!ag`H[#`�W%[ k∏

i=1

ai = 1W�V4e

k∑

i=1

∑̀

j=1

ai+j−1

m

= n�

w `2U*]?U [#`2Ux^gs!u�^ya#]�RZoI[?^W�]?U�[?W�z�U*Vmt k eIs2d kk�

{%s4o4o k ^yUq[#`�W�[ jlk ]|^ k t_U(k, `, m)

[#`2UqRZVI[?U�\%U%]2005

RL^8Ugn�o4]?U%^G^#R�uId�U w `IRpd�U1987

RZ^V k [ �~} RZV4e>[#`2U k ]?e4U%]?U�e>[c]�RZo�d�U(k, `, m)

�.�"*Kh32:Z9i"�5�+�-|��"2(yKA.�D?0IK;"�'�+2(� h�*<S�E1�-*'#9;)4(=<SP>�2<��r.@���

��U w RLdpd�^y` k w [#`�W%[(74, 3, 3)

W�V4e(16, 5, 3)

W*]?U�[#`2U k V�d X8[ w k o k ^g^gRZu�RpdLR�[?R�U%^jlk ]4[#`2U�[c]�RZo�d�U(k, `, m)

�A� `IRZ^ w RLdpd*u4Ue2U�eIs!a#U*e j ] k t�[#`2U j W�a?[4[#`�W�[nRL^rUgn�o4]?U%^G^#R�uId�Ujlk ]

(k, `, m)R j W*V4e k V�d�X>R j

n ≥ k`m�

ï�ï:�

ð4A<K<K<?@5+1"E)J@3ZEnFH5�1)ä�K0CG1:5(5(FHL@/H1�gq?@C

(k, `, m)c���J0149 L:=�CG14Kn1434EG16;PA<571�?BgBE)J01d�� � � � à^9<1:�@A<34/DFRE�=���z�1PJ@3��@1

n =

k∑

i=1

∑̀

j=1

ai+j−1

m

≥k∑

i=1

`

∏̀

j=1

ai+j−1

1`

m

= `mk∑

i=1

∏̀

j=1

ai+j−1

m`

≥ k`m

k∏

i=1

∏̀

j=1

ai+j−1

m`

1k

= k`m

∏̀

j=1

k∏

i=1

ai+j−1

mk`

= k`m

∏̀

j=1

1

mk`

= k`m c

@?@Ca ≥ 1

� /H1ZEf(a)

L01�E)J01��B34/DAn1�?Bg k∑

i=1

(

∑̀

j=1

ai+j−1

)m z�J01:9�z�1N5+1�E(a1, a2, . . . , an) = (a, a−1, 1, . . . , 1)

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)

��K0CG1 e FD5+1Z/R=­z�J01:91987 < k`m ≤ 2005

c à^9 E)JBFD5 e 3:5+1B�3`3 ≤ k`m ≤ 2005

�"57?áE)J@3ZE3 ≤ ` ≤

⌊(2005/3)1/3

⌋= 8

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

` = ab�b ≥ 3

3:9<;nFH5

(k, l, m)1)ä�K0CG1:5(5(FHL@/H1@��E)J01:9

nFH5�34/H57?

(kam, b, m)1)ä�K0CG1:5(5(FHL@/H1<c½ð4FDgDEGFD9<âNE)J01�B3:/|An1:5"?Bg

k�`�@3:9<;

mCG1+�@143:/D5VE)J@3ZE

(k, `, m)FD5"14FRE)J01:C

(74, 3, 3)?@C

(16, 5, 3)c

��������� �����&� ��� ������# 6 �&���&� �������� ����� � �!��"$#%�&������'1������ � �;F ��� ��� � F�� ����' ����� � �������;F�� �����&� ���� � � � ��(

������8�����������>�����*�2�4������� �! g"%$&"�'@(�)v+�- F 9�0��1�Kl-m�r(@5JD��=(=<O�E �1*'@"�@<�&"��_1�5I9�12� j

a�b�c� W*V4e

dW�]?Uxo k ^#Rp[?R T�U RZVI[?U�\%U%]�^ � o4] k T�U [#`�W�[

45

(1

a + b + c + d + 1− 1

(a + 1)(b + 1)(c + 1)(d + 1)

)

≤ 4 +∑

� ����� � �

[1

a + 1+

1

(a + 1)(b + 1)

] �

ï�ï6ï

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�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)

�O3:9<;½36K0K0/ =@FD9<â�E)J01 d�� � � �à^9<16�BA<3:/|F|E^=��6z�1�J@3��@11

(a + 1)(b + 1)(c + 1)(d + 1)≥ 256

(a + b + c + d + 4)4=

1

(x + 1)4�

� ����� � �

1

a + 1≥ 4

4√

(a + 1)(b + 1)(c + 1)(d + 1)

≥ 16

a + b + c + d + 4=

4

x + 1

�3:9<; ∑

� ����� � �

1

(a + 1)(b + 1)≥ 4

(a + 1)(b + 1)(c + 1)(d + 1)

≥ 64

(a + b + c + d + 4)2=

4

(x + 1)2c

à2E�E)J014CG14gq?@CG1{5(A�� e 145VEG?�5+J0?6z E)J@34E45

(1

4x + 1− 1

(x + 1)4

)

≤ 4 +4

x + 1+

4

(x + 1)2�

?@C16x5 + 39x4 − 86x2 + 84x + 12

(4x + 1)(x + 1)4≥ 0

càqg

x ≤ 1��E)J01:9

16x5 + 39x4 − 86x2 + 84x + 12

= 16x5 + 39x4 + 84x(1 − x) + 2(1 − x2) + 10 > 0c

àqgx > 1

��E)J01:916x5 + 39x4 − 86x2 + 84x + 12

> 55x3 − 86x2 + 84x + 12

= 43x(x − 1)2 + 12x3 + 41x + 12 > 0c

��J014CG14gq?@CG1>E)J01{FD9<16�BA<3:/|F|E^=}J0?B/H;05X3:9<; E)J01�1:�@A<34/DFRE�=N;<?0145"9<?�E�c� � �� ����J����' "��@� � &LD �� � �&��;� * � ���� �� ��� � ��������� ��� �����@��� ����� � �����E�/�!�E6 � � �4� *3. �5��� ��

� ���4!4* *8��� & � &;-�� �� � ' ���� ����� �:9 �;��� ����� ����= ������� �:9 �;���;��? ������� �%����' � �BA ��=���� � � � ����' ������6���=�6 ����(

ï�ï6�

��è Ñ � c ) �6�6� � ç ï�ï:� � ï6ïZò �£­#¯�°,±�°�² ´�Ó Â ¼�Ô ¸�� Å�´ » º µ Ä µP¸�»e» ´*½ Æ�° Ò�´ ¹ ½ � ¯bµ�¹/� ´ c�:1ZE {Ln} L<1�E)J01 �ZA e 365N5716�BAn1:9 e 1á;<1Z�n9<1:;ÇL:= L0 = 2

�L1 = 1

�"3:9<;Ln+1 = Ln + Ln−1

gq?@Cn ≥ 1

c{.<CG?4�@1PE)J@3ZE+�<gq?@CV34/D/�9<?@9<�)9<16âZ3ZEGFR�@1PFH9BEG16â414C�5n�z�1�J@3��@1

bn/2c∑

k=0

(n + 1

2k + 1

)L2k

2n=

bn/2c∑

k=0

(n − k

k

)Lk

22k

cÊ�°+» Ò�Ä ¸(°�¹I ¼\Ä&Å�´ ±�¯�°,±�°�² ´ ¯ ½�´ � ±�µ+¹ Ó¬´+Ó Â ¼\Ä&Å�´a´�Ó ¸ Ä °�¯*Ç

à2E FH5Sz�14/|/ �69<?6zP9 E)J@34ELk = αk

1 + αk2

gq?@Ck = 0

�1�2�

. . .�Vz�J014CG1

α1 = 12

(1 +

√5) 3:9<;

α2 = 12

(1 −

√5) 3:CG1SE)J01NCG?0?�E�5{?Bg

x2 − x − 1 = 0c��J01{C�F2â4J�E�5(FH;<1{?Bg<E)J01>K<CG?@K<?@5+1:; FH;<149BEGF|E^=SE)J0149N5)Anâ4â41:5IEG5�E)J01>FD9BEICG?0;@A e EGFH?@9}?Bg<E)J01

Kn?B/ =09<?@M�FH34/H5Pn(x) =

bn/2c∑

k=0

(n − k

k

)(x

4

)k �6gq?@Cn = 0

�1�2�. . .cQà2E¥FH5QCG1:36;@F|/R=

5+1:1:9 E)J@3ZEP0(x) = P1(x) = 1

c��1 e /D3:FDM�E)J@34E

Pn(x)573ZEGFH5)��145QE)J01>gq?B/D/H?6z{FH9<â�CG1 e A<CGCG1:9 e 1�CG1Z/H3ZEGF2?@9`ç

Pn+1(x) = Pn(x) +x

4Pn−1(x)

� �^ï �gq?@CQ34/D/

n ∈ �c �0?}145)E�3�LB/DFD5+J �^ï �)�@9<?�EI1>�nC�5IEOE)J@3ZEx

4Pn−1(x) =

b(n−1)/2c∑

k=0

(n − 1 − k

k

)(x

4

)k+1

=

b(n−1)/2c+1∑

k=1

(n − k

k − 1

)(x

4

)k càqg

nFD5"1+�@1:9 � 5(3Z=

n = 2`�I��E)J01:9

b(n − 1)/2c + 1 = b(n + 1)/2c = bn/2c = `c

�P149 e 1B�

Pn(x) +x

4Pn−1(x) =

∑̀

k=0

(n − k

k

)(x

4

)k

+∑̀

k=1

(n − k

k − 1

)(x

4

)k

=∑̀

k=1

[(n − k

k

)

+

(n − k

k − 1

)](x

4

)k

+ 1

=∑̀

k=1

(n + 1 − k

k

)(x

4

)k

+ 1

=

b(n+1)/2c∑

k=0

(n + 1 − k

k

)(x

4

)k

= Pn+1(x)c

�"5(FD9<âVE)J01�573:M�1�3:CGâ�A<M�149BE�z{F|E)J{MNFD9<?@C�M�?0;BFD� e 3ZEGF2?@9PCG1:âZ36CG;BFH9<âVE)J01�A<K0Kn14C�/|FHM�F|E�5?BgnE)J01P5)A<M}MN34EGFH?@905Z�:z�1 e 369}14365)FD/ =�5+J0?6z E)J@34E �qï � FD5X3:/D5+?�EICGAn18z�J01:9nFH5"?0;0;�c

ï�ï6ñ

� ?6z���5(A<K0Kn?@571�E)J@34Ex > −1

c~ð4FH9 e 1�E)J01 e J@36C�3 e EI14CGFD5)EGF e 16�BA<34EGFH?@9½?BgOE)J01CG1 e A<CGCG1:9 e 1{CG1Z/H3ZEGF2?@9�â�F �@1:9SFD9 �^ï �`FH54w2 − 4w − x = 0

�:E)J01 e J@3:C�3 e EG1:C�FH5IEGF e CG?0?�E�536CG1q1 = 1

2

(1 +

√1 + x

) 369<;q2 = 1

2

(1 − √

1 + x) c

BCG?@M ��9<?6zP9NE)J016?@C^=��0E)J01 â4149<1:C�34/`5+?B/|A@EGF2?@9�EI? E)J01SCG1 e A<CGCG1:9 e 1 CG1Z/H3ZEGF2?@9}FD9�^ï �OFD58â�F �@1:9~L:=Pn(x) = c1qn+1

1 + c2qn+12

�Bz�J01:CG1c13:9<;

c236CG1{EG?}L<1S;<1�EI14C)�MNFD9<16;`c�ð6?B/ �@FD9<â

c1q1 + c2q2 = P0(x) = 13:9<;

c1q21 + c2q2

2 = P1(x) = 1�6z�1�n9<;�E)J@34E

c1 = 1/√

1 + x369<;

c2 = −1/√

1 + xc �:1�E

z =√

1 + xc ��1{E)J01:9J@3Z�@1

Pn(x) =1

z

[(1 + z

2

)n+1

−(

1 − z

2

)n+1]

=1

2n+1z

[n+1∑

k=0

(n + 1

k

)

zk −n+1∑

k=0

(n + 1

k

)

(−z)k

]

=1

2nz

bn/2c∑

j=0

(n + 1

2j + 1

)

z2j+1 =1

2n

bn/2c∑

k=0

(n + 1

2k + 1

)

(1 + x)k c��J014CG14gq?@CG1B��z�1PJ@3Z�@1P1:5IEG36L@/|FH57J016;SE)J01{FH;<149BEGF|E^=

bn/2c∑

k=0

(n − k

k

)(x

4

)k

=1

2n

bn/2c∑

k=0

(n + 1

2k + 1

)

(1 + x)k c �^� �ð4FH9 e 1

1+ αi = α2i

gq?@Ci = 1

369<;i = 2

�BL:=S5(AnL@5)EGFREGA@EGFH9<âαigq?@C

xFD9 �q� �)�4z�1�57161E)J@34E

1

2n

bn/2c∑

k=0

(n + 1

2k + 1

)

α2k1 =

bn/2c∑

k=0

(n − k

k

)αk

1

4k

�^ñ �

3:9<; 1

2n

bn/2c∑

k=0

(n + 1

2k + 1

)

α2k2 =

bn/2c∑

k=0

(n − k

k

)αk

2

4k

c �Dò ���J01PK0CG?@Kn?@5716; F2;01:9BEGFRE�= gq?B/D/H?6zP5"L:=�3�;0;@FD9<â �qñ �¥3:9<; �Dò �7c� � �������J����'�"��#� � &BD �� � �&��;� * � ���� �� �4� � ����������� � � �&� ��� ����� � �����E�1�!�E6 � � �4� *3. � � � �$� ����'

!%��* � �#� ��� � �)��� � � � =��;� � � �J�� � (�����$6���E6 �������# ��(' ��' � � ���F ���# �����$6��� �;F=!�J���� ��" ������ � � ����'�� � � �������� � � {Ln} ����� " ���:6 � ��� ��' "������%������ � ���� � {Un} ����� � � F���� �4= � ��� ������� � ����� Un+1 = Un+Un−1

����'�� � 6��� � ��� �!� ��L�"��%� �������&" ��� � ���������� ���������(

������8�|�?�*���I�>�������I�4������� �! #"%$&"�'�(*)_+�-mN �=9;)�9h3��!3* g)E3�9Z<�A5�9 �*(� h'g9;:l-H"���%"*K;(*)4"�<�*"*Ki(�)4"�<�N��<��r.@���

� U�[nu2U�W�V RLVI[cU*\%U*] w R�[#`

n ≥ 2� V� n

� d�U@[Eu2U [#`2U�^GU�[ kIj o k RLVI[?^

(x1, x2, . . . , xn)^gs!ag`H[#`�W%[

xi ≥ 0jlk ]�W*dpd

iW�V4e

0 < x1 + x2 + · · · + xn ≤ 1�

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∆ = max{∆z1�∆z2

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0 < x1 + x2 + · · · + xn ≤ zz{F|E)Jxi ≥ 0

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V (z) = zn/n!c���J01

k��� 5)/DF e 1�?BgVE)J01�5(FDM}K@/21)ä

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�Qz�JBF e J�J@365S�@?B/|A<M�1V (zk) − V (zk−1)

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(z∗k, 0, 0, . . . , 0)

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S =s∑

k=1

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FH5I3:5

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z

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∫ 1

0

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z

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∫ 1

0

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z

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∞∑

j=1

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1j+1

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3

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r = C0V0��3:9<;�E)J01N5(Kn1:16;@5P?Bg E)J01 e 3:C8369<;~E)J01}L@FDCG;�L:=

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vb

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V0A

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vc

vb

c� 1)ä:E+�6/H1ZE

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V CA369<;

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vc

· rc

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vc· rFH5PE)J01~M}3¯ä6FDMNA<M

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e ?BFD9 e F2;01:5�z{F|E)JV0F

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C0V0

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vb

vc· r

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vb

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4

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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� � � * � ��'$�;� � � �#��� � ( �!�����%����� ������ ��� ��# 6 ���������������� ����� �!��"$#%�&������'�(

& � ����� ����' " �#6 ��� ���&��' � � ��� � ����������6��� � ��� ����� � �;F������ ���� �Sx

����' {12�22�

. . .�

n2} F���n ∈ {15

�16�23�24�31�32�39�40} �������� � ����� � ����� �5=����� �!� � F�����LK���# 6 �&���

{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}

� � ��� ��'$#%� ��� ��"��&��6��� � �&� �&��� '�� � ������NF ���# � ��� ������=!�J���� ��" �����(

ï�ï �

��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�

C��CG1457K<1 e EGF �@14/ =�c �:1ZE

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

� � �� ����J����' "�� ������� ����� ���/����� �!������ � � �#� � �$���� ! � �$! � ��� � ��������+ & -������ ����� � ����� �;F���� �:9 �;���;�/���� �:9 �;���;��? ����� � � ����' � �BA ��= �� � � �

(# ����&� � �&��� �

)��� ./� ? ��� ? ���%� � � ��"$�������8��� �����h�

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(���� ����' ����&� � �&���

)� � .�!

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(���������' �������� �����

)�3!%��* � �#� �@� � ����� � � � =��E� � ���B� � � � ��&��N�

0MC. ����� �N�����# -�����"O�� � � � ��# ����� � � ����' � ����6���E6 ����(

ï�ï+�

��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

c� � �� ����J����' "�� *3.�����* *,!"� �$� ���& ��� � � � ���" � ��B=;� �� ����� � ������������ ��� ���������8��������

� � ��� � �$��* & + � !%� ��*,!"��+ � ��� � . ME� �E� � �!� ���� = ��� �;� * �!� ��������E���)6���� � � �� ! � �$! � ��� � ��������+ & -�4�� ���J��� � �&� � �;F ���� ��9 �;���;� ���� ��9 �;���;�I? ����� � � ����' � �BA ��= �� � � � � � ./� ? ��� ? ���%� � � ��"$�������� ���J��� � �&� �;���$��� ��L� � ��"$������� �>* &L� ��!"� ? � � � & ��4! � � � � ��� �� ����� � �2� & . � �#! ? � & �4!%� � !"� �& !� ��* D ? !"�4� � ����' � ������! � � ( � & * & � � & ! � � �4=�� �&� � � � �&� ��� ����� � �����E�5����� � �4=�� �&�;� ���/� ��� ��* & � -����� ? !"� � 0�! ��?� ������� � ��# ����� � � � � . * & �#� � � ����+ � � = ���$�� ? � � � � ���&�# � � =��>� ����� ��� �M � � � � ���4=E� M ������� ��# � � � &BD �� � ��&J�E� * � ���� �� � ��� � � ����� � � � �&�,��� ����� � �����E���!�E6 � � �4� *3. � ��� ����� � ��� � !"�#�$!%� �8&��� ����" ��� � ����(' �)� � � ./! ��. � ���� �>D��� � ������ � � * � ��� ��8�$!%! �h� � & ��� �C�� ���= � ���=;� � � � � � � � ���!"* *8��� & � & -�� �� � ' ���� � ���� ��9 �;��� ����� ����= ��� ���� ��9 �;���;� ? ������� � ����'� �BA ��=���� � � ��� �%� +./� �4�? * & �". �� !%M & -� ����' + ( * & �)����. ME� ���� � "�� � ��M !%� � �)� � ( * � � � � �� ����L�#D � �5� � �$� �4� ��� � ? . �4� . � . *,!%� . * -���� � !10 �5� ���J��� � �&'$��' ' � M ��� � ��' ��� ��'1� M �!� � � ' ��� �&'1��E6��!� � � � ( �$( � *,!%!4�#� � 0 �!�&��" ��# # � � � � ��� � ��� ��� � ����'$� � D ����. � ! ( �1� � . ���1� .�+$�". �C� � ������ � �+ ������ � ��+$!N./� +$! �1� � D ��� & � &J�E� � =� � � ���;� + ������ � � ��� ��� � � +. � � * �����!�&���&�#����� ����= ���/��� ($D � ��� �* � �/��� �� D ! � !%�>� (%� .�. � ? �&��� ����� ����� � �����E� � � * � � '$�E�$� � �/� � �$� �& �C� 0MC. ����� �N������# -�����"O�� � �� ��# ����� � � ����' ������6���E6 ����(

�������q�m���������v�4�����I�A��� �%� �! g"%$&"�'@(�)C+�- � > ; 04 �9L' �*9L'y0E(� 1�5E) 1� �Ki(@- 6>(='%:i"�52<�r5I9��*(@ �'#9h:�-q"���&(@��9�521*< &(@��9�521*<A.����� U�[

A0B0C0u2U8W*VqRZ^ k ^yacU%dZU*^�[c]?RLW�V4\@dZU w ` k ^GUxW�o4Ugn�W�V4\@dZU

A0RL^V k [

120◦�

��U e4U 9 V4U W ^yU :�s!U%V4a#U kIj [c]?RLW�V4\@dZU*^AnBnCn

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

� W�V4e ^ k_jlk ]�[#`� � 8 ] k T�U�[#`�W�[W*dpd2[#`�]?U�U8W*V4\@d�U%^�W*o4o2] k W�ag`

60◦ W*^ n → ∞�

��2)�> �� `IRL^ o4] k uId�U%t RZ^�Wm^Go4U�acRLW*d&acW*^yU kIj W*V k o4U*VCo4] k uId�U%t e2U*^Ga#]?RZu4U*e�u*X� s!e2W�`8{�ag` w W�]h[��rRLV�� �JW*V [?U�ag`�V k d k \yX8`2U�dZoxs4^&t_W�z�UM[#`2U�t_W�[#`2U*tmW%[?RZac^Sacs4]c]�R�acs2dps4tRZVI[?U%dpd�U*a?[?s4W*dpd�Xx^#[?RLtms2dLW%[?RLV4\W*V4e8^ k acRLW*dpd�X ]?U*^go k V2^#R�uId�U���� ���i5J:;(@ c541�:Z9;"�521%K%��"!3* c541�K!"��; "��>$O32:i(� h' �;"� F 1�:l0J( �_12:Z9iD�1�K��I(�1* c5�9Z52�I�

4� ������ � � o4o �������I����4� �

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

�!°�¯sµP»�»nc ) ¾¬Ó Ç * ��JBFH5�FHMNM�1:;@FD34EG14/ =�1+�@? �6145�E)J01�FHMNKn?@C�EG3:9BE�@An145)EGFH?@9á?Bg¥z�J01�E)J01:C�5(A e J�3:9<â�/2145

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

��1 e 369 9<?6z ;01ZEG1:CGMNFD9<1�E)J01 �+K<CG?0JBFHL@FREI1:;á�B3:/|An1:5 �}gq?@Cx0c ��/H1:3:CG/ =��QE)J0116�BA<34EGFH?@9

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

��1á3:CG1á9<?6z â4?BFD9<â�EG?�5+J0?6z�E)J@3ZE+��gq?@CN143 e Jx0 ∈ (0, 2)

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<;

2c~ð4FH9 e 1

xn ∈ (0, a)369<;a < 1

�BE)J01:9LM�A<5)EXL<1

0c���JBFD5"FD5"3 e ?@9BEIC�3�;BF e EGFH?@9��nL01 e 3:A<571�3:9}FD9 e CG1:3:5(FD9<â5+1:�@An149 e 1�z{FRE)JSEI14CIMN5XFD9

(0, a)e 36909<?�EVJ@3Z�@1>/DFDMNFRE

0c

pnÕ:ë Ø y<cf(a) < x0 <

√3c��J0149

f(x0) < f(f(a)

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(f(a)

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;<1 e CG1:3:5(FD9<âS?@9 [a,

√3] c�5@=}K<CG?@K<1:C�E�= P �Bz�1Sâ41ZE x1 < a

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xn < agq?@C"1+�@14C^=

n ∈ �z{FRE)J

n ≥ 1�0z�JBF e J}/2143�;@5�A<5�EG?�3e ?@9BEIC�36;@F e EGF2?@9��@5)FHM�FD/D36C¥EG?SE)J01�?@9<1P?0L�EG34FH9<1:;�FH9}��36571 ï c

pnÕ:ë Ø�� c √3 ≤ x0

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5)A e J E)J@3ZExn0

<√

3c��J0149

xn < agq?@CV1+�@14C^=

n ∈ �z{FRE)J

n ≥ n0 + 1�0369<;Sz�1{3�âZ34FH9}?0LBE�3:FD9}3e ?@9BEIC�36;@F e EGF2?@9��@5)FHM�FD/D36C¥EG?SE)J0?@571�?0L�EG34FH9<1:;�FD9}��3:5+145 ï 3:9<; � c

pnÕ:ë Ø � c √3 ≤ xn

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gq?@C{1+�@1:C^=x ∈ (1, 2)

c��{1:9 e 1@� E)J01~5+1:�@An149 e 1 {xn} FD5�5)EIC�F e EG/ = ;016�e CG1:3:5(FD9<â>3:9<; E)J014CG14gq?@CG1B��E)J014CG1�1)ä6FD5)E�5lim xn = L

�6z{FRE)JL ∈

[√3, 2

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LFD5

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2c

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5(A e JE)J@34Exn0

∈ I3:9<;SE)J0149

xn ∈ Igq?@C�1+�@1:C^=

n ∈ �z{F|E)J

n ≥ n0�B5)FH9 e 1

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gq?@C 1+�@1:C^=n ∈ �

c���J0149yn ∈ Igq?@CP1+�@14C^=

n ∈ �c �X14�<9<1}E)J01�gHA<9 e EGFH?@9

h : I → ��z{FRE)Jh(x) = f

(f(x)

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I�05(FD9 e 1

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Ic

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gq?@CP1+�@1:C^=n ∈ �

c ��J015+1:�@An149 e 145 {y2n} 3:9<; {y2n+1} 3:CG1�M}?@9<?�EI?@9<10c � @?@C¥1(ä:3:M}K@/21B�6FDg y0 ≤ y2�4E)J01:9

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yn ∈ Ic���J01:CG1Zgq?@CG1@�<E)J01:CG1�1)ä6FD5)E

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ð4FHM�FD/D36C�/R=��L2 = h(L2)

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xL<1�E�z�16149b369<;

f(b)c~ð4FD9 e 1

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3:9<;x ∈ I

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�z�J01:CG1`(x) = x6 + 3 +

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c � ?6z`(x)

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I?Bg

h(x) = xc

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�z�JBF e J e ?@MNK0/H1ZEG1:5QE)J01{K<CG?0?BgGc

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BcQð4FH9 e 1

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