WholeIssue 30 6 - CMS-SMC

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Å/ÆnÇ�È�ZA\N\�É�ÆL\�ÊLËOÌqÍ�Ë6Î�ÆL\N\{Æ/È�Z1Ï<ÐfÌ{Ë�ÏOVYX�Æ9ÑL\{Ë�ÒMÉ)Ì�Æ�Ð>Ë�ÌqÍ�Ë�X�ÓLÉ�ÓfÌ{Ë�ÓLÒ�Ô�U�Ì@ÌqÍ�ËË�Ï�Õ�Æ�Î9Ön`�Ò�Z1ÏuÇ/Ì{Ë�É�]�ÈPË�È�ZN\A\u×EÆL\N\{Ë�×5ÌYØ�ÆnÇLXWÓ�Ï<É�ÈPË�X_É,Í�Ë�Ë�ÌDÔ�Å/ÆnÇ�X�Ì�Ë�Ó�ÒÙÈ�ZA\N\9X�ËL×GË,Z�ÊLËÚ V�ÆLZ1ÏYÌJÉ@Î�Æ�XsË�Ó>×DÍf×EÆ�XEX�ËL×5Ì�ÓLÏ/É�ÈPË�X¡ÔÛ b c \1Ó<Ì[ÌJZ�×EË�V�ÆLZJÏuÌ�CN4@26H9;>CJ7>B

(x, y) � }6F/0L?D0�BGF/0 d ;/;<?D8>CJ7/2�BE034s2L?D06I/;�BGF�CN7>BE0L�30L?[43bÜ<;<?!0� L2LKOH<.10 � (3, −4)2L798

(5, 0)2L?D0+.N23B[BDC d 0+H9;>CJ7>B[4 � I>=g�.N23B[BDC d 0�H9;>CJ7>B[4R;<7'BGF/0 d CJ? d =9K�gq0L?D037 d 0Ý;>g�BGF/0d CJ? d .J0x2 + y2 = 25

bÞ bT�L0�B

S =22 − 1

22× 32 − 1

32× 42 − 1

42× · · · × 20042 − 1

20042

b�ß3 �H/?D0L4�4S2L4�2g1?[2 d BDCJ;<7 � ?D0�8>= d 0L8�BD;~.1;�}6034GB_BD0L?DKO43bà b c VYÓ3\AZ1Ï�Õ>X�Æ�ÒfËPCN4�2)7<=9KOI903?!BGF<2�B@?D032�8<4@BGF/0)4�2�KO0�gq;<?�}�2�?D8<4�2�798RI<2 d ��}�2�?D8<4 �4�= d F�2�4

83382L798

50705b�L0�B

A2L798

BI90ágq;>=9?G��8<CJ��CAB�H923.NCN798/?D;<KO0L4 � 2L798�.10�B C

I/0j2â�<�<0L�58>C1��CQBH923.NCN798/?D;<KO09b!ãqgA + B = C � 890�BE03?EK�CJ790�23.N.nH9;<454GC1I>.10��>2L.A=n0L4s;>g C b½ b��F/0 d CJ? d .J0�}+CABGF�0��>=923BDCJ;<7

x2 +y2 = 1CN7>BE03?[4,0 d BD4�BGF/0s.NCN790

y = 7x+523BuB�}6;8<CN4GBDCN7 d B_Hn;>CN7>BD4 � A 2L798

Bb��L0�B

OI90sBGF/0 d 0L7>BE?D0�;>g<BGF/0 d CN? d .10/bäÜ�CN798�BGF/0PKO0L2L4�=9?D0;>g

∠AOBb

å b@¥@0�BD0L?DKMCN790�23.N.9CN7>BE0L�30L?[4x4�= d F)BGF<2�B

(x2 − 3x + 1)x+1 = 1b

æ b��L0�Bf(a, b)

8/0L79;�BD0@BGF/0s4G=9Kç;>g�BGF/0@CN7>BE0L�30L?[4WI90�B�}60�037a2L798

b � CN7 d .N=94GCQ�<0/bWÜ<;<?0� L2LKOH<.10 � f(1, 5) = 1+2+3+4+5 = 152L798

f(3, 6) = 3+4+5+6 = 18b¥@0�BD0L?DKMCN790�BGF/0P�>23.N=n0�;>g

f(133333, 533333)b

�� b c F/0G L2��3;<7O2�798M2�7R0��>=/CN.N23BD0L?[23.9BE?DCN2�79��.J0)F<2��<0)0��>=92L.�Hn03?DCNKR0�BE03?[4Lbsãqg�BGF/062L?D0L2;>gnBGF/06F/0G L2��3;<7�CJ4

6√

345�<=92L?D0+=97<CQBD4 � }6F<23BäCN4!BGF/0�2�?D0326;>gnBGF/0�BE?[CJ2L79��.10��

� b@¥@0�BD0L?DKMCN790�23.N.<�>23.N=n034�;>gxgq;<?_}6F>C d F

(1999x − 99)3 = (1234x − 56)3 + (765x − 43)3b

� b!Ü�CN798M23.N.n45;>.N=<BDCJ;<7/4(x, y)

CJ7M?D032L.n7<=9KRI/0L?[4�BD;1

x+

1

y=

5

6�

x2y + xy2 = 30b

Û�� b!Ü�CN798~BGF/0+7<=9KRI/0L?�;>gY?D032L.n4,;>.A=<BDC1;<7/4äBE;)BGF/0�0��>=923BDCJ;<7sin(x) =

x

315

b

�P0G LBÝ}60 ��C �<0�BGF/0 ;�� d CN2L.O4,;>.A=<BDC1;<7/4'BE;çBGF/0 �L�� 2�?[CABDCNKR0 ��2�BGF/0LKM23BDC d 4 ;<KMHn0�BDCABDCJ;<7� �L�� <��¢��>��¢�£�� b

Þ �� à�³�s¿n²q±�²E¼�¸¯³�_±�µ+¸@¼´_±�²G°�Ã��M«+¼´À�¸_±u²q±�²G«�Ä� ��<�� !#"%$'&�(*)!,+-$.&�/10��2!,�3��!,�4"%$5��/6&�/7+8!,��9Y��:

Û b � F/037�2)g123BGF/03?�8>CJ4EBE?DCJI<=<BD0L4�2~7<=9KOI903?�;>g d 2L798<CJ0L4�2LKR;<79�)F>CN4 d F>CN.J8/?D037 � 0L2 d Fd F>CN.J8M?D0 d 03C �<0L4 ��¢6d 2�798>C1034s2�798~BGF/03?D0+CJ4s;<790+.J03gNB�;L�<03?�b�ãqg � F/;�}60,�<0L? � B�}6;)g1?DCJ0L798<4;{;>CJ7~BGF/06��?D;>=9HO2�798~BGF/0 d 2L798<CJ0L4s2L?D0�?D0L8<CN4GBE?[C1I>=<BE0L8 � BGF/037R032 d F d F>CN.J8M?D0 d 0�CQ�<034 ��d 2L798<CJ0L4@2�798~BGF/03?D0�2�?D0PBGF<?D0�0�.10�gNB�;L�<03?�b � F<2�BäCJ4!BGF/0PBE;�B[2L.n7<=9KOI903?@;>g d 2L798<CJ0L4<�= ÆL\�Ç/ÌJZ{Æ�Ï?>_�L0�B

xI90@BGF/0�7<=9KOI903?W;>g d 2�798>C1034ä2L798�.10�B

yI90@BGF/0�7<=9KOI903?W;>g d F>CA.18<?D0L7�179;�BäCJ7 d .A=n8<CN79�PBGF/0PB�}6;~g1?[C103798/4 � b��YF/037~}60�F<2��<0

x = 15y + 12L798

x = 11(y + 2) + 3b

�YF>=94 � 15y+1 = 11(y+2)+3}6F>C d F�:<CJ03.J8/4

y = 6b!��;�}

x = 15(6)+1 = 91b��037 d 0 � BGF/0PBE;�B[2L.n7<=9KOI903?@;>g d 2L798<CJ0L4�CJ4 91

bÞ bu-<;>=9? d F<2��<=n06037>BDC103?!4EBE?DC d BE03KR037>B!Hn;<4GCABDCAg

n � Hn;<45;<7/4f(n) =

(

4(1)2 − 1)

×(

4(2)2 − 1)

× · · · × (4n2 − 1)b

->2�?!0� L2LKOH<.10 � f(1) = 30�B

f(2) = 3 × 15 = 45b�9?D;>=>�<0L?9BD;>=<BE034Y.1034n�>2L.J03=9?[4�890

nHn;>=9?9.1034,�>=n03.A.1034

f(n)034GB9=97 d 2L?E?�@0�H/2�?[g12LCQB�b

= ÆL\�Ç/ÌJZ{Æ�Ï >n-/=/CN4,�>=n04k2 − 1 = (2k)2 − 1 = (2k − 1)(2k + 1) � ;<7M2

f(n) =(

2(1) − 1)(

2(1) + 1)

×(

2(2) − 1)(

2(2) + 1)

× · · ·× (2n − 1)(2n + 1)

= (1)(3) × (3)(5) × (5)(7) × · · ·× (2n − 3)(2n − 1) × (2n − 1)(2n + 1)

= 3252 · · · (2n − 1)2(2n + 1)

=(

(3)(5) · · · (2n − 1))2

(2n + 1)b

��

-<;>=9?W�<=n0f(n)

4,;>CQB�=97 d 2�?D?�@0�H92L?Dg123CAB � CA.>g12L=<BW0�B�CN.<4G=.�MBW�>=n0 2n+145;>CAB�=97 d 2L?E?�@0H92L?Dg123CAB�b�3C�.10f79;<KRI<?D0fCJKMH923CJ?

2n + 1034GB�=97 d 2�?D?�@0�H/2�?[g12LCQB � CN.@0L4EB�.10 d 2L?E?�@0�8Y� =9779;<KRI<?D0@CNKOH/2LCN? � 8>CJ45;<7/4 2a+1

b<¥�2L7/4 d 0 d 2�4 � 2n+1 = (2a+1)2 = 4a2+4a+1 �89;<7 dn = 2a(a+1)

b<¤ @0 d CNH9?D;/�>=n0LKO0L7>B � 4GC n = 2a(a+1)H9;>=9?�=97 d 03?[B[2LCN7�037>BDC103?Hn;<4GCABDCAg

a � 2L.J;<?[4 2n+1 = (2a+1)2034GB�=97 d 2L?E?�@0PH/2�?[g12LCQB�b9�L0L4��>2L.J03=9?[4!8/0

nH9;>=9?.1034,�>=n03.A.1034

f(n)034GB@=97 d 2�?D?�@0~H92L?Dg123CABs45;<7>B�8/;<7 d 8/;<7/7�@0�034PH92L?

n = 2a(a + 1) �a = 1 � 2 � 3 � . . .

bà b c

10�GKR0�BE?D0f.N2�8/8903?�?D0L4EBD4~2��2LCN7/4GB�2R�<03?[BDC d 23.W}�23.N.�b'�YF/0fK�C18/��H9;>CJ7>B6;>g�BGF/0.J2�898/0L?äCJ4!B�}+C d 0�2L4@g12�?äg1?D;<K�BGF/06��?D;>=9798O2L4@CQBäCJ4@g1?D;<K�BGF/0P}�23.N.�b � F<23B�F/03CJ�3F�B@;<7BGF/0P}�2L.A.�8/;/0L4!BGF/0+.N2�8/8903?!?D032 d F.�

= ÆL\�Ç/ÌJZ{Æ�Ï?>��L0�B�BGF/0�Hn;>CN7>BD4P2�B�}6F>C d FOBGF/0~.N2�8/8903?@BD;>= d F/0L4BGF/0!}�23.N.L2�798�BGF/0��?D;>=9798�I/0A2�798

B � ?D0L4�Hn0 d BDCQ�<0�.Q: � 2L798.10�BCI/0�BGF/0sH9;>CJ7>Bu23B9}6F>C d F�BGF/0!}�23.N.�KO0�0�BD4uBGF/0��?D;>=9798�b�L0�B

PI90�BGF/0�KMCJ89�GHn;>CN7>Bä;>g<BGF/0�.J2�898/0L? � 2�7986.J0�B_H9;>CJ7>B DI90M;<7RBGF/06}�2L.A.�45;OBGF<2�B

PDCN4�H/2�?[23.N.J03.YBE;OBGF/0M��?D;>=9798�bÜ�CN7/2L.A.Q: � .10�B E

I90RBGF/0�Hn;>CN7>B6;<7ÝBGF/0f��?D;>=9798'4G= d FTBGF<2�BPE

CJ4@H/2�?[23.N.J03./BE;6BGF/0P}�23.N.�b�L0�BxI906BGF/0~.J0L79�¡BGFf;>g

DPb+�YF/037�BGF/0).10379�¡BGF�;>g

EPCJ4

2xb_�3CN7 d 0sBGF/0s.J2�898/0L?uCJ4

10KO0�BE?D034W.J;<79� � }60�F<2��<0

|AP | = |BP | = 5b

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A

BC

D

E

P

�P;�}∠DAP = ∠EPB

2L798∠DPA = ∠EBP � }6F/0L7 d 0 � 4DAP

CJ4d ;<79��?[=n0L7>B�BD; 4EPBb6�YF>=94 � |DA| = |EP | = 2x

2�798 |EB| = |DP | = xbc H9H<.Q:>CJ79��BGF/0�->:�BGF<2��3;<?D032�7~�YF/0L;<?D0LK BE; 4DAP � }60�;/I>B[2LCN7 (2x)2 + x2 = 52 �}6F>C d F6:<CJ03.J8/4

x =√

5bW�3CJ7 d 0 |DA| = 2x

2�798 |CD| = |EP | = 2x � }606�30�B|AC| = 4x = 4

√5b

�YF/03?D03gq;<?D0 � BGF/0+.N2�8/8903?!?D032 d F/034s26F/03CJ�3F�B�;>g 4√5KR0�BE?D0L4s;<7~BGF/0P}�23.N.�b

½ b��9?D;>=>�<03?O=97 79;<KOI/?D0 �2 4�C d F>C��u?D034�89;<7>BM.10'H/?D0LK�C103? d F>C��Y?D0á034GB � 0�B��>=/C890,�>C1037>B�BE?D;>CJ4Pgq;>CJ4�H<.N=94��?[2L798�4GC�.J0MH/?D0LK�C103? d F>C��u?D0O034GBP8�@0LH<.J2 d @0 �2M.{� 23=<BE?D0RI/;>=<BHn;>=9?�8/0,�<0L7<CN?ä.10 d F>C��u?D068/0L4�=97<CQB @0L43b= ÆL\�Ç/ÌJZ{Æ�Ï >��;>CQB

1abcde.10�79;<KOI/?D0O0L7��>=n0L4EBDC1;<7�b��@7�2

1abcde = 105 + x; �=

x = abcdeb��L;<?[45�<=n0+.J0�H/?D0LK�C103? d F>C��Y?D060L4EB�8�@03H/.N2 d @0 �2�.�� 2L=<BE?D06I/;>=B

abcde1 = 10x + 1b 0T79;>=>�<032L=j79;<KOI/?D0Ý034GB6BE?D;>CN4�gq;>CJ4MH/.A=94R��?[2L798j�>=n0�.10H9?D03KMCJ0L? � d 06�>=/CY8/;<7/790

10x + 1 = 3(105 + x) �7x = 3 × 105 − 1 �x = 42857

b�L0�79;<KOI/?D0 d F/03? d F�@060L4EB�8/;<7 d

142857b

å b!ß��>23.N=92�BE03

√

5 + 2√

13 +3

√

5 − 2√

13b

��£

= ÆL\�Ç/ÌJZ{Æ�Ï?>��L0�Bx =

3√

5 + 2√

132�798

y =3√

5 − 2√

13 � 2�798�.J0�B s = x + yb�YF>=94 � s CN4!BGF/0�?D0L�<=/CN?D0�8M4�=9K�b � 06F<2��<0

s3 = (x + y)3 = x3 + 3x2y + 3xy2 + y3 = x3 + y3 + 3xysb

�P;�}x3 = 5 + 2

√132L798

y3 = 5 − 2√

13 � 2�798xy =

3

√

(5 + 2√

13)(5 − 2√

13) = 3√

25 − 4(13) = 3√

−27 = −3b

��037 d 0 �s3 = 5 + 2

√13 + 5 − 2

√13 + 3(−3)s �}6F>C d F�4GCJKMH/.ACN�n0L4äBD;

s3 + 9s − 10 = 0b@©<:~CJ7/4�Hn0 d BDC1;<7 � s = 1

CN4@2�?D;/;�B!;>g9BGF>CJ40��>=923BDCJ;<7�b��F>=94 � s − 1CN4!2Pg12 d BE;<?W;>g

s3 +9s − 10b@©<:6.J;<79�P8>CQ�>CJ4GC1;<7 � }60�;/I>B[2LCN7

s2 + s + 102�4!BGF/0�;�BGF/0L?Wg12 d BD;<?�b��F/0L?D0�gq;<?D0 �

(s − 1)(s2 + s + 10) = 0b

�P;�}s2+s+10 = 0

F<2L4W79;�?D032L.�?D;/;�BD43bW��037 d 0 � BGF/0�;<7<.Q:+4,;>.A=<BDC1;<7PBE;PBGF/0�2�I/;L�<00��>=923BDCJ;<7MCN4s = 1

b��YF/03?D03gq;<?D0 �3

√

5 + 2√

13 +3

√

5 − 2√

13 = 1b

æ b<�9?D;>=>�<0L?WBD;>=<BE034�.1034@H923CJ?D034�8�� 037>BDC103?[4sHn;<4GCABDCAg14(x, y)

BE0�.N.J0L4s�>=n0x2 − 11y! = 2003

b�[->2L?�8�@0��n7<CQBDC1;<7 � 1! = 1 � 2! = 1 · 2 = 2 � 3! = 1 · 2 · 3 = 6 � 0�B d b �= ÆL\�Ç/ÌJZ{Æ�Ï >_�3C

x034GB6H/2LCN? � 8>CJ45;<7/4 x = 2k � 23.1;<?[4 x2 = (2k)2 = 4k2

0L4EB�=97KM=/.QBDCJH<.10�890��<=92�BE?D09bW¥�� 23=<BE?D0�H/2�?�B � 4�C x0L4EBsCNKOH/2LCN? � 8>CJ45;<7/4 x = 2k + 1 � 2L.J;<?[4

x2 = (2k + 1)2 = 4k2 + 4k + 1 = 4(k2 + k) + 10L4EBä=97OKM=/.QBDCJH<.1068/06�<=92�BE?D0H/.A=94W=97�b �@7 d ;<7 d .N=<B_�>=n0�Hn;>=9?�=97 d 03?[B[2LCN7)0L7>BDCJ0L?

m � x2 = 4m;>=

x2 = 4m+1b¥@0�H/.A=94 � .1;<?[45�<=n0 y ≥ 4 � y!

d ;<7>BDC1037>B!.J0+g12 d BE0�=9? £ b��L0�KO0LKOI/?D068906��23= d F/0890O.{�7@0L�<=92�BDC1;<7 � d � 0L4EB �2f8<CN?D0x2 − 11y! � 034GB+8/;<7 d 45;>CABP=97TKM=/.QBDCJH<.10�8/0��<=92�BE?D0 �4,;>CQB�=97RK�=/.ABDCNH/.J0~8/0~�>=923BE?D0)H<.N=94s=97�b�->2�? d ;<7>BE?D0 � .106KO0LKOI/?D0~8/0~8<?D;>CABD0 � d � 0L4EB �28<CN?D0

2003 � 0L4EBs=97fKM=/.QBDCJH<.10�890��<=92�BE?D0MH<.N=94�BE?D;>CJ43bW�>�7@0��>=923BDCJ;<7f7Y� 2O8/;<7 d H/2�4�8904,;>.A=<BDC1;<7/4s�>=92�798y ≥ 4

b�@7)��@03?DCA��0�=97OH/2�?W=97M.J0L4 d 2�4

y = 1 � 2 � 3 b�3Cy = 1 � ;<7á2 x2 − 11(1!) = 2003 � 8/;<7 d x2 = 2014

b �@7��@03?DCA��02LCN4 @03KR037>B��>=n02014

7Y� 0L4EB+H/2�4�=97 d 2�?D?�@0RH/2�?[g12LCQB�b!ãq.ä7Y� 2f89;<7 d H/2�46890R45;>.N=<BDCJ;<7/4.1;<?[45�<=n0y = 1

b�3Cy = 2 � ;<7Ý2 x2 − 11(2!) = 2003 � 8/;<7 d x2 = 2025

b 0�B[BE0Ogq;>CN46;<7BE?D;>=>�<0x = 45

bß<7<�97 � 4�C y = 3 � ;<7O2 x2 − 11(3!) = 2003 � 89;<7 d x2 = 2069b �@7~��@03?DCA��0�<=n0

20697Y� 0L4EB�H/2�4�=97 d 2L?E?�@0TH92L?Dg123CAB�b�ãq.�7Y� 2�89;<7 d H92L4R8/0Ý45;>.N=<BDCJ;<7/4�.1;<?[45�<=n0

y = 3b�>�7@0L�<=92�BDC1;<7MHn;<4�4 �0L890�8/;<7 d .J2�4,;>.A=<BDC1;<7~=97<CJ�<=n0

(x, y) = (45, 2)b

��¢

�P0G LBP}60��CQ�<0OBGF/0f;�� d CN2L.ä4,;>.A=<BDC1;<7/4+BD;�BGF/0 �L�� b §3b@©9.A=97989;<7â��2�BGF/0LKM23BDC d 4 ;<7>BD0L4EB �L�� L£/� � �3£/�� b

� µ�¸ �� ¸@Ä@±u²G¸_±�µ�� [º6Ä��«�Ä³�_±�µ+¸@¼´_±�²G°�Ã��M«�Ä@±�¸@Ã�±Û bW��;>.Q�<0 �

log2(9 − 2x) = 3 − xb

= ÆL\�Ç/ÌJZ{Æ�Ï?>�¦s4GCJ79��BGF/068/03�97<CABDCJ;<7O;>gnBGF/0�.1;/��2L?DCQBGF<K�gJ=97 d BDCJ;<7 � }606F<2��<0log2(9 − 2x) = 3 − x �

9 − 2x = 23−x �9 − 2x = 8 · 2−x �

9 · 2x − 22x = 8 �(2x)2 − 9 · 2x + 8 = 0 �(2x − 1)(2x − 8) = 0 �

2x = 1;<?

2x = 8 �x = 0

;<?x = 3

b F/0 d ��CN79�@BGF/034,0sCJ7+BGF/0�;<?DCJ��CJ7/23.<0��>=923BDCJ;<7 � }60s�n798�BGF<23BuBGF/0,:62L?D0PI/;�BGF�4,;>.A=<BDC1;<7/43bÞ bW��F/;�}�BGF<23B

(√

5 + 2)13 − (

√5 − 2)

13CN4@?[23BDCJ;<7/2L. � 2L798��9798�CAB[4ä�>2L.A=n09b

= ÆL\�Ç/ÌJZ{Æ�Ï?>ä�L0�Bx = (

√5 + 2)

13 − (

√5 − 2)

13b��YF/037

x3 = (√

5 + 2) − 3(√

5 + 2)23 (

√5 − 2)

13

+ 3(√

5 + 2)13 (

√5 − 2)

23 − (

√5 − 2)

= 4 − 3(√

5 + 2)13 (

√5 − 2)

13

(

(√

5 + 2)13 − (

√5 − 2)

13

)

= 4 − 3(1)(x)b

��037 d 0 � (√5 + 2)13 − (

√5 − 2)

13CJ4@2��0L?D;�;>g

P (x) = x3 + 3x − 4 = (x − 1)(x2 + x + 4)}6F>C d F�F<2�4x = 1

2L4�CAB[4W;<7<.Q:+?D0L23.��03?D;ubu�YF/03?D03gq;<?D0 � (√5+2)13 −(

√5−2)

13 = 1

b��/Õ�Ôu�YF>CN4�H/?D;/I<.J0LK CJ4�4GCJK�CN.N2�?9BD;PH/?D;/I<.J0LK ¢ ;<7�BGF/0 ��>� ��2L?DCQBDCJKO0P��2�BGF/0LKM23BDC d 4 ;<KMHn0�BDCABDCJ;<7 � gq;<?_}6F>C d F�2�4,;>.A=<BDC1;<76}�2�4s��C �<0L7O;<7~BGF/0�H/?D0 d 0�8>CJ79��H92��309b �à b!ãqg

a3 + b3 = 42�798

ab = 2

3� }6F/0L?D0 a

2L798b2�?D0+?D0L23. � �9798 a + b

b= ÆL\�Ç/ÌJZ{Æ�Ï?>W��;�BE0��n?[4EBWBGF<23B

(a + b)3 = a3 + 3a2b + 3ab2 + b3 = a3 + b3 + 3ab(a + b)b

�L0�Bx = a + b

b��YF/037x3 = 4 + 3

(

2

3

)

x � }6F>C d F6:>C10�.18<4x3 − 2x − 4 = 0 �

(x − 2)(x2 + 2x + 2) = 0b

.J0L2L?D. : � x = 2CJ4!BGF/0�;<7<.Q:�?D0L23.�?D;/;�B�bä��037 d 0 � a + b = 2

b

��½ b!Ü�CN798

x � y � 2L798 z4�= d F6BGF<23B�}6F/037M2�7�:�;<790�;>g9BGF/0LK�CN4@2�898/0�8)BE;6BGF/0+H/?D;/8<= d B;>gnBGF/06;�BGF/03?WB�}6; � BGF/0�?D034�=/.QBäCJ4 2

b= ÆL\�Ç/ÌJZ{Æ�Ï?>�©<:�4[:<KOKO0�BE?{: � }60fK�=94GB)F<2��<0 x = y = z

bá��F/0L7 0L2 d F'0��>=923BDCJ;<7I90 d ;<KO0L4x + x2 = 2 �

BGF<2�B+CN4 � x + x2 − 2 = 0 � ;<? (x − 1)(x + 2) = 0b�YF>=94 � x = 1

;<?x = −2

b!Ü�CJ7/23.N. : � x = y = z = 1;<?

x = y = z = −2b

å bäãqga � b � 2L798 c

2�?D0sBGF/0sBGF<?D0�0s��0L?D;<4!;>gP (x) = x3 −x2 +x−2 � �n798 a+ b+ c2�798

a2 + b2 + c2b

= ÆL\�Ç/ÌJZ{Æ�Ï?>_ãqga � b � 2L798 c

2L?D0�BGF/0+?D;/;�BD4s;>gnBGF/0��CQ�<037 d =nI<C d � BGF/037x3 − x2 + x − 2 = (x − a)(x − b)(x − c)

= x3 − (a + b + c)x2 + (ab + ac + bc)x − abcb

ß/�>=923BDCN79� d ;/0�� d CJ0L7>B[4P;>gYBGF/0x2� BD0L?DKO4 � }60)450�0+BGF<23B a + b + c = 1

b��<=92L?DCN79�BGF>CJ4 � }60)�30�B a2 + b2 + c2 + 2(ab + ac + bc) = 1b�©9=<Bs0��>=923BDCN79� d ;/0�� d CJ0L7>B[4;>gnBGF/0

x� BD0L?DKO4s��CQ�<034

ab + ac + bc = 1b��+0L7 d 0 �

a2 + b2 + c2 + 2(1) = 1 �a2 + b2 + c2 = −1

bæ b!ãqg

sin x + cos x =

√

2 +√

3

2� }+CQBGF 0 < x < π

2� �n798 x

b= ÆL\�Ç/ÌJZ{Æ�Ï?>��>=92�?[CJ79�PBGF/0��CQ�<037R0L�<=92�BDC1;<7 � }60��30�B

sin2 x + 2 sin x cos x + cos2 x =2 +

√3

2�

1 + 2 sin x cos x = 1 +

√3

2�

sin 2x =

√3

2�

}6F/0L?D00 < 2x < π

b~�YF/03?D03gq;<?D0 � 2x = π/3;<?

2x = 2π/3b��F/0L7

x = π/6;<?x = π/3

b� b�-9?D;3�<0+BGF<2�BäB�}6; d ;<7/4,0 d =<BDCQ�<06;/8/8OH9;<4�CQBDCQ�<0+CN7>BE0L�30L?[4 d 2L7/79;�B@F<2��<0�2 d ;<KOKO;<7g12 d BD;<?�;�BGF/03?WBGF<2L7

1b

= ÆL\�Ç/ÌJZ{Æ�Ï?>n�>}6; d ;<7/450 d =<BDC �<0P;/8986H9;<4�CQBDCQ�<0sCJ7>BD0��303?[4 d 2L7~I/0@}�?DCQB[BE037)2L42k−1

2L7982k + 1

gq;<?ä4,;<KO0+Hn;<4GCABDC �<0�CJ7>BD0��303?kbW�3=9H9H9;<4,0

pCN4s2 d ;<KMKR;<7�g12 d BE;<?!;>g

2k − 12�7982k + 1

b_��F/0L72k − 1 = mp

2L7982k + 1 = np �

gq;<?�4,;<KO06H9;<4�CQBDCQ�<0+CN7>BE0L�30L?[4m2L798

nbs�3=nI�BE?[2 d BDCN79��BGF/0��n?[4EB@0��>=923BDCJ;<7Og1?D;<K�BGF/04,0 d ;<798O��C �<0L4

2 = (n − m)pb_�YF>=94 � p = 1

;<?p = 2

b�©9=<Bp 6= 2 � 4GCJ7 d 0 2k − 12�798

2k + 12�?D06;/8/8�b��F/0L?D0�gq;<?D0 � p = 1

b

�� b6�9?DCN2�79��.J0

ABCF<2�4��<03?[BDC d 034

A(3, 1) � B(5, 7) � 2�798 C(1, y)bMÜ�CJ798f2L.A.

y45;BGF<23B!2L79��.10

CCJ4@2�?[C1�3F�B!2�79��.J09b

= ÆL\�Ç/ÌJZ{Æ�Ï?>BC ⊥ AC �

mBC =−1

mAC�

7 − y

4=

−2

1 − y�

7 − 8y + y2 = −8 �y2 − 8y + 15 = 0 �

(y − 5)(y − 3) = 0 �y = 5

;<?y = 3

b -

6

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

A(3, 1)

B(5, 7)

C(1, y)

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

� b ã{7jBGF/0'8<CN2��?[2LK BE; BGF/0'?[C1�3F�B � PQ = 8 �TS = 12 � 2L798 QS = 20

báÜ�CJ798QR

4,;�BGF<2�B∠PRT

CN4@2�?DCJ�3F�B�2L79��.10/b= ÆL\�Ç/ÌJZ{Æ�Ï?>ä�L0�B

QR = xb��YF/037

(PT )2 = (PR)2 + (RT )2

= (64 + x2) +(

144 + (20 − x)2)

= 2x2 − 40x + 608b

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P

Q R S

T

c .J45; � (PT )2 = 202 + 42 = 416b��037 d 0 �

2x2 − 40x + 608 = 416 �x2 − 20x + 96 = 0 �

(x − 8)(x − 12) = 0 �x = 8

;<?x = 12

b ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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P

Q R S

T

U

8 8

4

x 20 − x

20

Û�� b c 4,�>=92�?D0�;>g!4GC18/02CN4�H/.N2 d 0�8f}+CQBGFÝ;<790O4�CJ890R;<7Ý2MBD2L79�30L7>BPBD;�2 d CN? d .10R;>g?[2�8>CN=94

545;�BGF<23BnBGF/0@4,�>=92�?D0�.AC1034_;>=<B[4�CJ890äBGF/0 d CJ? d .J0 � 2�798+;<790ä�<03?[BD0� �;>gLBGF/0@45�<=92L?D0.NCJ0L4+;<7RBGF/0 d CN? d .10/b c .NCN790~CJ4+8<?[2�}�7fg1?D;<K¯BGF/0 d 0L7>BE?D0O;>gBGF/0 d CJ? d .J0�BGF<?D;>=n�3F~BGF/0P�<0L?�BE0G �;>gnBGF/0�45�<=92L?D0�BGF<23BäCJ4@79;�B;<7�BGF/0�BD2L79�30L7>B�2L798�79;�B�;<7PBGF/0 d CJ? d .J09bu�YF>CN4�.NCN790 d =<BD4�BGF/0BD2L79�30L7>B�23B�2�H9;>CJ7>B

Tb~ãqg�BGF/06BD2L79�30L7>B�KR0L0�B[4�BGF/0 d CJ? d .J023B

S � �9798~BGF/0+.J0L79�¡BGFM;>gnBGF/0�.NCN790�450��KO0L7>B TSb

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S T= ÆL\�Ç/ÌJZ{Æ�Ï?> 4TQR2L798 4TSO

2�?D0+4�CNKMCA.J2L?�b��YF>=94 �T Q

2=

T Q + 6

5�

5TQ = 2TQ + 12 �TQ = 4

b��037 d 0 � TS = 6 + 4 = 10

b ............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ..

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K�b�©<:�=94GCJ79�27;>g>BGF/0�=97<CQB d =nI/0L4 � 2 d =nI>C d g1?[2�KO0�}6;<?E� d 2�7)I90 d ;<7/4GBE?[= d BE0L8+BGF<23B_F<2L4ä4�CJ890 3

K�bß<2 d F�BDCNKR0�BGF/0�=97<CAB d =nI9034�2�?D0 d ;<KOI<CN790�8 � BGF/0L?D062�?D068<=9H<.NC d 2�BE06?D;/8/4s2L.J;<79�BGF/0�0L89�3034�}6F/03?D0�BGF/0 d =nI9034O�<B)BE;/�30�BGF/0L?¡bç�YF/0Ý8<=9H<.NC d 2�BE034RKM2�:âI90T?D0LKO;L�<0L8�J.J0L2��<CN79��;<790+?D;/8)}6F/03?D0�BGF/03?D0P}60L?D0+H9?D0,�>C1;>=94G.Q:)B�}6; � 2�798M?D0L�G=9450�8M03.N4,0¡}6F/0L?D0/b��;�} .N2�?D�30~2 d =nI>C d g1?[2�KO0�}6;<?E� d 2L7�I90 d ?D0323BD0�8�CN7RBGF/0�KO2L7/790L?�8/0L4 d ?DCJI90L82�I/;L�<0PCAg/BGF/0�BD;�BD23./.J0L79�¡BGF~;>g/BGF/0�2��>2LCA.J2�I<.J0P?D;/8/4_}6;>=/.18 d ;<7/790 d B_BGF/0�0L2L?[BGF�BE;�BGF/0KR;/;<7 � 2L454G=9KMCN79��2)8>CJ4EBD2L7 d 0�;>g 384 000��K#I90�B�}60�037~BGF/0LK �

��V W�XºZ]\2^ �ubfe��'b��]g

³ �+à b+¼nX�Æ3V�Æ�É�Ë�ÕRÑ�ØMÌqÍYËqË Ó�Ø�Í�Ë�Ò = ÌJÓ��WbÜ�C �<0~I/23.N.N4�7<=9KOI903?D0�8

1BD;

52L?D0)H/=<B@CN7>BE;O2�I9;� nb c I<2L.A.YCJ4�8/?[2�}�7�23B@?[2L79�89;<K � CAB[4s7<=9KRI/0L?@?D0 d ;<?D8/0�8 � 2�798�BGF/0)I<2L.A.Y?D0�BD=9?E790L8MBD;~BGF/0)I9;� nb!�YF>CN4�H9?D; d 0L4�4sCN4?D0LH90L2�BE0L8O=97>BDCA.��>�<067<=9KOI903?[4PF<2��<0)I90L0L7O?D0 d ;<?D890L8�b�ãqg�BGF/064�=9KÙ;>g�BGF/06?D0 d ;<?D890L87<=9KRI/0L?[4�CJ4

15 � }6F<23BPCN4�BGF/0RH9?D;/I<2�I>CN.ACAB{:�BGF<2�B�BGF/0R7<=9KRI/0L? 3}�2L4)8<?[2�}�7Ý0L2 d FBDCJKO0 �

= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �_Ë�ÏYË3Ê¡ZËLÊLË �LÓ3\{Æ�ÏYÕ9Ë¡]�Ë'ÓLÉ�É�Ë�Ø3]��N¾)b�YF/0�B[2�I>.10+I/03.J;�}�4,F/;�}�4WBGF/0�7<=9KOI903?!;>g<}�2�:/4�;>g�8/?[2�}+CJ79��>�<0+I<2L.A.J4_}+CABGF)24�=9K#;>g

15b_��F/0�BE;�B[2L.n7<=9KOI903?�;>g9}�2�:<4�CJ4

381b_��F/0L?D0�gq;<?D0 � BGF/0�H9?D;/I<2�I>CN.ACAB{:~BGF<2�B0,�<0L?{:O8<?[2�} }�2L4@2

3CJ4 1

381

b©n23.N.N4 �s=9KRI/0L?�;>g � 2�:/4 ©n23.N.N4 �s=9KRI/0L?�;>g � 2�:/4

5 � 5 � 3 � 1 � 1 5!

2!2!= 30 5 � 3 � 3 � 2 � 2 5!

2!2!= 30

5 � 5 � 2 � 2 � 1 5!

2!2!= 30 4 � 4 � 4 � 2 � 1 5!

3!= 20

5 � 4 � 4 � 1 � 1 5!

2!2!= 30 4 � 4 � 3 � 3 � 1 5!

2!2!= 30

5 � 4 � 3 � 2 � 1 5! = 120 4 � 4 � 3 � 2 � 2 5!

2!2!= 30

5 � 4 � 2 � 2 � 2 5!

3!= 20 4 � 3 � 3 � 3 � 2 5!

3!= 20

5 � 3 � 3 � 3 � 1 5!

3!= 20 3 � 3 � 3 � 3 � 3 5!

5!= 1

�� "!#� �$��&%('��)��*�,+ ��-�.³ �@½ b+¼nX�Æ3V�Æ�É�Ë�ÕRÑ�ØMÌqÍYËqË Ó�Ø�Í�Ë�Ò = ÌJÓ��Wb

;<7/4GC18/0L?YBGF/0@B�}6;�gJ=97 d BDC1;<7/4f(x) = x2 −2ax+1

2�798g(x) = 2b(a−x) �}6F/0L?D0

a � b � x ∈ / b � 0s}+CA.N. d ;<7/4�CJ8903?W0L2 d F)H/2LCN?W;>g d ;<7/4GB[2�7>B[4 a2L798

b2�4ä2�Hn;>CN7>B

(a, b)CJ7)BGF/0

ab� H/.N2�790/bs�L0�B

AI90PBGF/0+450�B!;>g�Hn;>CN7>BD4

(a, b)gq;<?�}6F>C d F)BGF/0��?[2LHnF<4;>g

y = f(x)2L798

y = g(x)8/;MÏYÆ9Ì�CN7>BE03?[4,0 d B�b!Ü�CN798~BGF/0+2�?D032);>g

Ab

= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �äÇ3É3ÌJÓ<ÊLÆ�ÄuX[Z1Ò10<Ë�XD] ��Ï>Z�ÊLË�X[ÉGZ�Õ>Ó>Õ}Â�U �EÂ ��]3Æ_Ç<Ë�ÏYÆ�ÉWUäZJX�Ë¡É5]<U�X�Ð<Ë�ÏuÌJZJÏ9Ó/b�YF/0��?[2�H9F<4�;>g

y = f(x)2�798

y = g(x)89;~79;�BWCJ7>BD0L?[450 d B�}6F/0L7~BGF/0�0L�<=92��BDC1;<7

x2 − 2ax + 1 = 2b(a − x)F<2L4�79;�4,;>.A=<BDC1;<7�b+�YF>CN4+F<2LH9H90L7/4PCNg_2L798�;<7<.Q:CNg

b2 + a2 < 1bP��037 d 0 � A CJ4@BGF/0~;<H90L7�8>CJ45�M;>gu?[2�8>CN=94

1d 037>BE?D0�8R2�B!BGF/0~;<?[C1��CN7�b�YF>=94 � BGF/0+2�?D032);>g A CN4

πb

23� ��4�� ,+ ��-6587:9;�<5��=��>?� � � ��@�� A; )!<� ��BC�� A�D)EF.³ � å b+¼nX�Æ3V�Æ�É�Ë�ÕRÑ�ØMÌqÍYËqË Ó�Ø�Í�Ë�Ò = ÌJÓ��Wb

Ü�CN798f26BE?DCN2�79��.J06}6F/;<450)CN7>BE0L�30L?s4�CJ89034P2L?D0)CJ7�2�?[CABGF<KO0�BDC d H/?D;/��?D0L4�4�CJ;<7O}+CQBGF2 d ;<KOKO;<7O8<C��03?D0L7 d 06;>g22L798)}6F>C d FMF<2�4@2�7M2L?D0L26;>g

336b

��

= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �äÇ3É3ÌJÓ<ÊLÆ�ÄuX[Z1Ò10<Ë�XD] ��Ï>Z�ÊLË�X[ÉGZ�Õ>Ó>Õ}Â�U �EÂ ��]3Æ_Ç<Ë�ÏYÆ�ÉWUäZJX�Ë¡É5]<U�X�Ð<Ë�ÏuÌJZJÏ9Ó/b�L0�B

a � a+2 � 2L798 a+4I/0@BGF/0s4�CJ89034W;>g�BGF/0@BE?[CJ2L79��.10/b�©<:)�+0L?D;<7Y� 4�Ü>;<?EK�=/.J2 �}606F<2��<0

336 =√

1

16(a + 6)(a + 2)(a − 2)(3a + 6)

b��037 d 0 � (a + 6)(a + 2)(a − 2)(3a + 6) = 16 · 3362 = 1806336 � }6F>C d FF<2�4�BGF/0+=97<CJ�<=n0�Hn;<4GCABDC �<0+CN7>BE0L�30L?!45;>.N=<BDCJ;<7

a = 26bW�YF/037�BGF/0�BE?[CJ2L79��.10�4GC18/0L4@2�?D0

26 � 28 � 2�798 30b

23� ��4�� ,+ ��-6587:9;�<5��=��>?� � � ��@�� A; )!<� ��BC�� A�D)EF.³ �Pæ b ¼nX�Æ3V�Æ�É�½Ë�VYÓ�X_\oÉ{½Ë"Ê�Ç3ZQV�Ë6Õ9ËzË'Ó�ØLÍYË�Ò�b

��;>CAB!890�=� )79;<KRI<?D0L4s0L7>BDCJ0L?[4@Hn;<4GCABDCAg14a0�B

bbu->2L?EK�Cn.J0L4@79;<KOI/?D034

a � 2a � 3a � . . . � (b − 1)a � ba

d ;<KOI<CJ0L7):O037M2��DBG�GCA.��>=/C�4,;>CJ0L7>B!8<C �<CN4�CJI<.J0L4�H92L?b�

= ÆL\�Ç/ÌJZ{Æ�Ï~VYÓLXÅ_Æ9Ñ/Ë�X,Ì8Æ9ZN\AZ1Ï<É<0�ZN]:�sÇ<ÌqX�Ë�ÒfÆ�ÏuÌ1]��ÂuÔ�3C

(a, b) = 1 � 23.1;<?[4�CN./:M264503=/.J0LKO0L7>B ba�<=/CY034GB�8<C �<CN4�CJI<.J0�H/2�?

bbYãq./:O0L7O289;<7 d

1b�3C(a, b) = b � 23.1;<?[4�BD;>=946.1034)79;<KRI<?D0L4)4,;<7>B68>CQ�>CJ4GC1I>.1034)H/2�? b

b�ãq.W:�0L7�289;<7 dbb�3C(a, b) = k � 2L.J;<?[4 a = ks

0�Bb = kt

2��<0 d(s, t) = 1

bs�32��>=n0L4EBDC1;<7890,�>C1037>B �� ->2L?EK�C3.J0L4Y79;<KRI<?D0L4s � 2s � . . . � (kt−1)s � kts � d ;<KOI<CJ0L7�4,;<7>B�8<C �<CN4�CJI<.J0L4H92L?

t��f¥�2�7/46.N2f4G=/CABD0f890�79;<KOI/?D0f;<7TBE?D;>=>�<03?[2

ts � 2ts � . . . � (k − 1)ts � kts �89;<7 dk79;<KRI<?D0L4!4,;<7>BW8<C �<CN4�CJI<.J0L4äH92L?

tb c =<BE?D03KR037>Bä8<CQB � CN.�:~23=9?[2PBD;>=�;{;>=9?[4 (a, b)79;<KRI<?D0L4�8>CQ�>CJ4GC1I>.1034@H/2�?

b8<2�7/4�.J2+.ACJ4EBE0/b

� �&��!<� �� "!��' ��!�� '<+ � �F� � B6@�� A��<�$+�� $-8'<-6E�2 � E � A;>;!<��23� ��A<2 ��#�� #.³ � � b+¼nX�Æ3V�Æ�É�Ë�ÕRÑ�ØMÌqÍYËqË Ó�Ø�Í�Ë�Ò = ÌJÓ��Wb

��0,�<0L7OH90�;<H<.10 � A � B � C � D � E � F � G � 2L?D06;<7R;<790�4GC18/06;>gu2�?DC �<0L?¡bä�/;O�30�B2 d ?D;<4�4_BGF/0�?DC �<0L?uBGF/0,:~F<2��<0P2P?D;�}6I/;<23B � I>=<B�CAB d 2L7~;<7<.Q:��<B�B�}6;�H90�;<H<.10�2�B_2�BDCNKR0/b�YF/0)BDCNKR034�BGF<23B!}6;>=/.18fI90~?D0��>=/CJ?D0L8�gq;<?@BGF/0~Hn0L;<H/.J0)BD;�?D;�} 2 d ?D;<4�4�CJ798>CQ�>C18>=92L.A.Q:2�?D01 � 2 � 3 � 5 � 10 � 15 � 2L798 20

KMCN7<=<BE034 � ?D0345H90 d BDC �<03. :�bá�+;�}60,�<0L? � }6F/0L7TB�}6;Hn0L;<H/.J0�2L?D0�CN7�BGF/0TI/;<23B � BGF/0fBDCJKO0fCAB+BD2��0L4)BGF/0LK%BE;'?D;�}�2 d ?D;<4�4~CJ4)BGF/0�452LKR02�4PBGF/0~BDCNKR0M790 d 03454�2�?{:�gq;<?sBGF/0O4G.1;�}603?�;>g_BGF/0~B�}6;RBD;�?D;�}�2 d ?D;<454PCJ798>CQ�>C18>=92L.A.Q:ubc 454G=9KMCN79�sBGF<2�BW79;);<790 d 2�7 d ?D;<454�}+CABGF/;>=<B�BGF/0�I9;<2�B � }6F<2�B_CN4WBGF/0PK�CJ7<CNKM=9KçBDCNKR0gq;<?!23.N.n450,�<0L7MH90�;<H<.10PBE;M�30�B!2 d ?D;<4�4!BGF/0�?[CQ�<03? �= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �_Ë�ÏYË3Ê¡ZËLÊLË �LÓ3\{Æ�ÏYÕ9Ë¡]�Ë'ÓLÉ�É�Ë�Ø3]��N¾)b

�YF/0�K�CJ7<CNKM=9K BDCNKR0PCN446KMCN7<=<BE034�2�798 d 2�7�I90�2 dGd ;<KOH<.NCN4,F/0L8~2�4!45F/;�}�7~CN7BGF/0�BD2�I<.J0�I90�.1;�} �

�� ?D;<4�4 �nCNKR0 ¤Y0�BD=9?E7 �nCNKR0A � B 2 A 1

F � G 20 B 2

A � B 2 A 1

D � E 10 B 2

A � C 3 A 1

A � B 2 �� "!#� �$��&%('��)��*�,+ ��-�.

³ �� b+¼nX�Æ3V�Æ�É�Ë�ÕRÑ�ØMÌqÍYËqË Ó�Ø�Í�Ë�Ò = ÌJÓ��Wbc 4,0�B

Sd ;<7/4�CN4GB[4P;>g�4�C �7<=9KRI/0L?[43b � F/0L7M}606B[2��0~23.N.uHn;<4�4�CJI<.J0)4�=nI<4,0�BD4P;>g

Sd ;<7>BD23CJ7<CN79�

50�.103KR037>BD4 � BGF/0~4�=9KM4�;>guBGF/0�0�.103KR037>BD4�;>guBGF/034,0)4�=nI<4,0�BD4�2�?D0 87 �

92 � 98 � 99 � 104 � 2L798 110 � ?D0L4�Hn0 d BDCQ�<0�.Q:ub@¥@0�BD0L?DKMCN790�BGF/0+4�C 67<=9KRI/0L?[4@CN7 Sb

= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø�Uä\N\{Ë�Ï ��É ��Ó�X[ÓL]�ÐLXDÓ>Õ9Ë S�`�É�ÌDÇ<Õ9Ë�ÏuÌ1]1�WÓ0>X[Z{Õ�Ð<Ë = Ë�×EÆ�ÏYÕ>ÓLXEØ = ×DÍYÆ/ÆL\N]�>Æ�ÏYÕ9Æ�Ï<]��N¾)b

�L0�BS = {a � b � c � d � e � f} b Ü�?D;<KÂBGF/0OCJ7<gq;<?DKO2�BDC1;<7ÝH/?D;L�>C18/0�8 � }60 d 2�7d ?D0L2�BE0�BGF/0�gq;>.N.J;�}+CJ79�P450�B�;>gY0��>=923BDCJ;<7/4 �

a + b + c + d + e = 87 � � � �a + b + c + d + f = 92 � � � �a + b + c + e + f = 98 � � � �a + b + d + e + f = 99 � � £ �a + c + d + e + f = 104 � � ¢ �b + c + d + e + f = 110

b � � �©<:�4G=nI>BE?[2 d BDCJ79��0L2 d F�0��>=923BDCJ;<7~g1?D;<K BGF/0P.J2L4GB � }60 d 2�7�?D0�.J2�BE0+0L2 d F��>2�?[CJ2�I<.J0PBD; a�

f = a + 23 � e = a + 18 � d = a + 12 � c = a + 11 � b = a + 6b

�P;�} }60 d 2L7�?D0�}�?[CABD0O;<790O;>g_BGF/0O0L�<=92�BDC1;<7/4�� BE;�8/0�BD0L?DKMCN790 a � 2�798�BGF/0L7;/I>B[2LCN7~BGF/0�?D034GB�;>g9BGF/0�7<=9KOI903?[4Lb�Ü>;<?!0� L2LKOH<.10 � }60 d 2L7M?D0�}�?[CABD060L�<=92�BDC1;<7O� � � 2L4(a + 6) + (a + 11) + (a + 12) + (a + 18) + (a + 23) = 110 �}6F>C d F d 2L7OI90+4�CNKOH<.NCA��0L8)BE; �

5a + 70 = 110 �g1?D;<K }6F>C d Ff}60��30�Ba = 8

b��YF/037b = 14 � c = 19 � d = 20 � e = 26 � 2L798

f = 31b_��F>=94 � S = {8 � 14 � 19 � 20 � 26 � 31} b23� �� $+��-&587 9;�<5��=� >?� � � ��@�� A )!<� ��BC�� A)D)E��393!�� ,� +*A � � '<-��4�� !<-�� A3>;!��<'<587

�<��!#��#��-8'��7�#��<� � A�>;!��<'<587 A?>;E��#'��- �!�� '<+ � �� B6@<�� A �F�#�,+ �� ,-8'#-4E�2 � E � A?>;!#��23� ��A2 ��#�� <'?.³ �� bR¼nXGÆ3V�Æ�É�½ËsVYÓ�XW\°É ½� Ê�Ç3ZQV�Ë6Õ9ËqË Ó�Ø�Í�Ë�Ò�b

�9?D;>=>�<0L?_BD;>=94!.1034@0L7>BDCJ0L?[4�H9;<4�CQBDCNg14xH9;>=9?W.1034,�>=n03.N4

x(x + 60)034GB_=97 d 2L?E?�@0H92L?Dg123CAB�b

��£

oGb = ÆL\[Ç<ÌJZ�Æ�Ï)VYÓ�XÅ�Æ9Ñ/Ë�X,Ì8Æ9ZA\NZJÏ/É#0�ZJ] �sÇ/ÌqXGË�ÒfÆ�ÏYÌ1] ��Âub-<;<4,;<7/4

x = n − 30 � 2L.J;<?[4 d F/0L? d F/03? x > 0BE0�.ä�>=n0

x(x + 60)4,;>CQB�=97d 2L?E?�@0�H/2�?[g12LCQBW?D0,�<CJ0L7>B �2 d F/03? d F/0L?

n > 30BE0�.9�<=n0

(n − 30)(n +30) = n2 − 9004,;>CQB_=97 d 2�?D?�@0�H92L?Dg123CAB�bn�P;<KMKR;<7/4k2d 0 d 2�?D?�@0�H92L?Dg123CABä�<=n0�.{� ;<7~?D0 d F/03? d F/09b c CJ7/4GC �

n2 − 900 = k2;>=RI>C1037

n2 − k2 = (n − k)(n + k) = 900 = 223252b�P;>=94�890,�<;<7/4PKO23CJ7>BD0L7/2L7>B�4 @03H92L?D0L?

900037�890�=� Mg12 d BE0�=9?[4

a0�B

bBE0�.J4��>=n0

n − k = a0�B

n + k = bb �@7'23=9?[2�2LCN7/4�C@�>=n0

n =a + b

2

0�Bk =

b − a

2

b-/=/CJ45�<=n0�79;>=94ä�<;>=/.1;<7/4s�<=n0n0�B

k4,;>CJ0L7>B!89034�037>BDC103?[4 � CA.9g12L=<B��<=n0 a

0�Bb4,;>CJ0L7>B890PK È0LKO0PH/2�?[CAB�@0/bn-/=/CJ45�<=n0�=97�8/0L4!890�=� �g12 d BE0�=9?[4�890

9008/;>CAB È0�BE?D0�H923CJ? � .1034�8/03=� .10P450L?D;<7>B�bn�L0PH/?D;/I<. �0LKO0P?D0,�<CJ0L7>BW89;<7 d �2�BE?D;>=>�<0L?W890�=� �g12 d BD03=9?[4

r0�B

s8/0

225�>=/C89;<7/7903?D;<7>B

n = r + s > 300�B

k = r − s > 0b �@?�.1034!H/2LCN?D0L4!8<C�� @03?D0L7>BD0L4�8/0�g12 d �BE0�=9?[4s890

22545;<7>B

(1, 225) � (3, 75) � (5, 45) � (9, 25)0�B

(15, 15)bnß>.A.1034@89;<7/79037>B

n ∈ {226 � 78 � 50 � 34} ;>=RI>C1037 x ∈ {196 � 48 � 20 � 4} bo�oGb = ÆL\[Ç<ÌJZ�Æ�Ï·Ñ�Ø �äÇLÉ�Ì1Ó>ÊLÆ ÄuX[ZJÒ10/Ë�XD] �uÏ<Z{ÊLË�X�É�Z{Õ>Ó<Õ Â�U ��Â ��] Æ_Ç>Ë�Ï�Æ�É UäZ1XGË�É�]U�X[Ð>Ë�ÏYÌJZ1Ï/Ó9b

�L0�BkI/0@2sHn;<4GCABDC �<0�CJ7>BD0��303?�bu��F/0�0L�<=92�BDC1;<7

x2+60x−k2 = 0F<2L4�Hn;<4GCABDC �<0CJ7>BD0��303?s4,;>.A=<BDC1;<7/4sCNg�2L798f;<7<. :OCNg √

302 + k2CN4�2L7RCJ7>BD0��303?�b��YF/037O}60)K�=94GB@�97982L.A.P->:�BGF<2��3;<?D032�7TBE?[CJH<.1034�;>gäBGF/0�gq;<?EK

(30, k,√

302 + k2)b'��F>CJ46CJ4~0L�<=/C �>2L.J0L7>BBE;R�9798<CN79�)2L.A.�H9?[CJK�CABDC �<0f->:�BGF<2��3;<?D032�7RBE?[CJH<.1034

(a, b, c)}6F/0L?D0

a | 30;<?

b | 30b�YF/034,0�BE?[CJH<.1034!2L?D0

(3, 4, 5) � (5, 12, 13) � (8, 15, 17) � 2�798 (15, 112, 113)b!��037 d 0 �

x = 20 � x = 48 � x = 4 � 2L798 x = 196 � ?D0L4�Hn0 d BDCQ�<0�.Q:ub23� �� ,+ ��- 587 93! �� ,� +*A � � '<-�� !<-�� A1>;!��<'<587 �<��!#�� <��-8' ��7 �<� ��<� � A

>;!��<'<587 A;>;EF.³ � � b+¼nX�Æ3V�Æ�É�Ë�ÕRÑ�ØMÌqÍYËqË Ó�Ø�Í�Ë�Ò = ÌJÓ��Wb

¥@0�BD0L?DKMCN790+BGF/0+.J2L?D�30L4EB�H9;<4�CQBDCQ�<0�CJ7>BD0��303?ngq;<?_}6F>C d F

2002n CN4s2�g12 d BD;<?�;>g2002!

b � F<2�B�F<2LH9H90L7/4@CAg2002

CJ4@?D03H/.N2 d 0�8OIL:2003

;<?2004

�= ÆL\�Ç/ÌJZ{Æ�ÏfÑ�Ø�Å!ÇLÉGZ�Ä�ÆL\{Ë3Ê�]�ÐLXDÓ>Õ9Ë)S�`�É3ÌDÇ>Õ9Ë�ÏYÌ1]�Æ_Ç�XEÏ/Ó/Ñ�Ø = ÆnÇ/ÌqÍ = Ë�×EÆ�ÏYÕ>ÓLXEØ = ×DÍYÆ/ÆL\N]Æ_Ç�XEÏ/Ó/Ñ�ØL]�Æ�Âub

�L0�BF = 2002!

bÝ�3CN7 d 02002 = 2 · 7 · 11 · 13 � CN7 2002!

BGF/03?D0M}+CA.N.äI90b2002/13c = 154

KM=/.QBDCJH<.1034);>g132�798 b2002/132c = 11

KM=/.QBDCJH<.1034);>g1322�798 b2002/133c = 0

KM=/.QBDCJH<.1034�;>g133

b��YF/03?D06}+CA.N.�I90~KR;<?D0~K�=/.ABDCNH/.J0L4P;>g11 �

7 � 2L798 2b��YF/03?D03gq;<?D0 � BGF/0~.N2�?D�3034GBsCJ7>BD0��303? n

4G= d FRBGF<2�B2002n 8>CQ�>C18/0L4 2002!

CN4n = 154 + 11 = 165

b_��F>=94 � n = 165gq;<?

2002b�3CJ7 d 0

2003CJ4�2+H/?DCNKR0�7<=9KRI/0L? � CJ7 2003!

BGF/0L?D0�}+CN.A.nI/0+;<7<.Q:�;<790�KM=/.QBDCJH<.10;>g2003

b_�YF>=94 � ;<7<. : 20031}+CA.N.�8>CQ�>C18/0 �L�� {b_��F>=94 � n = 1

gq;<?2003

b�3CJ7 d 02004 = 22 ·3 ·167 � CJ7 2004!

BGF/03?D0�2�?D0 b2004/167c = 12K�=/.ABDCNH/.J0L4;>g

167 � 2�798)4�CN7 d 0 1672 > 2004 � BGF/0L?D0P2�?D0 0KM=/.QBDCJH<.1034�;>g

1672b��F/0L?D0�gq;<?D0 � BGF/0.J2L?D�30L4EBäCJ7>BD0��303?

n4�= d F)BGF<2�B

2004n 8<C �<CJ89034 2004!CJ4

n = 12b

23� �� $+��- 587 9;�<5��=� >?� � � ��@�� A4 )!<� ��BC�� A D(E��(' ��-&23� �,�� ' � ' A�� '#-�� &�� !<-�� A �' @#� �$- �<� �#��-8'��7�<� ��<� � A��<��-��<A� ��:.

��¢³ � Û bz¼nXGÆ3V�Æ�É�ËLÕ�Ñ�Ø}Å_Æ9Ñ/Ë�X,Ì�ËâÆ�X�ËLÈPÆ/Æ/Õ�]NÆ_ÇLXEÏ9Ó<Ñ�Ø = ÆnÇ<ÌqÍ = Ë�×EÆ�ÏYÕ>ÓLXEØ = ×DÍYÆ/ÆL\N]Æ_Ç�XEÏ/Ó/Ñ�ØL]�Æ�Âub

�L0�BkI90f2�gq;>=9?E�58>C1��CQB+CJ7>BD0��303?�b ¥@0�BE03?EK�CJ790�2L.A.!Hn;<4�4�CJI<.J0M�>2L.A=n0L4~;>g

kgq;<?}6F>C d F

k20030L798<4�CJ7OBGF/0)gq;>=9?�8<CJ��CAB[4

2003b � F<2�B�F<2LH9H90L7/4�CNg

2003CJ4�?D0LH<.J2 d 0L8IL:

2002;<?

2004�

= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �_Ë�ÏYË3Ê¡ZËLÊLË �LÓ3\{Æ�ÏYÕ9Ë¡]�Ë'ÓLÉ�É�Ë�Ø3]��N¾)b�L;/;/��CN79��2�B!BGF/0)H/?D;/I<.J0LKÙKR;/8>=/.1;

10 � }6064,0L0+BGF<23B k2003 ≡ 3 (mod 10)b�YF>=94 � k ≡ 7 (mod 10)

b� 0f790� 3B679;�BDC d 0RBGF<2�B72 ≡ 49 (mod 100) � 73 ≡ 43 (mod 100) � 2L798

74 ≡ 1 (mod 100)b)�L0�B

k ≡ 10a + 7 (mod 100) � }6F/03?D0 aCJ4�2~4�CN79��.10)79;<79�790��2�BDCQ�<068>C1��CQB�b��YF/037 � =94GCJ79��BGF/0�©9CJ79;<K�CJ23.9�YF/0L;<?D0LK � }606�30�B

k2003 ≡ (10a + 7)2003 ≡ 72003 + 2003 · 10a · 72002

≡ 43 + 3 · 10a · 49 ≡ 43 + 70a (mod 100)b

�YF/037 � 4GCJ7 d 0 k200303798/4@CN7~BGF/068>C1��CQBD4

2003 � }60�F<2��<043 + 70a ≡ 3 (mod 100) �

70a ≡ 60 (mod 100) �7a ≡ 6 (mod 10) �a ≡ 8 (mod 10)

b��037 d 0 � k ≡ 87 (mod 100)

b � CQBGF62�.ACAB[BD.J0PF/0�.JH6g1?D;<K#��2�H<.10 � }60��9798�BGF<2�Bk ≡ 587 (mod 1000)

2�798k ≡ 587 (mod 10000)

bÝ��F>=94 � BGF/0f;<7<.Q: � gq;>=9?E�8<CJ��CAB ��7<=9KOI903?!CN4k = 0587

b� F/0L72003

CN4M?D0LH<.J2 d 0L8�}+CQBGF2002

;<?2004 � BGF/03?D0�CN4M79; 7<=9KOI903?)BGF<2�B452�BDCJ4G��034!BGF/0 d ;<79��?[=n0L7 d 06KO;/8<=/.J;

10b_��F>=94 � BGF/03?D0�2�?D0+79;�4,;>.A=<BDC1;<7/43b³ �+Þ b+¼nX�Æ3V�Æ�É�Ë�ÕRÑ�ØMÌqÍYËqË Ó�Ø�Í�Ë�Ò = ÌJÓ��Wb

c3 × 3

KM2��C d 45�<=92L?D0 d ;<7/4�CN4GB[4s;>g�7<CJ790�8<CN4GBDCN7 d B_�>23.N=n034 � 4�= d F6BGF<23B�032 d F�;>gBGF/0+?D;�}�4 � d ;>.N=9KM7/4 � 2�798�8<CN2��3;<7/23.J4@F<2��<0+2 d ;<7/4GB[2�7>BW4�=9K�b@©�0�.1;�} CN4�2L7M0� L2LKOH<.10;>gY23 × 3

KM2��C d 4,�>=92�?D0/b�3=9H9H9;<4,0~BGF<23B�2

3 × 3KM2��C d 45�<=92L?D0RF<2L4+2 d ;<7/4GB[2�7>B�4�=9KÂ;>g

Tbf�L0�B�BGF/0KMCJ898>.10�0L7>BE?{:O;>gnBGF>CN4@4,�>=92�?D06I/0

Eb@-9?D;3�<0�BGF<2�B

T = 3Eb

2 9 4

7 5 3

6 1 8

��

= ÆL\�Ç/ÌJZ{Æ�ÏfÑ�Ø�Å!ÇLÉGZ�Ä�ÆL\{Ë3Ê�]�ÐLXDÓ>Õ9Ë)S�`�É3ÌDÇ>Õ9Ë�ÏYÌ1]�Æ_Ç�XEÏ/Ó/Ñ�Ø = ÆnÇ/ÌqÍ = Ë�×EÆ�ÏYÕ>ÓLXEØ = ×DÍYÆ/ÆL\N]Æ_Ç�XEÏ/Ó/Ñ�ØL]�Æ�Âub

�L0�BWBGF/0+KO2��C d 45�<=92L?D06I/0a1 a2 a3

a4 a5 a6

a7 a8 a9�YF/037a5 = E

b �3CJ7 d 0fBGF/0�4G=9K ;>g�BGF/0�03.J0LKO0L7>B[4OF/;<?[CA��;<7>B[2L.A.Q: � �<03?[BDC d 23.N. : � ;<?8<CN2��3;<7/23.N. :�CN4s0��>=92L./BE;T � BGF/037

3T = (a1 + a5 + a9) + (a2 + a5 + a8) + (a3 + a5 + a7)

= (a1 + a2 + a3) + 3a5 + (a9 + a8 + a7)

= T + 3E + T = 3E + 2T �2�798OF/037 d 0 � T = 3E

b23� ��6�� $+��-C587&9;�<5��=�)>?� � � ��@�� A) )!<� ��BC�� A(D(E��! �� '#+�� B6@<�� A �F�#�,+ �� ,-8'#- E�2 � E � A

>;!<��23� ��A<2 ��#�� <'?.³ �+à b�¼nXGÆ3V�Æ�É�ËLÕ Ñ�Ød¿�Æ�É�½Ë �/Ç�ZJÉ��½� Ó��Æ�Ó�XEXGË�XGÆ�] ��Ï>Z�ÊLË�X[ÉGZ�ÌJÓ/Ì}¼nÆL\AZ�Ì;ËL×DÏ<Z{×[Ó Õ9ËÂ/Ó/ÌJÓ3\�Ç�Ï9Ø3ÓL]8Æ�ÓLX�×EË¡\{Æ�Ï9Ó3] = VYÓ�Z1Ïub

ã{7TBE?[CJ2L79��.10ABC � 4�=9H/Hn;<450OBGF<2�B tan A � tan B � tan C

2�?D0OCJ7�F<2L?EKO;<7<C dH9?D;/��?D03454GC1;<7�bW��F/;�}�BGF<23Ba2 � b2 � c2

gq;<?DK#2L7O2L?DCQBGF<KR0�BDC d H9?D;/��?D03454GC1;<7�b%¾�Æ9Ì{Ë�>

x � y � z 2L?D0�CN7+F<2L?EKO;<7<C d H/?D;/��?D0L4�4�CJ;<7PCAg 1

x� 1

y� 1z

gq;<?EK�2L7�2�?[CABGF<KO0�BDC dH9?D;/��?D03454GC1;<7�b �= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �äÇ3É3ÌJÓ<ÊLÆ�ÄuX[Z1Ò10<Ë�XD] ��Ï>Z�ÊLË�X[ÉGZ�Õ>Ó>Õ}Â�U �EÂ ��]3Æ_Ç<Ë�ÏYÆ�ÉWUäZJX�Ë¡É5]<U�X�Ð<Ë�ÏuÌJZJÏ9Ó/b

�L0�BABC

I/0O2�7�2 d =<BE0L��2L79��.10L8fBE?DCN2�79��.J09bT�L0�BAHa � BHb � 2�798 CHc

I90BGF/0�2L.QBDCABD=n8/0L4 d ;<?E?D0345H9;<798<CN79�PBD;�4�CJ89034a � b � 2L798 c � ?D0L4�Hn0 d BDCQ�<0�.Q:ub�3CJ7 d 0

tan A � tan B � tan C2�?D0�CN7jF<2L?EKO;<7<C d H/?D;/��?D0L4�4�CJ;<7 � BGF/03?D0�0G �CN4GB[4

d ∈ / 4�= d F)BGF<23B �1

tan A=

1

tan B− d � � � �

1

tan C=

1

tan B+ d

b � � �ã{7O2�898>CABDCJ;<7 � }60�F<2��<0�BGF/0�gq;>.N.J;�}+CJ79�P?D03.N23BDCJ;<7/4 �

1

tan A=

AHb

BHb

=AHc

CHc

� � � �1

tan B=

BHc

CHc

=BHa

AHa

� � £ �1

tan C=

CHa

AHa

=CHb

BHb

b � ¢ �

��

Ü>?D;<K#� � � 2�798O� £ �E� � £ � 2L798O� ¢ �G� � � � 2�798M� ¢ �G� ?D0L4�Hn0 d BDCQ�<0�.Q: � }60��30�B �1

tan A+

1

tan B=

AHc + BHc

CHc

=c

CHc

� � � �1

tan B+

1

tan C=

BHa + CHa

AHa

=a

AHa

� �� 1

tan A+

1

tan C=

AHb + CHb

BHb

=b

BHb

b � � �Ü>?D;<K#� � � 2�798O� � �E� � � � 2L798O�� G� 2�798M� � �G� � � � 2�798O� � �E� ?D0345H90 d BDC �<03. : � }606;/I�BD23CJ7

2

tan B=

c

CHc

+ d � � �2

tan B=

a

AHa

− d � � �L� �2

tan B=

b

BHb

b � �� L0�B

k =2

tan B

b!Ü�?D;<KÙ0L�<=92�BDC1;<7/4s� �G� � �L� � 2L798O� �� E� CQBägq;>.A.1;�}�4äBGF<23B �c2 = (k − d)2 · (CHc)

2 � � �� a2 = (k + d)2 · (AHa)2 � � �� b2 = k2 · (BHb)

2 b � ��£ ��L0�B∆I90PBGF/0�2L?D0L26;>g 4ABC

bW�3CN7 d 0∆ =

c · CHc

2=

a · AHa

2=

b · BHb

2�

}606F<2��<0 �(CHc)

2 =4∆2

c2� � ��¢ �

(AHa)2 =4∆2

a2� � ��

(BHb)2 =

4∆2

b2

b � � � � ;<KOI<CN7<CJ79�+� �� }+CABGF�� ¢ �E� � �� }+CABGF�� E� 2L798O� ��£ � }+CQBGFM� � � �E� }60�;/I>B[2LCN7 �

c2 = 2∆(k − d) = 2k∆ − 2d∆ �b2 = 2∆k �a2 = 2∆(k + d) = 2k∆ + 2d∆

b�YF/034,0+0L�<=92�BDC1;<7/4�45F/;�} BGF<2�B

c2 � b2 � 2L798 a22L?D0PCJ7~2�?[CABGF<KO0�BDC d H/?D;/��?D0L4�4�CJ;<7}+CABGF d ;<KOKO;<7O8<C��03?D0L7 d 0

2d∆b

�� _a`+bdcfe"\2^ b�� §Z]\�V b�� §Z

� ��"��Ý��"~(�� @�� "�*��Y*��Â&�� "!,&)&��03?D0�CN4W2�H9?D;/I>.103K }6F/0L?D0sCABuCJ4WH9;<454GC1I>.10�BE;�890�BE03?EK�CJ790sBGF/0�2�7/4D}60L?�}+CQBGF/;>=<B8<CN4 d ;L�<03?DCN79�+45;<KR0�;>g9BGF/0�?D0L23.N. :~CJ7>BD0L?D034GBDCN79��BGF>CJ79��4�BGF<2�B!2�?D0�F<2�H/Hn037<CJ79�� ¦ T ��037<C1;<?s��23BGF/03KO2�BDC d 4 F<2L.A.10379�30 �c 4,�>=92�?D0XABD

;>g/4�CJ890�.J0L79�¡BGF1CJ4ä8/?[2�}�76CJ7/4GC18/0�2 d CN? d .10�}+CABGF68<CN2�KO0�BD0L?

XY;>g<.10379�¡BGF

2b��F/0PH9;>CJ7>B

A.AC1034!;<7�BGF/0 d CN? d =9KMgq03?D0L7 d 0�;>g<BGF/0 d CN? d .10/b c 79;�BGF/0L?4,�>=92�?D0

Y CBECN4�8/?[2�}�7�b � F<23B!CJ4@BGF/06?[23BDCJ;O;>gYBGF/062�?D032�;>gu4,�>=92�?D0

XABDBD;BGF/0�2�?D032);>g�4,�>=92�?D0

Y CBE�

� CABGFOBGF>CJ4�B{:/H90O;>gWH/?D;/I<.J0LK �12O�30L;<KR0�BE?DC d 23._;<790)}+CQBGF�.J0L79�¡BGF<4+��CQ�<037 �G� K~:�n?[4EB�CJ7/4EBDCJ7 d B�45;<KR0�BDCJKO0L4�CJ4�BE;�=94,0 d ;/;<?D8<CN7/23BD0L4 � I/0 d 2L=9450P�<=/CQBE0�;>gNBE037�BGF/0�2�7/4D}60L?d 2L7�I90MF>CAB�=9H9;<7��<=/C d ��. :fCN7�BGF>CJ4�}�2�:�b6��F>CJ4PCJ4P79;�B�H92L?[BDC d =/.J2L?D. :�03.J0��2L7>B � I<=<B�CQB}6;<?G��4 �.Lx�y !��Nx�$%$-/=<B_BGF/0 d CJ? d .J0PCJ7)BGF/0 d ;/;<?D8>CJ7/2�BE0�H/.N2�790�}+CQBGF)CAB[4 d 0L7>BE?D0 � O � 2�B (0, 0) � X 2�B(−1, 0) � 2L798 Y

23B(1, 0)

bãqg�}60 ;{;>CJ7OBE;

A�Q}6F>C d F�CN4M;<7�BGF/0d CJ? d .J0 �E� BGF/037 OA = OX = 1 � 4�CN7 d 0MBGF/0,:2�?D0!?[2�8>CNC � 2L798 AX = 1 � I/0 d 2L=9450äBGF/0!4,�>=92�?D0F<2�4~4GC18/0�.J0L79�¡BGF

1b'�YF/03?D03gq;<?D0 � 4XAO

CN40��>=/CN.N23BD0L?[23.�BGF>=94 � ∠OXA = 60◦ b� CABGFáBGF>CN4�H/CJ0 d 0 ;>g�CN7<gq;<?EKM23BDCJ;<7e2�798;>=9?��79;�}+.10L89�30O;>g

30�60�90BE?DCN2�79��.J0L4 � }60d 2L7á890�BE03?EK�CJ790�BGF<23B�BGF/0 d ;/;<?D8<CN7/23BD0L4M;>g

A2�?D0 (−1

2,

√3

2

) bäã{7~;�BGF/03?Y}6;<?D8/4 � BD;)�30�B�g1?D;<KXBE;

A}60��3; 1

2

=97<CAB@BD;OBGF/0~?[C1�3F�B�2L798 √3

2=97<CAB[4@=9H�b

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� 0 d 2L7+4,;>. �<0!BGF>CN4�gq;<?q � =94�CN79� r = 1

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1 : (2 −√

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b��L0�BA � B � C � D I/0!BGF/0@Hn0L;<H/.J09b_ãqg�BGF/0,:�2L.A.��79;�}�;<790)2�79;�BGF/03? � BGF/0L7f0L2 d FRHn03?[4,;<7�2�B�BGF/06B[2�I>.10+}+CA.N.��79;�}�F>CN4P790�C1�3F/I/;>=9?[4 �79;�KO2�B[BE03?s}6F<23BP4,0323BDCN79�M2L?E?[2L79�30LKO0L7>BPCJ4�=94,0L8�b �!BGF/0L?�}+CJ450 � }60OKM2�:�4G=9H9H9;<4,0BGF<23B

A8/;/0L4@79;�B��79;�}

Cb��YF/037

A2�798

CKM=94EB@032 d FM��79;�}

B2L798

D � 2L798�BGF/04,0323BDCN79��2�?D?[2�79�303KR037>BA�B�C�DCN4@2645;>.N=<BDCJ;<7�b�P;�}#4�=9H/Hn;<450

N ≥ 3b ;<7/4GC18/0L?�BGF/0O4GCJKMH/.J0R��?[2LHnF�}6F/;<450~�<03?[BDC d 034�2�?D0BGF/0+H90�;<H<.10 � B�}6;~;>g/}6F>C d F~2L?D0 ;{;>CN790�8�IL:~2�7M0L89�30PCNgn2�798�;<7<.Q:)CAg9BGF/0,:��79;�} 0L2 d F;�BGF/0L?¡b��YF/0+H9?D;/I>.103K�CJ4äBE;~H/?D;L�<0PBGF<23B_BGF>CN4s��?[2�H9F d ;<7>B[2LCN7/4@26�>=92�8<?DCA.J2�BE03?[2L.��ABGF<23BCJ4 � 2 4

��d CN? d =/CAB � b c�dEd ;<?D8>CJ79�äBE; � � �q0G �0L? d CN4,0 � �� H�b �: �G� 0,�<03?{:+4�CNKOH<.10s��?[2LHnF�}+CQBGF2N

�<0L?�BDC d 0L4@2�798M2�Bä.J0L2L4GB N

2(1 +

√8N − 3)

0�8/�30L4 d ;<7>B[2LCN7/4@2)�>=92�8<?DCA.J2�BE03?[2L.�b�L0�BAI/0�2�7�: ;<790T;>g@BGF/0R�<0L?�BDC d 0L4 � 2L798'.10�B d(A)

I90fCAB[4M890L��?D0�0�CN7'BGF/0��?[2�H9Fub�Ü�?D;<K BGF/0��CQ�<037~CN7<gq;<?EKM23BDCJ;<7 � }60+F<2��<0 d(A) ≥ Nb_�3CJ7 d 0�2�7�:~0�8/�30

ABCJ4 d ;>=97>BD0�86CN7~I9;�BGFd(A)

2�798d(B) � }60�8/0�8>= d 0�BGF<23B�BGF/0sBE;�B[2L.<7<=9KRI/0L?W;>gn0�8/�30L4CJ4s2�B!.J0L2L4GB

N2bsã1B!CN4�032�4[:~BE;6�<0L?[CNgA:~BGF<23B

N2 ≥ N

2(1 +

√8N − 3)

gq;<?N ≥ 3

b�YF/0 d ;<7 d .N=94GC1;<7~gq;>.N.J;�}�4Lb��&P��#��$ �� /b9�L;L��@2�4E� � Â�Æ�Ò�ÑLZ1Ï/Ó/Ì{Æ�X[Z1Ó�\�¼nX�Æ9ÑL\{Ë�ÒMÉäÓ�Ï�Õ ��<Ë�X�×�ZNÉ�Ë¡É � ��;<?[BGF/�,��;>.A.J2L798 � �" � bæ b@-9?D;3�<0�BGF<2�BW2�7�:)79;<79��790L��23BDC �<0+0,�<0L7)CJ7>BD0��303? d 2L7�I90P=97<CJ�<=n0�.Q:~?D03H9?D034,037>BE0L8�2L4(x + y)2 + 3x + y

}6F/03?D0x2L798

y2L?D0�79;<79�G790��2�BDCQ�<0+CN7>BE0L�30L?[43b

= ÆL\{ÊLË�ÕâÑ�Ø[¼/Z�Ë�XEX�Ë]Æ�Æ�XEÏ/É ��Ì�Ë,Z1Ï<]zË'Ó3ZNÉ�Æ�Ï<É � �LÓ ��Ì[Ì{Ë�] ��XDÓLÏY×EË � ÓLÏYÕ Â/Í9X[ZNÉ3Ì{Æ3VYÍYË�X�¿LÔÆ�XDÓ>Õ�\�Ë�Ø3]8Æ�X[ZJÉ�Ì�ÆL\N]��ÄäÔ �~ËPÐ3Z{ÊLË¡Æ�Æ�XEÏ/É ��Ì�Ë,Z1Ï�É É@É�ÆL\�Ç/ÌJZ{Æ�Ï/]YÓ>Õ>Ó�V_Ì{Ë�ÕRÑ�ØMÌqÍ�Ë6ËLÕ�Z�Ì{Æ�X¡b

�L0�BnI/0+2�79;<79�G790��2�BDCQ�<0�0,�<0L7~CJ7>BD0��303? � 2L798~.10�B x

2L798yI/0+79;<79��790L��23BDC �<0CJ7>BD0��303?[4s4�= d F)BGF<2�B

(x + y)2 + 3x + y = nã1B�CN460L2L4[:fBE;�450�0�BGF<2�Bn = 0

CNgä2�798Ý;<7<. :�CNgx = y = 0

bTÜ>?D;<KÂ79;�}¯;<7 � }602�4�4�=9KO0�BGF<2�Bn ≥ 2

b�L0�BaI/0�BGF/0)H9;<4�CQBDCQ�<0+CJ7>BD0��303?@4�= d F�BGF<2�B

a2 ≤ n < (a + 1)2b��P;�BD0�BGF<2�B

a2 + a − 22�798

a2 + a2L?D0�B�}6; d ;<7/450 d =<BDC �<0+0,�<0L7~CN7>BE0L�30L?[43b ;<7/4,0L�<=n037>BD.Q: � BGF/0gq;>.N.J;�}+CJ79��B�}6; d 2�450L4s0G >F<23=94GB!23.N.nH9;<454GC1I>CN.ACABDCJ0L4 �

pE�"6��$�ba2 + a ≤ n < (a + 1)2

bãqgx + y ≥ a + 1 � BGF/0L7

n = (x + y)2 + 3x + y ≥ (a + 1)2 > n �2 d ;<7>BE?[2�8>C d BDCJ;<7�b!ãqgx + y ≤ a − 1 � BGF/0L7

n = (x + y)2 + 3x + y ≤ (a − 1)2 + 3(a − 1)

= a2 + a − 2 < a2 + a ≤ n �

��£�

2 d ;<7>BE?[2�8>C d BDCJ;<7�b��YF/03?D03gq;<?D0 � x + y = ab��YF/037

n = (x + y)2 + 3x + y = a2 + a + 2xb

��037 d 0 � x = 1

2(n − a2 − a)

2L798y = a − x = 1

2(a2 + 3a − n)

b�P;�}ç.J0�Bx2�798

yI90~8/03�9790�8�IL:MBGF/0L450~0� �H9?D03454GC1;<7/43b��3CN7 d 0

nCN4P0,�<037�2L798

a2 ≡ a (mod 2) � I9;�BGF x2L798

y2L?D0+CJ7>BD0��303?[4LbW�3CN7 d 0

a ≥ 1 � }60�F<2��<0a2 + 3a ≥ a2 + 2a + 1 = (a + 1)2 > n �g1?D;<K�}6F>C d F)}606890L8<= d 0+BGF<2�B

y > 0b!�3CJ7 d 0

n ≥ a2 + a�Ngq;<?äBGF>CN4 d 2L4,0 �G� }60�2L.N4,;F<2��<0

x > 0b��F>=94 � BGF/0)H923CJ? (x, y)

��C �<0L4�2)=97<CJ�<=n0~?D03H9?D034,037>BD2�BDC1;<7�;>gnCJ7OBGF/089034�CN?D0�8�gq;<?DKTb

pE�"6�� 9ba2 ≤ n ≤ a2 + a − 2

bãqgx + y ≤ a − 2 � BGF/0L7

n = (x + y)2 + 3x + y ≤ (a − 2)2 + 3(a − 2)

= a2 − a − 2 < a2 ≤ n �2 d ;<7>BE?[2�8>C d BDCJ;<7�b!ãqgx + y ≥ a � BGF/037

n = (x + y)2 + 3x + y ≥ a2 + a + 2x ≥ a2 + a > n �2 d ;<7>BE?[2�8>C d BDCJ;<7�b��YF/03?D03gq;<?D0 � x + y = a − 1b_��F/0L7

n = (x + y)2 + 3x + y = (a − 1)2 + a − 1 + 2x = a2 − a + 2xb

��037 d 0 � }60�F<2��<0 x = 1

2(n − a2 + a)

2�798y = a − 1 − x = 1

2(a2 + a − n − 2)

b�P;�}�.J0�Bx2�798

yI/0O8/03�9790�8�IL:�BGF/034,0M0� �H9?D03454GC1;<7/43b c 4PCJ7 2�450 � � I/;�BGF

x2�798

y2L?D0fCJ7>BD0��303?[4Lbe�3CN7 d 0

a2 ≤ n2L798

a > 0 � }60�F<2��<0 x > 0be�3CJ7 d 0

n ≤ a2 + a − 2�Ngq;<?sBGF>CN4 d 2�450 �E� }60�2L.N4,;fF<2��<0 y > 0

b~�YF>=94 � BGF/0MH/2LCN? (x, y)��CQ�<034s2+=97<C1�>=n0�?D0LH/?D0L450L7>B[23BDCJ;<7R;>gnCJ7~BGF/068/0L4GCJ?D0L8�gq;<?DKTb

� b�ã{7fBE?DCN2�79��.J0ABC � BGF/0M4�CJ89034+452�BDCJ4GgA: AB + AC = 2BC

b�-9?D;3�<0~BGF<23B�BGF/0I<CN4,0 d BE;<?�;>g∠A

CN4sH90L?DHn03798<C d =/.J2L?äBE;)BGF/0+.NCN790�4,0L��KR037>B ;{;>CN7<CJ79�PBGF/0+CN7 d 037>BE?D062L798d CJ? d =9K d 037>BE?D0);>gABC

b= ÆL\{ÊLË�Õ'Ñ�Ø Ë�Z{×DÍYË,\�Æ�Ó/ÌJÓ3ZA\N\{Ë�]§Å_ÆnÇ>Ë�Ï<] ��X[Ó�Ï�×GË � Â/Í/X[ZJÉ�Ì�Æ3VYÍ�Ë�X+¿LÔ�Æ�X[Ó<Õ�\{Ë�ØL]§Æ�X[ZJÉ�Ì�ÆL\N]��Ä � Ó�Ï�Õ �3Æ�É,Í>Z�Æ = Ë¡ZJÒ�ZqØ3Ó3]8Ä�Ó<ÈsÓ3É,ÓF0�ZJ]�¿,Ó�VYÓLÏ�Ô �~Ë�Ð3Z{ÊLË = Ë,Z1Ò�Z1Ø3Ó�É ÉsÓ�X�Ð9ÇLÒfË�ÏYÌDb

� 0+2L454G=9KR0�BGF<2�BAB 6= AC

b��L0�BI2�798

OI/0BGF/0�CJ7 d 0L7>BE?D0+2�798 d CN? d =9K d 0L7>BE?D0�;>g 4ABC � ?D0L4�Hn0 d �BDCQ�<0�.Q: � 2�798�.10�B D

I90MBGF/0RCN7>BE03?[4,0 d BDC1;<7�;>gAI

}+CABGFBC

b_�3CN7 d 0BI

2L798CI

I<CN4,0 d B∠ABD

2�798∠ACD �?D0L4�Hn0 d BDCQ�<0�.Q: � }606F<2��<0

AI

ID=

AB

BD=

AC

CD

=AB + AC

BD + CD=

AB + AC

BC= 2

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A

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AI = 2IDb ;<7/450��>=n0L7>BD. : �

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MI90WBGF/0!4,0 d ;<798�CJ7>BD0L?[450 d BDCJ;<7�;>g

AD}+CABGF@BGF/0 d CN? d =9K d CN? d .10�;>g 4ABC

b�YF/037∠MBD = ∠MBC = ∠MAC = ∠MAB

b�3CJ7 d 0

∠BMD = ∠BMA � }60��30�B 4MBD ∼ 4MABb!ã1Bägq;>.N.J;�}�4!BGF<2�B

DM

BM=

BM

AM=

BD

AB=

1

2

b�YF>=94 �

DM

AM=

DM

BM· BM

AM=(

1

2

)2

=1

4

b��037 d 0 � AM = 4DM � g1?D;<K }6F>C d F6}60�;/I>B[2LCN7

AD = 3DMb � � �Ü>?D;<K�� 2�798�� }60+�30�B ID = DM

b��F>=94 � AI = 2ID = ID + DM = IMb�YF/03?D03gq;<?D0 � OI ⊥ AM

b��F>CJ4�CJKMH/.AC1034!BGF<23BAI ⊥ OI

b��/Õ�Ô_��0�0sBGF/0�4,;>.A=<BDC1;<7�BE;M-9?D;/I<.J0LK �� .N23BD0L?uCJ7+BGF>CJ4WCN454G=n0 � 0L4�Hn0 d CJ23.N. :+BGF/00�8>CABD;<?DCN2L. d ;<KOKO0L7>B[4@gq;>.A.1;�}+CN79��BGF/0+4,;>.A=<BDC1;<7�b �

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.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................P (t = t1)

Q(t = t4)

R(t = 0) S(t = t3)X

Y (t = t2)

(t = t5)

(t = t6)

t3x |t5 − t3|x

|t4 − t1|w |t6 − t4|wt1u

(t2 − t1)u |t2 − t4|v

|t4 − t5|v-��

A

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b � � �¦s4�CN79�)�'0L790�.J23=94��F/0�;<?D03KÙ;<7 4PRX

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b � � �ß/�>=923BDCJ;<7�� CQ�<034 t1t2t3 + t1t3t5 + t2t4t5 = t1t2t5 + t1t3t4 + t2t4t5 �2�798�0L�<=92�BDC1;<7�� CQ�<034 t1t2t3 + t1t3t6 + t2t4t5 = t1t2t5 + t1t3t4 + t2t3t6

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b�©9=<Bt1 6= t2 �

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DKR0L0�B�b

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|G| <∑

x

d(x) �}6F>C d F d ;<7>BE?[2�8<C d BD4s� � � b��YF/037 n ≤ k � 2L4 d .J23CJKO0�8ub��'#��b ��F/0�CN790��>=92L.ACAB{:'CJ4)BGF/0TI9034GB)Hn;<4�4�CJI<.J0 � CJ7�BGF/0�gq;>.N.J;�}+CJ79�f450L7/4509b �L0�BS = {x1 � x2 � . . . � xn} b ãqg G1 = {x2 � . . . xn} � 2L798 Gi = {x1 � xi}

gq;<?i = 2 � . . . � n � BGF/0L7~BGF/0 d ;<798<CQBDC1;<7/4s;>gnBGF/0+H9?D;/I>.103K#2L?D0�452�BDCJ4G��0L8�2L798

k = nb

Û � b � 0 d F/;/;<4,02p−1

4G=nI/450�B[4!g1?D;<K�2�4,0�B�}+CQBGFp03.J0LKO0L7>B[4�4�= d F�BGF<23BW2�7�:6BGF<?D0L0F<2��<0�2 d ;<KMKR;<7M03.J0LKO0L7>B�b@-9?D;L�<0PBGF<23BWBGF/0,:�2L.A.�F<2��<0�2 d ;<KOKO;<7O03.J0LKO0L7>B�b

= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Øu¼/Z{Ë�XEXGË¡Æ�Æ�XEÏ/É ��Ì�Ë,Z1Ï<]�Ë Ó�ZJÉ�Æ�Ï/É�� LÓ��Ì[Ì�Ë¡] ��X[Ó�Ï�×GË�br:��¡��nb!�L0�B

n ≥ 1I90�2�7�CN7>BE0L�30L? � 2L798�.10�B S

I90�2�4,0�BY}+CQBGFn0�.103KR037>BD43b��L0�B

A1 �A2 � . . . � Ak

I/0P8>CJ4EBDCJ7 d B�4�=nI<4,0�BD4ä;>gS4G= d F�BGF<23B � gq;<?�2L.A. i 2�798 j � BGF/0sCJ7>BD0L?[450 d BDCJ;<7

Ai ∩ AjCJ4@79;�B�03KOH>B�:ub_��F/0L7

k ≤ 2n−1b

¼nX�Æ/Æ�Î!Æ�Î�\�Ë�ÒOÒOÓ9b~Ü>;<?i = 1 � . . . � k � .10�B Bi = S\Ai

b+��F/0L7B1 � B2 � . . . � Bk2�?D0R8<CN4GBDCN7 d B�4G=nI/450�B[46;>g

Sb�ãqg

Bi = Ajgq;<?P4,;<KO0

i2�798

j � BGF/037�}60M�9798�BGF<2�BAi∩Aj = Ai∩(S\Ai) = ∅ � }6F>C d F d ;<7>BE?[2�8<C d BD4uBGF/0sF�:<Hn;�BGF/034�CN4Lb_ã1BYgq;>.A.1;�}�4YBGF<2�BA1 � A2 � . . . � Ak � B1 � B2 � . . . � Bk

2L?D0)8<CN4GBDCN7 d B�4�=nI<4,0�BD4�;>gSb!��F>=94 � 2k ≤ 2n �BGF<23BäCN4 � k ≤ 2n−1

b� 0~79;�} H9?D;3�<06BGF/0�89034�CN?D0�8R?D0L4G=/.AB�IL:OCJ798>= d BDC1;<7Rgq;<?

p ≥ 3b+�{ãqg

p = 1;<?

p = 2 � BGF/0L7fBGF/0 d ;<798<CQBDC1;<7fBGF<2�B�2L7�:fBGF<?D0L0O4�=nI<4,0�BD4�F<2��<0O2 d ;<KMKR;<7T0�.103KR037>BCJ4��>2 d =n;>=94G.Q:�BE?[=n0 � 4GCJ7 d 0+}60 d 2L7 d F/;/;<4,0);<7<. : 2p−1 < 34G=nI/450�B[4Lb@��F/0)?D0L4G=/.AB�CN4g12L.N4,0�gq;<?_BGF/0L450 d 2�450L43b �ãqg

p = 3 � .J0�B S = {x1 � x2 � x3} b � 0 d F/;/;<450)gq;>=9?�450�B[4 A1 � A2 � A3 � A42�KO;<79� ∅ � {x1} � {x2} � {x3} � {x1 � x2} � {x1 � x3} � {x2 � x3} � {x1 � x2 � x3} �4�= d F~BGF<23BäBGF/0�CN7>BE03?[4,0 d BDC1;<7R;>gY2�7�:�BGF<?D0L0);>g�BGF/03K#CJ4s79;<79��0LKMH<B{:�b .1032�?[.Q: � 79;<790;>guBGF/06gq;>=9? d 2�7fI/0 ∅ b c .N4,; � BGF/0)450�B[4 {x1 � x2} � {x1 � x3} � {x2 � x3} d 2L7/79;�Bs2L.A.I90 d F/;<450L7 � I90 d 23=94,0OBGF/0�CJ?�CN7>BE03?[4,0 d BDC1;<7TCJ4)0LKMH<B{:�b ã1B+gq;>.A.1;�}�4+BGF<2�B�;<790f;>g!BGF/01� 03.J0LKO0L7>B�4,0�BD4WK�=94GB_I/0 d F/;<4,037 � 452�: {x1} b��YF/037~032 d F6;>g>BGF/0 d F/;<4,037)450�B[4WKM=94EBd ;<7>B[2LCN7

x1

b��YF>=94 � BGF/0 d ;<7 d .A=94�CJ;<7OF/;>.J8/4�gq;<? p = 3b�L0�B

p ≥ 4I90M2O�� 0�8fCJ7>BD0��303?�bR�3=9H9H9;<4,0�BGF<2�B�BGF/0O?D034�=/.QB�F/;>.18<4+gq;<?�2O4,0�B}+CABGF

p − 103.J0LKO0L7>B[4Lb@�L0�B

S = {x1 � x2 � . . . � xp} � 2�798~.10�B A1 � A2 � . . . � A2p−1I9068>CJ4EBDCJ7 d Bä4�=nI<4,0�BD4s;>gS4G= d F)BGF<23B

Ai ∩ Aj ∩ Ak 6= 0 � gq;<?ä2L.A.i < j < k

b�L0�B

npI/06BGF/0�7<=9KOI903?�;>g�4,0�BD4

AiBGF<23B d ;<7>B[2LCN7

xpb6ãqg

np < 2p−2 � BGF/0L7BGF/0�7<=9KOI903?_;>g>4,0�BD4Ai79;�B d ;<7>BD23CJ7<CN79�

xpCJ4

2p−1−np > 2p−1−2p−2 = 2p−2b�YF/034,0 4,0�BD4�gq;<?DK 2 d ;>.A.10 d BDC1;<7 ;>geKR;<?D0´BGF<2�7

2p−28<CN4GBDCN7 d B 4G=nI/450�B[4#;>g

{x1 � x2 � . . . � xp−1} 4�= d F�BGF<23B Ai ∩ Aj 6= ∅ gq;<?+2L.A. i2�798

j � d ;<7>BE?[2�8<C d BDCJ79�BGF/0+.103KOKM2nb��F>=94np ≥ 2p−2

b� CABGF�79;Ý.J;<454�;>g��303790L?[23.NCQB�: � }60�KM2�:�79;�}�4�=9H/Hn;<450�BGF<23Bxp

I/03.J;<79��4)BD;A1 � A2 � . . . � Anp

b

��¢��

pE�"6��$�bnp > 2p−2

b�L0�BA′

i = Ai \ {xp} gq;<? i = 1 � 2 � . . . � npb��YF/037

A′1 � A′

2 � . . . � A′np

gq;<?EK2 d ;>.A.10 d BDC1;<7);>g9KR;<?D0�BGF<2L72p−2

8<CN4GBDCN7 d B_4�=nI<4,0�BD4�;>g {x1 � x2 � . . . � xp−1} b_¦s4�CN79�BGF/0�.103KOKM2 � }60M890L8<= d 0�BGF<2�BsBGF/0L?D0O0G �CN4GB r2L798

s4G= d FRBGF<23B

A′r ∩ A′

s = ∅ bRã1Bgq;>.N.J;�}�4WBGF<23BAr ∩ As = {xp} b!Ü>;<?ä032 d F k 6∈ {r � s} � 4�CN7 d 0 Ar ∩ As ∩ Ak 6= ∅ �}60�F<2��<0

Ar ∩ As ∩ Ak = {xp} � g1?D;<K }6F>C d F6}60�890L8<= d 0�BGF<2�B xpCJ4@2 d ;<KOKO;<703.J0LKO0L7>B�;>gY23.N./BGF/0

Ai� 4Lb

pE�"6�� 9bnp = 2p−2

b�L0�BkI/0�4�= d FsBGF<23B

xp 6∈ Ak2�798�8/03�9790

Bk = Ak∪{xp} bu�YF/037 Bk 6= Akb�3=9H9H9;<4,0�BGF<23B

Bk 6= Ajgq;<?!2L.A.

jbW�YF/037�BGF/0�4,0�BD4

A1 � A2 � . . . � A2p−1 � Bkgq;<?EK2 d ;>.N.J0 d BDCJ;<7);>g

2p−1 +18<CN4GBDCN7 d B_4G=nI/450�B[4!;>g

S � 2�798+BGF/0�CN7>BE03?[4,0 d BDC1;<7~;>g/2�7�:6H923CJ?;>gu8<CN4GBDCN7 d B!450�B[4�CJ7~BGF>CN4 d ;>.N.J0 d BDCJ;<7~CJ4@79;<79��0LKMH<B{: � d ;<7>BE?[2�8<C d BDCJ79��BGF/0+.J0LKMKO29b�YF/03?D03gq;<?D0 � gq;<?O032 d F k4�= d FáBGF<23B

xp 6∈ Ak � BGF/0L?D0 0G �CN4GB[4 j4G= d FâBGF<2�B

Ak ∪ {xp} = Ajb��3CN7 d 06BGF/0)7<=9KRI/0L?�;>g

Ai� 4s}6F>C d F�89;R79;�B d ;<7>BD23CJ7

xpCN4�BGF/0452LKR0+2L4!BGF/0+7<=9KRI/0L?�;>g

Ai� 4!}6F>C d F d ;<7>BD23CJ7

xp � }60�890L8<= d 0�BGF<2�BWBGF/0 d ;>.A.10 d BDC1;<7;>gnBGF/0Ai� 4�CJ4

A1 � A2 � . . . � Anp� A1 ∪ {xp} � A2 ∪ {xp} � . . . � Anp

∪ {xp} �}6F/0L?D0xp 6∈ Ai

gq;<?i = 1 � 2 � . . . � np

b�P;�}A1 � A2 � . . . � Anp

2L?D02p−2

8>CJ4EBDCJ7 d B!4G=nI/450�B[4�;>g {x1, x2, . . . , xp−1}4�= d FMBGF<23B � gq;<?s23.N. i < j < k � }60~F<2��<0 Ai ∩ Aj ∩ Ak 6= ∅ b~©<:MBGF/06CJ798>= d BDC1;<7F�:/H9;�BGF/0L4GCJ4 � BGF/0,:64,F<2L?D0P2 d ;<KMKR;<7)03.J0LKO0L7>B � 452�: ab .J0L2L?D. : � a ∈ Ai ∪{xp} gq;<?0L2 d F

i � 2L798MBGF/0L?D0�gq;<?D0 � a CN4�2 d ;<KOKO;<7�03.J0LKO0L7>Bs;>g�23.N.nBGF/0 Ai� 43b@�YF>CN4P03798/4sBGF/0CJ798>= d BDC1;<7M4GBD0LHM2�798)BGF/0�H9?D;/;>gDb

Û � b@�L0�Ba � b 2L798 c

I/0�?D032L.n7<=9KRI/0L?[4!}+CQBGF�4�=9K � b@-9?D;3�<0�BGF<2�Ba7 + b7 + c7

7=

(

a5 + b5 + c5

5

)(

a2 + b2 + c2

2

) b= ÆL\{ÊLË�Õ'Ñ�Ø Ë�Z{×DÍYË,\�Æ�Ó/ÌJÓ3ZA\N\{Ë�]§Å_ÆnÇ>Ë�Ï<] ��X[Ó�Ï�×GË � Â/Í/X[ZJÉ�Ì�Æ3VYÍ�Ë�X+¿LÔ�Æ�X[Ó<Õ�\{Ë�ØL]§Æ�X[ZJÉ�Ì�ÆL\N]��Ä � ¼/Z{Ë�XEXGË Æ�Æ�XEÏ/É ��Ì�Ë,Z1Ï<]dË Ó�ZJÉ�Æ�Ï/É�� LÓ��Ì[Ì�Ë¡] ��X[Ó�Ï�×GË � Ó�Ï�Õ��nËLÕ�Ç3\JÓ ¾)Ô�ËeÇ�X,Ì1ØL]�PÆ>ÊLË�XD]�¼3U!]�� = UPÔ��~ËPÐ�Z�ÊLË¡Æ�X[Ó<Õ�\{Ë�Ø�É ÉsÉ�ÆL\[Ç<ÌJZ�Æ�Ïub

�L0�Ba � b � c I/0�BGF/0+?D;/;�B[4@;>g x3 + qx − r = 0 � 2L798~.J0�B Sk = ak + bk + ck b�YF/037

S1 = a + b + c = 02�798

S2 = a2 + b2 + c2

= (a + b + c)2 − 2(ab + bc + ca) = −2q�P;�} }60�F<2��<0S3 + qS1 − 3r = 0 ==⇒ S3 = 3r

S4 + qS2 − rS1 = 0 ==⇒ S4 = 2q2

S5 + qS3 − rS2 = 0 ==⇒ S5 = −3qr − 2qr = −5qr

S7 + qS5 − rS4 = 0 ==⇒ S7 = 5q2r + 2q2r = 7q2rb

�YF/037 S7

7= q2r =

S5

5· S2

2

b

��¢ �Þ Û bç��0L��KR037>BD4

AC2L798

BDCJ7>BD0L?[450 d B~2�B~Hn;>CN7>B

Eb�-<;>CJ7>B[4

K2�798

M � ;<74,0L��KR037>BD4AB

2�798CD � ?D0345H90 d BDC �<03. : � 2L?D0�4�= d F)BGF<23BWBGF/0�4,0L��KR037>B KM

H/2�4�4,034BGF<?D;>=n�3FEb@-9?D;L�<0PBGF<23B

KM ≤ max{AC � BD} b= ÆL\{ÊLË�Õ�Ñ�Ø�Â/Í/X[ZJÉ�Ì�Æ3VYÍ�Ë�X�¿LÔ4Æ�XDÓ>Õ�\�Ë�Ø3]�Æ�X[ZJÉ�Ì�ÆL\N]��Ä � Ó�Ï�Õ �3Æ�É,Í>Z�Æ = Ë¡ZJÒ�ZqØ3Ó3]�Ä�Ó<ÈsÓ3É,ÓF0�ZJ]¿,Ó�VYÓLÏ�Ô �~ËPÐ�Z�ÊLË = Ë,Z1Ò�Z1Ø3Ó�É ÉsÉ�ÆL\[Ç<ÌJZ�Æ�Ïub

�YF/0+gq;>.A.1;�}+CN79��.103KOKM2�CN4ä}603.A.��79;�}�7 � 2L798�CAB[4@H9?D;/;>g9}+CA.N.979;�B�I90��CQ�<037�br:��¡��nb ãqgP

CN4 2 H9;>CJ7>B ;<7 BGF/0´4�CJ890BC

;>g BE?DCN2�79��.J0ABC � BGF/0L7

AP ≤ max{AB, AC} bpE�"6��$�bAB ‖ CD

b_�q��0L0�BGF/068>CJ2��?[2�KÙ;<7)BGF/0+.10�gNB�I/03.J;�}Ob ��L0�BX � Y

I90�H9;>CJ7>B[4~;<7CD

4G= d F�BGF<23BKX ‖ AC

2�798KY ‖ BD

b'�3CJ7 d 0AKXC

2L798BKY D

2�?D0�I9;�BGF�H/2�?[23.N.J03.J;/��?[2�KM4 � }606F<2��<0KX = AC

2L798KY = BD

b�3CJ7 d 0

MCJ4@2�Hn;>CN7>B�;<7~BGF/0�450��KO0L7>B

XY � }60�F<2��<0 � IL:)BGF/0+.J0LKMKO2 �KM ≤ max{KX, KY } b�YF/03?D03gq;<?D0 �KM ≤ max{AC, BD} b

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f̃(X) = X2 + 2ABX + A2(B2 − 4AC)b��F/0gq;>.N.J;�}+CJ79��BGF<?D0�0�4EBD2�BE03KR037>BD4@2L?D060L�<=/C �>2L.J0L7>B �

�JC � f(X2)CN4@?D0�8>= d C1I>.10+CN7

Z[X] ��NCNC � f(X)CJ4�?D0�8>= d C1I>.10)CN7

Z[X] � ;<?�BGF/03?D0�0� LCJ4EBD4 S ∈ Z}+CQBGF

AC = S24�= d FBGF<2�B

A(2S − B)CN4@2645�<=92L?D0+CN7

Z ��NCNCAC � f(X)CN4u?D0�8>= d C1I>.10äCJ7

Z[X] � ;<?n45;<KR0ä4,�>=92�?D0�03.J0LKO0L7>BY;>g Z CN4u2�?D;/;�B�;>g f̃(X)b

¼nX�Æ/Æ�Î[Ô 03?[B[2LCN7<.Q:ÝBGF/0�BGF<?D0�0�4EBD2�BE03KR037>BD4�2L?D0�2L.A.!BE?[=n0�CAgf(X)

CN4�?D0L8<= d CJI<.J0�CN7Z[X]

b_��F>=94 � }60+KO2�:M2L454G=9KR0PBGF<2�B f(X)CN4�CJ?D?D0�8>= d C1I>.10+CN7

Z[X]b� 06�9?[4GB@4G=9H9H9;<4,0�BGF<2�Bs�JC � CJ4@BE?D=n0/b+ã1B�gq;>.A.1;�}�4sg1?D;<KÙ��F/0�;<?D03K � BGF<23B!BGF/03?D00� LCJ4EB!H9;>.Q:<79;<KMCN2L.N4

G(X)2L798

H(X) ∈ Z[X]2�798M2+=97<CQB

u;>g

Z4G= d F)BGF<23B

u(AX2 + BX + C) = G2(X) − XH2(X)b

�YF>=94 � G(X)CN4�.NCN790L2L?ä2�798

H(X)CJ4 d ;<7/4GB[2�7>B � 4�2�:

G(X) = αX + γ2L798

H(X) = βb

�YF/037uA = α2 � uC = γ2 � 2L798 2αγ − uB = β2

b6�+0L7 d 0 � AC = (u−1αγ)22�798u(2u−1αγ − B) = β2

b��YF/037R�NCNC � CN4!BE?D=n0�}+CABGF S = u−1αγ � I90 d 23=94,0A(2S − B) = (u−1)2(uA)(u(2S − B)) = (u−1αβ)2

b�P;�} 4G=9H9H9;<4,0@BGF<23B_�NCNC � CJ4�BE?[=n0 � BGF<23B�CN4 � BGF/0L?D0P0G �CN4GB S

2�798TCJ7

Z4G= d F+BGF<2�B

AC = S22�798

A(2S − B) = T 2b_��F/0L7O�JCACNC � CJ4!BE?[=n0P}+CABGF f̃(T 2) = 0 � I/0 d 2L=9450

f̃(X2) = (X2 + AB)2 − 4A2S2

= (X2 − 2AS + AB)(X2 + 2AS + AB)

= (X2 − T 2)(X2 + 2AS + AB)b

Ü�CN7/2L.A.Q: � 4G=9H9H9;<4,0MBGF<23B��NCNCAC � CN4+BE?D=n0/b'ã1B�CJ4 d .J0L2L?PBGF<2�BPBGF/0�8<CN4 d ?[CJK�CJ7/2L7>B�;>gf̃(X)

CN460L�<=923.�BE;16A3C � }6F>C d F�CN4�2R45�<=92L?D0OCN7 Z

I90 d 23=94,0f̃(X)

F<2�4�2O?D;/;�BCJ7Zb��F/0L7fBGF/03?D0O0� LCJ4EB

S2�798

TCN7

Z4G= d F�BGF<2�B

AC = S22�798

f̃(T 2) = 0b��037 d 0 �

0 = (T 2 + AB)2 − 4A2S2 = (T 2 − A(2S − B))(T 2 − A(2(−S) − B))b

�YF/03?D03gq;<?D0 � 4GCJ7 d 0 AC = S2 = (−S)2 � }60MKO2�:�2L454G=9KR0~BGF<23B AC = S22L798

A(2S − B) = T 2b�ã1Bägq;>.A.1;�}�4!g1?D;<K�BGF>CN4!BGF<23B

Af(X2) = A2X4 + ABX2 + AC = (AX2 + S)2 − T 2X2

= (AX2 − TX + S)(AX2 + TX + S)

= d2(A′X2 − T ′X + S′)(A′X2 + T ′X + S′) � � � �

� ��¢

}6F/0L?D0d = gcd(A, T, S)

2�798A′ = A/d � T ′ = T/d � S′ = S/d

bj�+0L7 d 0 �A′X2 −T ′X +S′ 2�798 A′X2 +T ′X +S′ 2L?D0PH/?DCNKMCQBDCQ�<0�Hn;>. :/79;<K�CJ23.J4!;>g Z[X]

b ;<7��<03?[4,0�.Q: � 2)}603.A.1��79;�}�7f?D0L4G=/.ABP;>gä¨_2L=94�4+�1450�04 � � H�b �� 0L4EBD2�I<.ACJ45F/0L4�BGF<2�BBGF/0MH/?D;/8<= d BP;>g_H9?[CJK�CABDC �<0�H9;>.Q:<79;<KMCN2L.N4PCN4�23.J45;�2MH/?DCNKMCQBDCQ�<0�H9;>.Q:<79;<KMCN2L.�b6��F/0L7 �4�CN7 d 0

f(X2)CN4@H/?DCNKMCQBDCQ�<0 � g1?D;<K#� � �G� }60��30�B A = d2 � g1?D;<K }6F>C d FM�NC � gq;>.A.1;�}�43b 2

��'#��b��YF/0�H9;>.Q:<79;<KMCN2L.f(X2)

F<2L4+0G L2 d BD. :�gq;>=9?�?D;/;�B[4�CJ7Z� d ;>=97>BDCJ79�~K�=/.1�BDCJH<.NC d CABDCJ0L4 � CNguBGF/03?D0�0G �CN4GB S

2�798TCN7

Z4�= d FMBGF<23B

S2 = AC � f̃(T 2) = 0 � 2L798T 2 − 4AS

CJ4ä2�4,�>=92�?D0�CJ7Z �;�BGF/0L?�}+CJ450 � CABWF<2�4!79;6?D;/;�BD43b ��gn4�Hn0 d CJ23.<CNKOH9;<?[B[2�7 d 0CJ4@BGF/0 d 2L4,0+}6F/037

T 2 − 4ASCN4�79;�B@2~45�<=92L?D06CJ7

Z � 4�CN7 d 0 f(X2)BGF/0L7�F<2�4sB�}6;CJ?D?D0�8>= d C1I>.106�>=92�8<?[23BDC d g12 d BE;<?[4 � I>=<B�79;�?D;/;�B[4Lb�P;�}á}60sH9?D;3�<0 � }+CABGF/;>=<Bu=94GCJ79� d ;<KOH<.10G P7<=9KOI903?[4 � BGF<2�B�2�7�: � �303790L?[23.NCQ��0�8��I<CJ�<=92�8/?[2�BDC d Hn;>. :/79;<K�CJ23.�;>g / [X]

CN4@?D0�8>= d C1I>.10+CN7 / [X]b

p�x:#�x�yQy-�'#o* � �L0�BmI90+2�7�:�Hn;<4GCABDC �<0�CJ7>BD0��303? � 2L798�.10�B

F (X) = AX4m + BX2m + CI90�2�H9;>.Q:<79;<KMCN2L.9CN7 / [X]}+CABGF

A 6= 0b_��F/0L7

F (X)CN4@?D0�8>= d C1I>.10+CN7 / [X]

b¼nX�Æ/Æ�Î[Ôäã1BYCJ4 d .1032�? � �<CN2@BGF/0 d F<2L79�30�;>gL�>2L?DCN2�I>.10 Y = Xm � BGF<23BYCABu4G=.� d 0L4�BD;�H9?D;3�<0BGF/0 d 2L4,0

m = 1b � 0�KM2�:�2L.N4,;R2�4�4�=9KO0

A > 0b��P;�}�4G=9H9H9;<4,06BGF<23B

F (X)CN4CJ?D?D0�8>= d C1I>.10~CN7 / [X]

b�ã1Bsgq;>.A.1;�}�4�g1?D;<K �JCAC � ;>g��YF/0L;<?D0LK � BGF<23BB2 − 4AC < 02�798�BGF<2�B � gq;<?�2L7�:T?D032L.W7<=9KRI/0L? S � }60RF<2��<0 AC < S2

;<?A(2S − B) < 0

b�3CJ7 d 06BGF>CJ4�CN4 d .1032�?[.Q:�0L�<=/C �>2L.J0L7>B�BD;MBGF/0 d ;<7>BE?[2�8<C d BDC1;<7B2 < 4AC

2�798C < 0 �BGF/0�H9?D;/;>g�CN4 d ;<KMH/.J0�BD09b

2

c 4)2�7Ý2�H/H/.AC d 23BDCJ;<7';>g!�YF/0L;<?D0LK � � }60 d ;<7/4GC18/0L?��Ngq;<?�2L7�: a � b ∈ Z � BGF/0Hn;>. :/79;<K�CJ23.Fa,b(X) = X4 − 2(a + b)X2 + (a − b)2

b� 0��9?[4GB�890�BE03?EK�CJ790f790 d 03454�2�?{:Ý2�798�4G=.� d CJ0L7>B d ;<798<CQBDC1;<7/4~;<7

a2L798

b4,;BGF<23B

Fa,b(X)CN4P?D0�8>= d C1I>.10~CJ7

Z[X]�1�30L7903?[2L.ACA�ECN79�6BGF>CJ4s}�2�:fH/?D;/I<.J0LK c@� ;>g�BGF/0��>� -/=<BE7/2�K ;<KMHn0�BDCABDCJ;<7� � � H/H�b �� bãqg

ZF<2�4 d F<2L?[2 d BD0L?[CJ4EBDC d

2 � CQB�CJ4 d .1032�?YBGF<23B Fa,b(X) = (X2 +a−b)2b!��0� 3B4�=9H/Hn;<450+BGF<23BäBGF/0 d F<2L?[2 d BD0L?[CJ4EBDC d ;>g

ZCJ4�8<C��03?D0L7>B!g1?D;<K

2b�Ü�?D;<K#�YF/0L;<?D0LK � � CQBgq;>.N.J;�}�4�2�B�;<7 d 06BGF<2�B

Fa,b(X)CN4�?D0�8>= d C1I>.106CJ7

Z[X]CNg�2L798f;<7<. :OCNg�[email protected];<790;>g�BGF/0603.J0LKO0L7>B[4

4(

(a + b)2 − (a − b)2)

= 16ab � 2(a − b) + 2(a + b) = 4a �2�798 −2(a − b)+2(a+ b) = 4bCJ4ä2�45�<=92L?D0�CJ7

Z �BGF<23B�CJ4 � CNg92L798~;<7<.Q:�CNg92�B�.1032�4EB;<7906;>gnBGF/0�03.J0LKO0L7>B[4

a � b � 2�798 abCJ4@2�4,�>=92�?D0�CJ7

Zb� 0�79;�BD0�BGF/0�gq;>.N.J;�}+CJ79� d ;<7/4,0L�<=n037 d 034 �

� b��F/0~H9;>.Q:<79;<KMCN2L.Fa,b(X)

CJ4�?D0L8<= d CJI<.J0)CJ7��p[X]

gq;<?s2L7�:�H/?DCNKR0p � I/0 d 2L=9450 �IL:�BGF/0M.N2�}#;>g!�<=92�8/?[2�BDC d ?D0 d CNH9?D; d CQB�: � BGF/0MH9?D;/8>= d B+;>gä79;<79�G4,�>=92�?D034+CN7�� p

CN423.Q}�2�:/4+2R4,�>=92�?D0MCJ7��pb��N�YF>=94 � CAB�CJ4�0L2L4[:�BD;��9798�CN?E?D0L8<= d CJI<.J0RH9;>.Q:<79;<KMCN2L.N4;>g��

[X]}6F>C d F�2�?D0O?D0�8>= d C1I>.10MCJ7��

p[X]gq;<?�2L7�:TH/?DCNKR0MCJ7>BD0��303?

p �BGF/0 d 2L4,0

a = 5 � b = 41CN4 d ;<7/4GC18/0L?D0L8�CJ7� £ � H/H�b ��>��£ � ��L�� b

� � �

� bX4 − 2(Y + W )X2 + (Y − W )2

CJ4�CN?E?D0L8<= d CJI<.J0+CJ7��[X, Y, W ]

b� b)�L0�B

α2�798

β8/0L79;�BD0s2L?GI>CABE?[2L?{:+45�<=92L?D0s?D;/;�B[4_;>g

a2L798

b � ?D0L4�Hn0 d BDCQ�<0�.Q: � CN7+4,;<KO00G LBD0L7/4GC1;<7Ý��0�.18';>g!BGF/0f�n03.J8';>g��>=n;�BDC1037>BD4�;>gZ � 452�: QZ

�J4,0L08 � � H9H�b ��L� ��£�£�� bR�3CJ7 d 0 Fa,b(α + β) = 0 � }60OF<2��<0M2L.N4,;��C �<0L7�2�7�03.J0LKO0L7>B[2�?{:�H/?D;/;>gBGF<2�BFa,b(X)

CJ4sBGF/0�K�CJ7<CNKO23.�H9;>.Q:<79;<KMCN2L.�;>gα + β

;3�<0L?QZ

CAg_2L798�;<7<.Q:RCAg79;<790�;>g�BGF/0�0�.103KR037>BD4a � b � ;<? ab

CN4P2�4,�>=92�?D0)CJ7Z�A}6F>C d F d ;<7/4EBDCABD=<BD0L4 � gq;<?4�Hn0 d CN� d �>2L.A=n0L4+;>g

a � b ∈ � � 2)�<0L?{: d ;<KMKR;<7�0� �03? d CJ450�CN7�2L7�:�CN7>BE?D;/8<= d BD;<?{:d ;>=9?[4,0�;<7O0G LBD0L7/4GC1;<7��n03.J8/4 � b� CABGF)KR;<?D0+�30L7903?[2L.ACAB{: � IL:)B[2��LCJ79� A = 1}60+�30�BWg1?D;<K��F/0�;<?D03K � BGF/0�gq;>.J�.1;�}+CN79� d F<2L?[2 d BD0L?[CA�G2�BDC1;<7�;>g9BGF/0+03.J0LKO0L7>B[4@;>g�45;<KR0�23.1�30LI/?[23C d�d .J;<4�=9?D0+;>g

QZ�1450�0

£ � H9H�b ��¢�� ¢� �>�� }6F/;<4,0)KMCN7<CJKM2L.uHn;>. :/79;<K�CJ23.�;L�<03? QZCN4P2�I<CJ�<=92�8/?[2�BDC dHn;>. :/79;<K�CJ23.�;>g

Z[X]b

p�x:#�x�yQy-�'#o* £ �3=9H9H9;<4,06BGF<23BZ8/;/0L4P79;�B�F<2��<0 d F<2L?[2 d BD0L?[CJ4EBDC d

2bM�L0�B

QZI/0�2L7�:2L.J�30�I<?[2LC dPd .1;<4G=9?D0�;>g

QZ � 2�798R.10�B ξI/0�2L7�:f03.J0LKO0L7>B�;>g

QZb��L0�B

F (X)I/0)BGF/0KMCN7<CJKM2L.nHn;>. :/79;<K�CJ23.�;>g

ξ;3�<0L?

QZb��F/0+gq;>.A.1;�}+CN79��4EBD2�BE03KR037>BD4@2�?D0�0��>=/CQ�>23.1037>B �

�JC � F (X) = X4 + BX2 + CCJ4@26I<CJ�<=92�8/?[2�BDC d Hn;>. :/79;<K�CJ23.�;>g

Z[X]b

�NCNC � � 0 d 2�7)}�?DCQBE0 ξCJ7M45;<KR0�;>gnBGF/0�gq;>.N.J;�}+CJ79��gq;>=9?_}�2�:/4 �

±√

B ±√

B2 − 4C

2

b}6F/03?D0

B2�798

C2L?D0+03.J0LKO0L7>B[4@;>g

Z4G= d F6BGF<23B

B2 − 4CCN4�79;�BW2�45�<=92L?D0�CN7

Z � 2�798 B ± 2√

C2�?D0+79;�B!4,�>=92�?D034@CN7

ZCNg

CCN4@2645�<=92L?D0+CN7

Zb

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b2 + c2≤ 4R + r

4R

bÞ �Pæ@½ b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø�¿�Æ/Ë �6Æ/ÈsÓ�XGÕ�]8¼nÆ�X,ÌJÓ3\{Ë�É�]�¾qË�]�� = UPb��ã{7/4�H/CN?D0�8OIL:�-9?D;/I<.J0LK �� bq¥ � ��2�BGFubn¨_23��0�B[BE0 �� £ �� " : � � H�b �� b ��L0�B

x ∈(

0, π2

) bW��F/;�}�BGF<2�B ��12 � [2 + cos x

3

]

[

2(1 − cos x)

x2

]

>1 + cos x

2�

�1I � 2 + cos x

3<

√

1 + cos x

2<

2(1 − cos x)

x2

bÞ �Pæ å b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø ��Z�ÌDÇ§Á>ÊLÆ�Ï/Ó�X¡Ç3]8Æ_Ç>×DÍ9ÓLX�Ë¡É3Ì1]�Å_Æ�ÒOÓLÏ<ZJÓ9b

�L0�BABCD

I90�2�H/2�?[23.N.J03.J;/��?[2�K�bW¦s4GCJ79��;<7<.Q:~2L7�=97/KM2�?E��0�8�4GBE?[23C1�3F�BD0�8/�30 ��n798M2�Hn;>CN7>BM;<7

AB4�= d F)BGF<2�B

AM = 1

5AB

bÞ �PæPæ b�¼nX�Æ3V�Æ�É�Ë�Õ Ñ�Ø ËÝZ�0>Í/Ó<Ë,\�Ä�Æ9Ì�×DÍ�Ë�Ì{Æ<Ê�] ËâË�ÒfÆ�X[ZJÓ3\ ��Ï>Z�ÊLË�X[ÉGZ�Ì1Ø�Æ�Î¾�Ë�È@Î�ÆnÇ�Ï�Õ�\1ÓLÏYÕ�] = ÌDÔ�¿�Æ�Í/Ï�É É�]�¾ �/b

;<7/4GC18/0L?679;<79�GCN7>BE03?[4,0 d BDCJ79�f2L798 79;<79� d ;<79��?[=n0L7>B d CN? d .1034Γ1

2L798Γ2

}+CQBGFd 037>BE?D0L4O1

2�798O2 � ?D0345H90 d BDC �<03. :�b��L0�B Q

I90~BGF/0OHn;>CN7>B�;>gWCJ7>BD0L?[450 d BDCJ;<7Ý;>g_BGF/0B�}6; d ;<KMKR;<7�BD2L79�30L7>B[4 � t12�798

t2 � }6F>C d FÝ89;T79;�B+CN7>BE03?[4,0 d BPBGF/0R.NCN790�4,0L��KR037>BO1O2

b c�d ;<KMKR;<7�BD2L79�30L7>B � tc � }6F>C d F6CJ7>BD0L?[450 d B[4WBGF/0P4,0L��KR037>B O1O2

KO0�0�BD4WBGF/0BD2L79�30L7>B[4t12�798

t22�B

E1

2�798E2 � ?D0L4�Hn0 d BDCQ�<0�.Q:ub�L0�B

PI/0~BGF/0MKMCJ89�GHn;>CN7>B�;>g�BGF/0�.NCN790M4,0L��KR037>B

O1O2

b�-9?D;3�<0~BGF<23BP � Q �

E1 � 2�798 E2

2L?D0 d ;<7 d : d .AC d b

� � �Þ �Pæ � b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø �<Ó3É�ZA\�Ë¡Â�� X,Ì�Æ�Ó��Ë¡]��Ï>Z�ÊLË�X[ÉGZ�Ì1Ø�Æ�Î�¼/\�ÆLZ{Ë�É�ÌJZJ]�Å�Æ�ÒOÓ�Ï>Z1Ó/b

�L0�Ba1 � a2 � . . . � an

I/0�H9;<4�CQBDCQ�<0+?D032L.n7<=9KOI903?[4 � 2L798�.J0�B

En =

n∑

i=1

(n−1∑

j=0

aji

)−1 b

ãqgr = n

√a1a2 · · · an ≥ 1 � H9?D;3�<0�BGF<23B En ≥ n

(n−1∑

j=0

rj

)−1 gq;<? ��12 � n = 2 � �qI � n = 3 � � d � ? n ≥ 4

bÞ �Pæ�� b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø �<Ó3É�ZA\�Ë¡Â�� X,Ì�Æ�Ó��Ë¡]��Ï>Z�ÊLË�X[ÉGZ�Ì1Ø�Æ�Î�¼/\�ÆLZ{Ë�É�ÌJZJ]�Å�Æ�ÒOÓ�Ï>Z1Ó/b

�L0�Ba1 � a2 � . . . � an

I/0�H9;<4�CQBDCQ�<0+?D032L.n7<=9KOI903?[4 � 2L798�.J0�BEn =

1 + a1a2

1 + a1

+1 + a2a3

1 + a2

+ · · · +1 + ana1

1 + an

b�L0�B

r = n√

a1a2 · · · an ≥ 1 � b�12 � -9?D;3�<0�BGF<2�B En ≥ n(1 + r2)

1 + r

gq;<?n = 3

2L798n = 4

b�1I � ? -9?D;3�<06;<?!8>CJ4�H9?D;3�<0�BGF<23B

En ≥ n(1 + r2)

1 + r

gq;<?n = 5

bÞ �Pæ�� b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø �<Ó3É�ZA\�Ë¡Â�� X,Ì�Æ�Ó��Ë¡]��Ï>Z�ÊLË�X[ÉGZ�Ì1Ø�Æ�Î�¼/\�ÆLZ{Ë�É�ÌJZJ]�Å�Æ�ÒOÓ�Ï>Z1Ó/b

�L0�Ba � b � c � d � 2L798 r

I/0sH9;<4�CQBDCQ�<0@?D0L23.�7<=9KOI903?[4_4�= d F�BGF<23Br =

4√

abcd ≥ 1b-9?D;L�<0PBGF<23B

1

(1 + a)2+

1

(1 + b)2+

1

(1 + c)2+

1

(1 + d)2≥ 4

(1 + r)2b

Þ � � � b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø ��Z�ÌDÇ§Á>ÊLÆ�Ï/Ó�X¡Ç3]8Æ_Ç>×DÍ9ÓLX�Ë¡É3Ì1]�Å_Æ�ÒOÓLÏ<ZJÓ9bãqg

m2L798

n2�?D0~H9;<4�CQBDCQ�<06CN7>BE0L�30L?[4P4G= d FOBGF<2�B

m ≥ n � 2L798RCNg a � b � c > 0 �H9?D;3�<0�BGF<23Bam

bm + cm+

bm

cm + am+

cm

am + bm≥ an

bn + cn+

bn

cn + an+

cn

an + bn

bÞ � � Û b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø�ËÝZ�×DÍ�Ë¡\�Æ�Ó/ÌJÓ3ZA\N\{Ë�]8Å�ÆnÇ<Ë�Ï/] ��XDÓLÏY×EË>b

Ü<;<?a � b � c ∈ (0, 1) � �9798�BGF/0�.1032�4EB!=9H/Hn03?sI/;>=9798R2L798�BGF/0)��?D0L2�BE034GB!.1;�}603?I9;>=9798O;>ga + b + c + abc � 4G=nI�;{0 d BWBE;)BGF/0 d ;<7/4EBE?[2LCN7>B ab + bc + ca = 1

b

� � �Þ � � Þ b ¼nXGÆ3V�Æ�É�ËLÕRÑ�Ø �<Ó3É�ZA\�Ë¡Â�� X,Ì�Æ�Ó��Ë¡]��Ï>Z�ÊLË�X[ÉGZ�Ì1Ø�Æ�Î�¼/\�ÆLZ{Ë�É�ÌJZJ]�Å�Æ�ÒOÓ�Ï>Z1Ó/b�12 � -9?D;3�<0�BGF<2�BäCNg 0 ≤ λ ≤ 4 � BGF/0L7 � gq;<?ä2L.A.nHn;<4GCABDC �<0+?D0L23.�7<=9KOI903?[4 x � y � z � t �

(t2 + 1)(x3 + y3 + z3) + 3(1 − t2)xyz

≥ (1 + λt)(x2y + y2z + z2x) + (1 − λt)(xy2 + yz2 + zx2)b

�1I � Ü>;<? t = 1

4

2L798λ = 4 � BGF/0�2�I/;L�<0�CJ790L�<=923.NCQB�:MI90 d ;<KO0L4

17(x3 + y3 + z3) + 45xyz ≥ 32(x2y + y2z + z2x)b

Ü�CJ798�2L.A.nHn;<4GCABDC �<0��>2L.A=n0L4s;>gδ4�= d F)BGF<2�BWBGF/0+CN790��>=92L.ACAB{:

x3 + y3 + z3 + 3δxyz ≥ (1 + δ)(x2y + y2z + z2x)F/;>.J8/4�gq;<?!23.N.x � y � z }6F>C d F�2L?D0 � �NC � H9;<4�CQBDCQ�<0+?D032L.n7<=9KRI/0L?[4 � �JCAC � 4GC18/0+.J0L79�¡BGF<4;>g�2+BE?[CJ2L79��.10/b

Þ � � à bM¼nXGÆ3V�Æ�É�ËLÕ Ñ�Ø ¿LÇ�ÓLÏ ��Æ�Æ�É�×EÆhÅ_Æ�ÒfË�X�Æ Ë ½Ó�X%Ê�Ç<Ë �G] �uÏ<Z{ÊLË�X�É�Z{Õ>Ó<Õ Õ9Ë�<Ó3\A\1Ó>Õ9ÆL\NZ{Õ�] �<Ó�\N\JÓ<Õ9ÆL\AZ�Õ�] = VYÓ�Z1Ï

(Õ9Ë�Õ�Z{×[Ó<Ì�ËLÕ�Ì{Æ �3Æ�É,Í>Z�Æ = Ë¡ZJÒ�ZqØ3Ó

)b

�L0�BABC

I90�2�79;<79�ECJ45;<4 d 03.J0L4~?[C1�3F�B+BE?DCN2�79��.J0}+CABGFf?[C1�3F�B�2�79��.J0M2�BA2�798

AC > ABbf�L0�B

DI/0BGF/0Rgq;/;�B�;>gWBGF/0�23.ABDCQBD=n890Og1?D;<K

ABD;�BGF/0�4GC18/0

BCb�L0�B

GI90�BGF/06H9;>CJ7>B@;>gYCN7>BE03?[4,0 d BDC1;<7O;>g�BGF/0�.ACJ790

AD�q0G LBD0L798/0�8 � }+CQBGFOBGF/0~.ACJ7906BGF<?D;>=n�3F C}6F>C d FRCJ4PH/2�?E�2L.A.10�.<BE;

ABb��L0�B

EI/0PBGF/0�Hn;>CN7>Bä4�= d F6BGF<2�B

ACGECJ4~2�?D0 d B[2�79��.J0 � 2�798�.J0�B FI/0�BGF/0�H9;>CJ7>B64G= d FÝBGF<23B

BFGECN4P2�?D0 d BD2L79��.10/bM�L0�B

HI/0)BGF/0)CN7>BE03?[4,0 d BDC1;<7;>g

AG2�798

BFbs�L0�B

O1

I90PBGF/0+CJ7>BD0L?[450 d BDCJ;<7O;>gnBGF/08<CN2��3;<7/23.J4ä;>g<BGF/0P�>=92�8<?DCA.J2�BE03?[2L.CDHF � 2L798�.10�B O2I90�BGF/0�CJ7>BD0L?[450 d BDCJ;<7~;>g<BGF/0�8>CJ2��3;<7/2L.N4�;>g<BGF/0P�<=92�8/?[C1�.J2�BE03?[2L.

BDGEb .............................................................................................................................................................................................................................................

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

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

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

A

B

C

D

E

F

G

H

O1

O2

-9?D;L�<0PBGF<23BWBGF/0PBE?DCN2�79��.J0L4ABC � DFE � 2�798 DO1O2

2L?D0�4GCJK�CN.N2�?¡bÞ � � ½ bM¼nXGÆ3V�Æ�É�ËLÕ Ñ�Ø ¿LÇ�ÓLÏ ��Æ�Æ�É�×EÆhÅ_Æ�ÒfË�X�Æ Ë ½Ó�X%Ê�Ç<Ë �G] �uÏ<Z{ÊLË�X�É�Z{Õ>Ó<Õ Õ9Ë�<Ó3\A\1Ó>Õ9ÆL\NZ{Õ�] �<Ó�\N\JÓ<Õ9ÆL\AZ�Õ�] = VYÓ�Z1Ïub

�L0�BPI/0+2�7�:~Hn;>CN7>B!;<7)BGF/0�.NCN790

BCCJ7 4ABC

b��L0�BA1

I90PBGF/0PCJ7>BD0L?[450 d �BDC1;<7�;>gAP

�JHn;<4�4�CJI<. :�0� 3BE0379890L8 � }+CQBGF�BGF/0).ACJ790�BGF<?D;>=n�3F B}6F>C d FOCN4�H92L?[2L.A.10�.�BD;

AC � 2�798M.10�B A2

I/0�BGF/0�CJ7>BD0L?[450 d BDCJ;<7R;>gAP

�JHn;<4�4�CJI<. :R0� 3BE0379890L8 � }+CQBGF�BGF/0�.ACJ790BGF<?D;>=n�3FC}6F>C d F~CJ4@H/2�?[23.N.J03./BE;

ABb-9?D;L�<0�BGF<2�B�BGF/0O2L?D0L2�;>g 4ABCCN4�BGF/0R�30L;<KR0�BE?DC d KO0L2L7Ý;>g_BGF/0O2�?D032�4�;>g

4A1BC2L798 4A2BC

bÞ � �På bM¼nXGÆ3V�Æ�É�ËLÕ Ñ�Ø ¿LÇ�ÓLÏ ��Æ�Æ�É�×EÆhÅ_Æ�ÒfË�X�Æ Ë ½Ó�X%Ê�Ç<Ë �G] �uÏ<Z{ÊLË�X�É�Z{Õ>Ó<Õ Õ9Ë�<Ó3\A\1Ó>Õ9ÆL\NZ{Õ�] �<Ó�\N\JÓ<Õ9ÆL\AZ�Õ�] = VYÓ�Z1Ïub

¨_C �<0L7�2L7�CJ7/4 d ?DCJI90L8 d ;<7��<0G f�>=92�8<?DCA.J2�BE03?[2L.u}+CABGF�4GC18/0L4�;>g_.10379�¡BGFm � n � p �

q � BD2��0L7~CJ7M;<?D8903?!2L?D;>=9798~BGF/0��>=92�8<?DCA.J2�BE03?[2L. � 2�798M8>CJ2��3;<7/2L.N4s;>gn.10379�¡BGF d2�798

d′ �H9?D;3�<0�BGF<23B √mp + nq ≤ 1

2(d + d′)

b

� � ��

¾�Æ�VYXGÆ9ÑL\�Ë�Ò·ZJÉ�Ë3ÊLË�X+V�Ë�XEÒOÓLÏYË�ÏuÌJ\1Ø'×�\�Æ�É�Ë�Õ�Ô �LÍ�ËTËLÕ�Z�Ì{Æ�X�ZNÉMÓ3\{ÈsÓ�Ø3É~Vn\�Ë�ÓLÉ�Ë�ÕâÌ{Æ×GÆ�Ï<É�Z{Õ9Ë�XäÎ�Æ�X�VWÇ/ÑL\AZ�×�Ó/ÌJZ{Æ�ÏOÏ�Ë�ÈâÉ�ÆL\�Ç/ÌJZ{Æ�Ï/ÉPÆ�X!Ï�Ë�ÈáZJÏ/ÉGZJÐLÍuÌNÉ�Æ�Ï)VYÓ3É3ÌnVYX�Æ9ÑL\{Ë�ÒMÉ3Ô� 0�F<2��<0�8>CJ4 d ;3�<0L?D0L8RBGF<2�B�H/?D;/I<.J0LK �" �� CN4�CJ89037>BDC d 2L.YBD;RH9?D;/I>.103K � � �� }6F/;<4504,;>.A=<BDC1;<7MF<[email protected]?D0L2�8�:MI/0�037MH/?DCN7>BE0L8� ��>� �>��¢ �� b ;<7/450��>=n0L7>BD. : � H9?D;/I>.103K�� CJ4 d .J;<4,0L8 � 2L798~79;)45;>.N=<BDCJ;<7/4ägq;<?_CQB�}+CA.N.9I/0�2 dGd 0LH>BE0L8�b � 0P2�H9;>.1;/��CQ��0�gq;<?�BGF>CJ4;L�<03?[4�CJ�3F�B�bÞ �Pæ@½ b� ��>� �<�� ¼nX�Æ3V�Æ�É�Ë�ÕRÑ�Øu¼<Ó�Ï�Æ�É��nÔ �¡É5Ó<ÆnÇ3É5É�Æ�Ð3\{ÆnÇL]/U�ÌqÍYË�Ï/É�]��uX�ËLË�×EË>b

ãqga � b � c 2L?D0�BGF/0+4�CJ89034s;>gY2�7M2 d =<BE0+2�79��.J0�8~BE?[CJ2L79��.10 � H/?D;L�<0�BGF<2�B

∑

� ��

√

a2 + b2 − c2√

a2 − b2 + c2 ≤ ab + bc + cab

oGb = ÆL\[Ç<ÌJZ�Æ�ÏfÑ�ØuÂ/Í9X[ZNÉ3Ì{Æ3VYÍYË�X3¿LÔRÆ�XDÓ>Õ�\�Ë�Ø3]8Æ�X[ZNÉ3Ì{ÆL\J]��8Ääb�L0�B

x2 = b2 + c2 − a2 � y2 = c2 + a2 − b2 � 2�798 z2 = a2 + b2 − c2 �}6F/0L?D0x � y � z > 0

b¯��F/0L7a2 = 1

2(y2 + z2) � b2 = 1

2(z2 + x2) � 2L798

c2 = 1

2(x2 + y2)

b��YF/0��CQ�<037MCN790��>=92L.ACAB{:OI/0 d ;<KR0342(xy + yz + zx) ≤

∑

� ��

√

y2 + z2√

z2 + x2 �;<? ∑

� �� (yz + zx) ≤

∑

� ��

√

y2 + z2√

z2 + x2 � � � �}6F>C d F~CJ4!BE?[=n0 � 4GCJ7 d 0 (y2 + z2)(z2 + x2) − (yz + zx)2 = (z2 − xy)2 ≥ 0

bß/�>=92L.ACAB{:�F/;>.18<4�CN7T� � � CNgW2�798�;<7<. :fCNg z2 = xy � x2 = yz � 2L798 y2 = zxb�YF/034,0 d ;<798>CABDCJ;<7/4s2�?D060L�<=/C �>2L.J0L7>BäBD;

x = y = zbs�+0L7 d 0 � 0L�<=923.NCQB�:OF/;>.18<4@CJ7�BGF/0H9?D;<H9;<4,0L8+CN790��>=92L.ACAB{:+CAg<2L7986;<7<. :�CNg

x = y = z �BGF<23BYCJ4 � CAg<2L7986;<7<. :�CNg a = b = c

bo�oEb = ÆL\�Ç/ÌJZ{Æ�ÏfÑ�ØRÅnÇ3Î�Ë,Z0ÁLÍ/Ó<Æ�]�É�ÌDÇ<Õ9Ë�ÏuÌ1] ��Æ�Ï¢Ë�ZA\N\NÉ�Â�ÆL\N\{Ë�Ð�Z1Ó<Ì�Ë¡ÇqÏ/É�ÌJZ�ÌDÇ<Ì�Ë¡] �LÆ�XGÆ�ÏuÌ{Æ�]�N¾)b

©<: 2L= d F�:�� 4Pã{790L�<=923.NCQB�: � gq;<?ä2L7�: x � y � z � w ≥ 0 � }606F<2��<0√

xy +√

zw ≤√

(x + z)(y + w)b

��037 d 0 �∑

� ��

√

a2 + b2 − c2√

a2 − b2 + c2

=1

2

∑

� ��

(

√

a2 + b2 − c2√

a2 − b2 + c2

+√

c2 + a2 − b2√

c2 − a2 + b2

)

≤ 1

2

∑

� ��

√

(2a2)(2c2) =∑

� ��

acb

� � £

o�o�oGb = ÆL\[Ç<ÌJZ�Æ�ÏfÑ�Ø ��ZEÁ3ÍYÆnÇ3]�¼nÆL\�0¢Â�Æ�ÒOÒ�ÇLÏ<Z�ÌqØuÂ�ÆL\N\{Ë�Ð>Ë�] ��Z1ÏYÌ�Ë�X ��Ó<ÊLË�Ï/] � �3]�� = UPb©<:~BGF/0 d ;<7��<0G �CQB�:M;>g √

x � }60�F<2��<01

2

(

√

a2 + b2 − c2 +√

a2 − b2 + c2)

≤

√

(a2 + b2 − c2) + (a2 − b2 + c2)

2= a

b��037 d 0 � ∑

� ��

√a2 + b2 − c2 ≤ a+b+c � }6F>C d F+I90 d ;<KO0L4uBGF/0�8/0L4GCJ?D0L8�CJ790L�<=923.NCQB�:=9Hn;<7M4,�>=92�?[CJ79���<� ' ' ��- � ��*� �#�#+�� ' ��E � � � ><2��2�� A:9;��!<��#A � ' �� E ��E ��9�� A�� ! � � �<��!#�� #� '#�"��E�� $� �<�*A�� ?� � �#A�� ;A � �*2 � ! "�� 9#�� AC�� !<-�� A$��,��%� �$��#� � '�� F�#�,+ �� ,� 7 A ��' � '�B6'&�*�<� A�� A � �*2 �'� � �: �$ 2(9#� A��=� ' �,��A �(� A � �*2 �$ 2 �� 9)��2 � � � A �� ! � � �� 7�B6�<'��,!�B�A%�$�<��58�*!#� @�A12 ! ��$� � ' �*� ��9�9?2!� �?. ��2�� #� ��A�F�#�,+ �� ,� 7:��2��,5��=� ' A � -8BC��,��<A�2(> � � � �,+ $�2��2 ��A��4�� A�E �#� �<' �# � ��2 �:.-� ��9��A� �#+�� A.� 23A � ��2 � � E��82�9#� �� E �/��A )�"�"� + + �� + �!#�� ,�"@�� <�$+�� $�$7 A0� ' �#-��5?!�� A ;��B6' �?7 �><2�>1�� -� � 9 � ��A4E ��' � @��,-8' A �� 2� � � � 93�;.4$6 ) ;A >?�$� � ' �F�#�,+ �� ,� 7 A � '.� � � '#-8' A:E�2�A � �*2 ��5�6� ��75 � ��)2(9��A;>;!#��<'�� A;9;��B6' �<� ' ��'��-6�� .

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a2 + b2 + c2�"� 5��<� �

��,-��A�� % �4�<5�� ' � �∑

� ��

√

a2 + b2 − c2 ≤ a + b + c�

� �% � ��#% �,��K

'��-s-��<�"� �� '��' '��-6�� B�� BC��"�� ;� ��4� � � ' � � �,�*A��C!��

a2 + b2 − c2 = 2ab cos C = 4(

12

ab sin C)

cot C = 4K cot C

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cot A +√

cot B +√

cot C ≤ s

K

�

�*' ��! � �<5�� ' � ��-6� ��"� � �$�#%)� � � � ��:9?! ' � �,� 7 % �<�$�� )�� #��(��<'��&�� -6��:=∑

� �� a(b + c) cos A ≤ ab + bc + ca

�

�F�� !#58�� $��!#� �$�� B�� '��3�,� � �� <� �,!<� �,��/�?'#5��#+��*A��"� � �$�#% ��- 587 E<'<!<� �?7 � ��$��:9?!�' � �$�$7'��-4��#% ��1� ��'��>� ��:9?! ' � �,� 7 A�� ' B6@8� �:�� '#5�� - ��3BC�� <�� ' �� ! �$�� '#�5�"��F' � �

n ≥ 2A

∑

� ��

n√

an + bn − cn n√

an − bn + cn ≤ ab + bc + ca,

��#+��$-��-6� ��9?! '�� $� �$�� ! ��-��(� �� '<-��$�"' � �)' �� ' � �?�� + �� '<� �,+ �#.

� � ¢Þ �Pæ å b ��>��3�" � � ¼nXGÆ3V�Æ�É�ËLÕ)Ñ�Ø �_ËLÆ�X[Ð>Ë�Æ�Ó�\�Æ�Ð�\�ÆnÇ3] = ��¾~Å �_É�ÈPË¡Ð<Æ�] �_É�ÈPË¡Ð<Æ�]¾~Å/]�� = UPb�3=9H9H9;<4,0sBGF<23B

D � E � F 2�?D0sBGF/0�H9;>CJ7>B[4ä2�Bu}6F>C d F+BGF/0 d ;<7 d =9?E?D037>B_.ACJ79034AD �

BE � CFKR0L0�B+BGF/0�4�CJ89034M;>gs2T��CQ�<037�BE?DCN2�79��.J0

ABCb �L0�B

p1

2L798p2

I/0fBGF/0Hn03?DCNKR0�BE03?[4M2L798δ1

2L798δ2

BGF/0�2L?D0L2L4M;>g 4ABC2�798 4DEF � ?D0L4�Hn0 d BDCQ�<0�.Q:ub-9?D;L�<0PBGF<23B

�J2 � 2p2 ≤ p1

CNgAD � BE � 2L798 CF

2�?D0+2�79��.J06I>CJ450 d BD;<?[4��qI � 2p2 ≤ p1

CNgAD � BE � 2L798 CF

2�?D0+2L.QBDCABD=n8/0L4�� d � 3p2 ≤ 2p1

gq;<?ä2L.A.D � E � F CAgY2�798M;<7<.Q:~CNg 4ABC

CJ4s0L�<=/CA.J2�BE03?[2L.��q8 � 4δ2 ≤ δ1

gq;<?!23.N.D � E � F 2�798�2�?EI<CQBE?[2�?{: 4ABC

b= ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø ��Z0ÁLÍ�ÆnÇL]�¼nÆL\ 0¢Â�Æ�ÒOÒ�Ç�Ï>Z�Ì1Ø2Â�ÆL\A\�Ë¡Ð<Ë¡] ��ZJÏuÌ{Ë�X ��Ó>ÊLË�Ï<] � �3]�� = UPb

�12 � �YF>CN4�CJ4 ¤�¦ � H9?D;/I>.103K 2502 �� 3£9¢

��>� �>¢��>¢�£�� b

�qI � ��F/0M}6;<?D8>CJ79�O;>gäBGF/0RH/?D;/I<.J0LK 4G=n�3�30L4EBD4+BGF<2�B 4ABCCN462 d =<BE0/bT�J��F/0?D0L4G=/.ABM89;/034R79;�BMF/;>.18 gq;<?�79;<79��2 d =<BE0�BE?[CJ2L79��.1034Lb � ��F>CJ4OH9?D;/I>.103K CN4R��79;�}�7j2L4Ü>2��7/2�79;�� 4�-9?D;/I>.103K � �� ;>gY2L.A.9BGF/0�BE?DCN2�79��.J0L4@CN7 d 4 d ?DCJI90L8MCJ7 4ABC � BGF/06;<?[BGF>C dBE?DCN2�79��.J0s2�B[BD23CJ7/4uBGF/0sK�CJ7<CNKM=9K Hn03?DCNKR0�BE03?�bWã{7�H/2�?�BDC d =/.N2�? � BGF/0�;<?[BGF>C d BE?[CJ2L79��.10sF<2�4Hn03?DCNKR0�BE03?!79;M��?D0L2�BE03?WBGF<2L7 1

2p1

b� d � ãqg 4ABC

CJ4+79;�BP0��>=/CN.N23BD0L?[23. � 2�798 BCCN4PBGF/0M.J;<79�30L4EBP4GC18/0 � BGF/0L7�}60d 2L7 d F/;/;<4,0

DBE;fI90)BGF/0�KMCJ89�GHn;>CN7>B�;>g

BCbO�L0�B

E → C2�798

F → BBD;��30�B

p2 → 2BC > 2

3p1

b ;<7��<03?[4,0�.Q: � 4G=9H9H9;<4,0 4ABC

CN4ä0L�<=/CA.J2�BE03?[2L.3}+CABGFAB = 1

b��L0�Bu = AF �

v = BD � 2�798 w = CEb��F/0L7

0 < u � v � w < 12L798

p2 = f(u, v, w) =∑

� ��

√

u2 + (1 − w)2 − u(1 − w)b

�YF/037∂f

∂u=

2u − (1 − w)

2√

u2 + (1 − w)2 − u(1 − w)+

v − 2(1 − u)

2√

v2 + (1 − u)2 − v(1 − u)

b��0�B[BDCN79� ∂f

∂u= 0 � }606�30�B

[2u − (1 − w)]2 · [v2 + (1 − u)2 − v(1 − u)]

= [v − 2(1 − u)]2 · [u2 + (1 − w)2 − u(1 − w)] �}6F>C d F�?D0�8>= d 034�BD;u(u + w − 1)v2 = (1 − u)(1 − u − v)(1 − w)2

b � � ��YF>CN4 d 2�7�gJ=9?[BGF/03?�I90+KO2L7<CJH<=/.J2�BE0L8�CJ7>BD;uv(1 − u)(1 − w)(1 − v − w) = 0 � � � �

� � �

IL:�=94�CN79� 0,�>2<� 4s?D0�.J2�BDC1;<7uvw = (1 − u)(1 − v)(1 − w)

b � � ��/Õ�b_��0�0@BGF/0�79;�BE0sgq;>.N.J;�}+CJ79�!BGF/0�.NCN4GB�;>g/45;>.Q�<03?[4!I/03.J;�}Ob � �+0L7 d 0 � 1−v −w = 0 �BGF<23B)CN4 � v = 1 − w

2�798w = 1 − v

b �YF>=94 � u = 1

2

b ©<: 4[:/KMKR0�BE?{: � }60d ;<7 d .A=n890 (1

2, 1

2, 1

2

) CJ4sBGF/0�;<7<. : d ?DCQBDC d 2L.uHn;>CN7>B�;>gfb6�P;�BD06BGF<2�B

f(

1

2, 1

2, 1

2

)

= 3

2

bc .J45; � 2�4PBGF/0 d ;<7 d =9?D?D0L7>B�H9;>CJ7>BP2�H/H9?D;<2 d F/0L4�2O4�CJ890O;>g 4ABC � BGF/0MgJ=97 d BDCJ;<7 fd .1032�?[.Q:T2�H/H9?D;<2 d F/0L4�CAB[4)KM2� LCJK�=9K �>23.N=n0�;>g2b ��037 d 0 � 1

2p1 ≤ p2 ≤ 2

3p1

gq;<?79;<79�58/0��303790L?[2�BE0 4DEFb

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uv2(1 − u − w) + (1 − u)(1 − u − v)(1 − w)2 = 0,

uv2(1 − u − w + uw) − u2v2w

+(1 − u)(1 − u − v + uv)(1 − w)2 − uv(1 − u)(1 − w)2 = 0,

uv2(1 − u)(1 − w) − u2v2w

+(1 − u)2(1 − v)(1 − w)2 − uv(1 − u)(1 − w)2 = 0,

uv2(1 − u)(1 − w) − uv(1 − u)(1 − v)(1 − w)

+ uvw(1 − u)(1 − w) − uv(1 − u)(1 − w)2 = 0,

uv(1 − u)(1 − w)[

v − (1 − v) + w − (1 − w)]

= 0,

2uv(1 − u)(1 − w)(v + w − 1) = 0�

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� ��

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�3CJ7 d 0MA2 = MB2 � }60�4,0L0sBGF<2�B M

CN4!;<7+BGF/0�?[2�8>C d 2L.<2¡ �CN4ä;>gΓ1

2�798Γ2

bã1B�gq;>.A.1;�}�4�BGF<2�BMP · MC = MQ · MD � 4,;fBGF<2�B CDQP

CJ4 d : d .AC d�F/0L7 d 0 �

∠MPQ = ∠CDQb@�L0�B

O1

2L798O2

890379;�BE0PBGF/0 d 037>BE?D0L4s;>gΓ1

2�798Γ2 � ?D0345H90 d �BDCQ�<0�.Q:ub��YF/037�BGF<?D;>=n�3FM45;<KR0+2L79��.10 d F<2�4GCJ79� � }60�F<2��<0

∠APQ + ∠QBA = ∠APM + ∠MPQ + ∠QBA

= 180◦ − ∠CPA + ∠CDQ + ∠QBA

= 1

2∠AO1C + ∠DBQ + ∠QBA

= ∠AO1O2 + ∠DBA

= ∠AO1O2 + ∠O1O2B = 180◦ �2�798~BGF/0+?D0L4G=/.ABägq;>.A.1;�}�43bo�oEb = ÆL\�Ç/ÌJZ{Æ�Ï�Ñ�Ø �@Ó�Ï>Z�Ë,\�Å�Ë¡ZNÉ �G] �/ZJÏY×EË¡\A\�Ë¡É5] ��X[Ó�Ï�×GË�b��/Õ�Z�Ì{Æ�X�É É�×EÆ�ÒOÒfË�ÏYÌDÔW��F/0+gq;>.N.J;�}+CJ79�PH9?D;/;>g�=9450L4ä�<0L?{:MK�= d F~BGF/0�4�2�KO0�2L?D��=9KR037>B2�4�CJ7��;>.A=<BDC1;<7'ã � 0G d 03H<BPBGF<2�B�F/0L?D0OBGF/0�2L79��.103462�?D0f8>CJ?D0 d BE0L8�b'ã1B�CJ4�CN7>BE03?D0L4EBDCJ79�BE; d ;<KOH/2�?D0OBGF/0O=94,0�;>g!=9798<CN?D0 d BD0�8Ý2�798�8>CJ?D0 d BE0L8Ý2�79��.J0L43b Ü>;<?PBGF/0O=9798<CN?D0 d BD0�82�79��.J0L4

∠XY Z2�798

∠XY ′ZCN7/4 d ?[C1I/0�8jCN7 2 d CJ? d .J0 � ∠XY Z = ∠XY ′Z

;<?∠XY Z = 180◦ − ∠XY ′Z

2 dGd ;<?D8<CN79�R2�4Y2L798

Y ′ .NCJ0f;<7TBGF/0�452LKR0�2L? d ;<?;<7OBGF/0~;<H/Hn;<4GCABD0)2L? d 4P8/0�BD0L?DKMCN790�8�IL:X2�798

Zb�Ü<;<?äBGF/0~8<CN?D0 d BD0�8R2�79��.J0L4�g1?D;<K

Y XBD;

Y Z2L798fg1?D;<K

Y ′XBD;

Y ′Z}60MF<2��<0

(Y X, Y Z) = (Y ′X, Y ′Z)� CN7¤Y03CN4G�¡� 4s79;�B[23BDCJ;<7 � gq;<?ä2�7�:~gq;>=9?!Hn;>CN7>BD4

X � Y � Y ′ � Z ;<7M2 d CN? d .10/b �Å_Ë�ÒOÓ�X%Ê�Ç<Ë¡ÉLÔ� b

MP · MC = MA2 = MB2 = MQ · MD � 89;<7 d .1034)�>=923BE?D0RHn;>CN7>BD4C � P � Q � D 4,;<7>B d ; d : d .NCJ�<=n034Lb

� b)-<;>=9?_8/0L4W?[2LCN4,;<7/4ä890�4[:/K4@0�BE?[C10��J.N2�8/?D;>CQBE0�89034 d 0L7>BE?D034O1O2

034GB�K4@0�8>CJ2�BE?DC d 0d ;<KMKM=9790 � .1034 d ;<?D8/0L4 AC0�B

BD45;<7>B!H/2�?[23.N. �03.J0L43b

� b �@7M79;�BE03?[2(d, d′)

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d0�B

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(PA, PQ) = (PA, PC) + (PC, PQ)

= (CA, CD) + (DC, DQ)

= (CA, DQ) = (DB, DQ) = (BA, BQ) �8Y� ; �=M.N2 d ; d : d .AC d CQB @0�8/0L4@Hn;>CN7>BD4A � B � P � Q bo�o�oGb = ÆL\�Ç/ÌJZ{Æ�Ï Ñ�Ø �9× 0�Ó�XGÕ = V�Ë�×DÍYÌ1] �@Ì[Ì�Æ �GÊLÆ�Ï � �äÇ<Ë�X[Z�× 0<Ë �uÏ<Z{ÊLË�X�É�Z�ÌqØ3]Ë Ó3Ð<Õ9Ë�ÑnÇ�X�ÐL]

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b

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M;>g

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WholeIssue 30 6 - CMS-SMC

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