64
8 6 480 420 576 1680 7 100 9 3 4 7 5 (x 1 2 - x 2 3 ) 7 Kx 4 K K 7 -7 35 -35 x 2 + y 2 =1 1 1 4 1 3 1 2 2 3 m y = x - 2 y = mx +3 x y m =1 m< 1 m> - 3 2 - 3 2 <m< 1 f (x) f (x)+2f (-x) = sin x x f ( π 2 ) -1 - 1 2 1 2 1

WholeIssue 33 3 - CMS-SMC

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Page 1: WholeIssue 33 3 - CMS-SMC

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Page 2: WholeIssue 33 3 - CMS-SMC

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Page 3: WholeIssue 33 3 - CMS-SMC

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Page 8: WholeIssue 33 3 - CMS-SMC

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a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a a» Ã�ÔNÔ a!³ � � � � �� � ���}´ � � � ��µ � �5¶ � � � ¸ # �A¹ � � � ����� · � � ��� �!� � � ��� � � � � ¼&» � �5¹ � �µ � ½�¾���� #p¿ � � »lÀ2aØ 5È�m�\A<CG3 d FK�9;23�7)7,?BA<7�Im3�A@E %3\E�CG3 d ?BA<I+�3�3�36:e;<3�A�ÉrL�?@C d 365�A�É�$@A*� ?@:eI<36A@EC#�3658CGC�58:<�636C�I<?BA<C�fp?@CGL 36C�A<: d 5�C�C#�3@¡ d ?@L�L�3�1K3£L�?@:BE�CG3�1K3�7,3 d ?@:<;�;23�7)7)F�: a

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Page 9: WholeIssue 33 3 - CMS-SMC

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Page 14: WholeIssue 33 3 - CMS-SMC

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Page 15: WholeIssue 33 3 - CMS-SMC

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Page 16: WholeIssue 33 3 - CMS-SMC

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[0, 1]}�HBF d H�7)56EGF�7)fD=�E*H23£58J<?6�@3 d ?@:<;@F�EGF4?@:27�J@A@Eu58CG3:<?�E�FK;<36:BEGF d 5�1D1�= �)3�CG? ×Ô a 0<CG?6�@3�E*H@56E¢58:�=NI2CGF�L 3�:@A<L�J<36C

22n

+1d 5�:2:<?�E�J23�CG3�I2CG3�793�:BEG38;N587�5#;@F �/36CG3�: d 3?BfmEÊ}�?!�2f�E*H�Im?8}�36C�7�?BfmFK:BEG38�636C�7 a

� }£58:BE2E�?�58I<?B14?2�\F �)3�fp?@C¢?6�@3�C�14?2?2��FK:<�U7,?@L�3u79?B1�A@EGFK?@:27tf4CG?@L ��F d H2361 Î 56E�5�FD1�1K3@¡� ?BAm3�:t¡ Ó C�58: d 3@¡@}�HBF d H�}�3�CG3£14?@7�E�FK: L!=��21�F�:<��7�=@7*EG3�L a Î 56E�5�FD1�1K3�� 7�:258L�3!79H2?BA214;H@5��@3�J23836:�58;<;238;QE�?QE*H23S1�F�7*Eu?Bft79?B1��@36C�7�fp?@CUE*H23Sfp?B1�1K?8}�FK:<�NI<CG?2JB1436L�7��� � Am�6?@7)1�5�� ) A<5�1DF�� d 5�EGF4?@: ��� � � ?BA<:<;¢¡m0<CG?2J@1K3�L � � ������� � � � ������� � � ��� �

�� ��� ! ��&" � � 1�=@L�I@FK58;<3 Î 361K�63@¡¢0<CG?2J@1K3�L � � ����� � � � ��������� ��� �����

�� Î ?@79:@F�5�5�:<;Q�£36C �)3��6?��BFK:25@¡ � 56EGFK?@:25�1 � 1 =2LMI2F�5�;£�83�143 d EGF4?@: � 3�7�E,¡20<CG?2J@1K3�LM7� 58:<; � � ����� � � � � ������� � � ���

�� � � �D� � C�FK79H � 1�=@L�I@FK58;t¡�0<CG?2J@1K3�LM7 � ¡ � ¡ � ¡ ¡>58:<; � � ������� � � � � Ù � �8� ������ � � �# ¡ ��� ¡ ��� a

Page 25: WholeIssue 33 3 - CMS-SMC

�����

� ?8} }�3�EGA<C�:�E�?�7,?B1DA@EGF4?@:27!f4CG?@L ?BA<CQCG365�;23�C�7!E�?�I<CG?2JB1436L�7�?Bf#E*H23 ��� �D�MUC�58;<3 � ?@LM58:@F�58:��r56E*H236L�5�EGF d 5�1 � 1 =2LMI2F�5�;M�\F��@36: � ����� � � � Ù �� aÇ a�Ø 3\E

OABCJ23�5NEG3�E�C�58H238;@CG?@:M7*A d HQE*H@5�E

OA ⊥ OB ⊥ OC ⊥ OA¡�143\E

rJ<3E*H23MC�5�;BF�A<7£?BfOF�EG7NFK:27 d CGFKJ<3�;�7)ImH236CG3@¡/5�:<;�143\E

HJ23!E*H23�?@C�E*H2? d 3�:BE�CG3�?Bf�E�C�FK5�:<�\143

ABCa 0<CG?��@3NE*H@56E

OH ≤ r(√

3 + 1) a

µ ����� �!�"��# ���¾�� �� À � � � � � ���;¼�´���� � � � � � ¸ # �I¹ � ��� � ��� �T³ ��� ������ � # �"� � � � � � ����� ��# � �´���� � � � ��# � � µ � � �!#4½Ø 3�E∆A

¡∆B

¡∆C

¡85�:<;∆O

J<3#E*H23>5�CG3�5S?BfBE�CGF�58:<�\1K3�7OBC

¡OAC

¡OAB

¡58:<;ABC

¡8CG3679I<3 d EGF �@361 = a/� 5���FK:<�#F�:BE�?£5 d*d ?BA<:BE/E*H@56E�E�CGF�58:<�\1K3�7OBC

¡OAC

¡85�:<;OAB

5�CG3£I2CG?8Ö{3 d EGFK?@:27�?BfABC

F�:MLMA@EGA<561�1 =�?@CÊE*H2?2�6?@:25�1¢;@F�CG3 d EGFK?@:27�¡�}�3�H@5\�@3∆2

O = ∆2A + ∆2

B + ∆2C

ab I<I@1�=BFK:<�>E*H23 b � Ù ) � � :<3�$BA<5�1DFDE{=t¡8}�3£�63�E

∆2O = ∆2

A + ∆2B + ∆2

C ≥ 13(∆A + ∆B + ∆C)2

¡58:<;�H236: d 3B¡

∆A + ∆B + ∆C

∆O

≤√

3a Ì � Í

�6FK: d 3OH

F�7#Im36C�I<3�:<;BF d A21�58C�E�?!E�C�FK5�:<�\143ABC

¡BE*H23S�@?B1DA<L 3�?Bf¢E*H23�EG3�E�C�58H238;@CG?@:OABC

FK713OH · ∆O = 1

3r(∆A + ∆B + ∆C + ∆O)

¡f4CG?@L }�HBF d H�}�3£�63\E

OH = r∆A + ∆B + ∆C + ∆O

∆O

= r

(

1 +∆A + ∆B + ∆C

∆O

) aÓ F�:25�1D1�=¢¡BA<7*FK:<� Ì � Í ¡8}�3£?2JBE�5�F�:

OH ≤ r(

1 +√

3) a

à a£� H23 d ?@L�I@143*É�:@A<L�J<36C�7z1¡z2¡. . .

¡z5H@5\�@3QE*H23�7)58L�3�:<?@:<� �)3�CG?~L ?2;BA21�A<7\¡

58:<; 5∑

i=1

zi =5∑

i=1

z2i = 0

a 0<CG?6�@3£E*H@56Ez1¡z2¡. . .

¡z558CG3£E*H23 d ?@LMI21K3)É d ?2?@CG;BF4�

:256EG3�7#?BfmE*H23N�@36C�EGF d 367�?Bf¢5�CG3��\A21K5�CuIm36:BEG58�6?@: aµ ���I¹ � � �� · �"� � � �N´�� �!� � � � � � à � � � # � � � ��# � ��� � # � ¾�� �� À � � � � � � �;¼�´���� � � � � �¸ # �A¹ � � � ���!� ��³ ��� � � �� � # � ��� � � � � ����� ��# � � ´���� � � �"��# � � µ � � �!#4½ � � Ä �I¹ � ´�� ����� � � � ¿ �������� �!�"��#4½

Ø 3�ErJ23£E*H23 d ?@LML ?@: L�?2;@A21DA<7N?Bf

z1¡z2¡. . .

¡z55�:<;

p = z1 · z2 · · · z5J<3SE*H23�FKC�I<CG?2;BA d E a�� ?�E�3NE*H@56E |p| = r5 6= 0a� H23 d ?@L�I@143*É�:@A<L J23�C�7

z1¡z2¡. . .

¡z55�CG3SE*H23�CG?2?�EG7#?Bf<E*H23£Im?B1 =2:<?@L�FK561

5∏

k=1

(z − zk) = z5 − σ1z4 + σ2z3 − σ3z2 + σ4z − p¡

Page 26: WholeIssue 33 3 - CMS-SMC

����

}�H23�CG3σk

Ìk = 1

¡2¡3¡4Í 58CG3 E*H23 A<7)A<561�7�=@L�L�3�E�C�F d fKA<: d EGFK?@:27�?BfuE*H23�CG?2?�E�7

z1¡z2¡. . .

¡z5a Î =�H8=2I<?�E*H23�7*FK7\¡

σ1 = 0a �6FK: d 3

2σ2 = σ21 −

5∑

i=1

z2i

¡<}�3�5�1�7,?H@5��@3

σ2 = 0a�� 7*FK:<�

zi · zi = r2 ¡8}�3 d ?@LMI2A@EG3 �σ3 =

zizjzk =∑

i<j

p

zizj

=p

r4

i<j

zizj =p

r4· σ2 = 0

σ4 =∑

zizjzkzl =

5∑

i=1

p

zi

=p

r2

5∑

i=1

zi =p

r2· σ1 = 0

a� E�fp?B1�1K?8}£7�E*H@56E

z1¡

z2¡

. . .¡

z558CG3~E*H23��2f�E*H�CG?2?�EG7�?Bf�E*H23�:<?@:<� �)3�CG? d ?@LMI21K3)É:@A<L J23�C

p5�:<;t¡�5�7�7)A d H¢¡\58CG3UE*H23 d ?@L�I@143*É d ?2?@CG;@F�:256EG3�7/?Bf6E*H23O�@3�CÊEGF d 3�7/?Bf�5�CG3��\A21K5�CIm36:BEG58�6?@: a

� H23 :<3*É�EeJ@1K? d �Ë?Bf�7,?B1DA@EGF4?@:27i5�CG3Âfp?@C�I2CG?2J@1K3�LM7c?Bf~E*H23 � � �D� O�?@CG3658:�r56E*H236L�5�EGF d 5�1 � 1�=@L�I@FK58;M5�I<I<3�5�CGF�:<� � ����� ��� � Ù � aÇ a/� H23 d ?@LMI2A@EG3�C�7OFK:�5 d ?@L�I@A@E�36C/1K58J£58CG3 d ?@:2:<3 d E�3�;QJ�= d 5�JB14367U587Ofp?B1D14?8}£7�� �@5 d Hd ?@LMI2A@EG3�C#FK7S;BFKCG3 d EG1�= d ?@:2:<3 d E�3�; E�?�3*É�5 d EG1 =�E*H@CG383M?�E*H236C d ?@L�I@A@E�36C�7��@F�5 d 58J@1K3�7 a� H236CG3�FK7�56E�L ?@7�Eu?@:<3 d 58J@1K3>Ö{?BFK:@F�:<�>EÊ}�? d ?@LMI2A@EG3�C�7�58:<;�58:�=�I25�F�Cu?Bf d ?@L�I@A@E�36C�7FK:�E*H23�1�5�J d 58:È3*É d H@58:<�63�;25�EG5 a Ì � }�? d ?@LMI2A@EG3�C�7A58:<;

Bd 58:�3)É d H@5�:<�63 ;25�EG5F�f�E*H236CG3�3*É8F�7*E�7S5M793�$BAm3�: d 3�?Bf d ?@L�I@A@E�36C�7�7�EG5�C�EGF�:<�Qf4CG?@L

A58:<;�3�:<;BFK:<�!5�E

BF�:}�HBF d H�EÊ}�? d ?@LMI2A@EG3�C�7>:<3)É6E�EG? 3�5 d H~?�E*H236C�FK:ME*H23Q7,3�$@Am36: d 3Q58CG3!;BFKCG3 d EG1�=!Ö{?BFK:<3�;J�=M5 d 5�JB143 a ÍØ 3�E

kJ23QE*H23M7)L�561�1K3�7�E�:@A<L�J<36CN?Bf d ?@L�I@A@E�36C�7SF�:~E*H23!1K58J�}�H2?@7,3�CG36L ?6�B5�1CG3�7*A21DE�7OFK:�14365��BFK:<��Ö4A<7*EO?@:<3 d ?@L�I@A@E�36C/F�:�E*H23#1K58J�?@C/5>I25�F�C�?Bf d ?@L�I@A@E�36C�7�:<?�E/5�JB143E�?�3)É d H@5�:<�63 ;@56E�5�58:�=�L ?@CG3 a�Ø 3\E

`J23!E*H23M79LM5�1D14367*E>:@A<L�J<36CS?Bf d 58J@1K3�7>}�H2?@793;<3�143\EGF4?@:!CG367)A21�EG7�FK:�E*H23�3)É�FK7�E�36: d 3S?Bf2EÊ}�? d ?@L�I@A@E�36C�7uE*H@5�E d 5�:2:<?�Eu3*É d H@58:<�63�;25�EG558:�=ML�?@CG3 a �8H2?8}�E*H@5�E

k = `a

µ ����� �!�"��# ���¾���� # ³�½ � � � � �!���# �u· � � ��� � � � � � � ��� � � Ä � � µ � ½|³ � � � ��· ��# # � ��� �!� �¸ µ ��½� : E*H23�1�58:<�\A<58�63i?Bf!��C�58I<HcE*H23�?@C{=t¡£E*HBF�7ÈI2CG?2J@1K3�L 587)7,36C�E�7�E*H@56E��@3�CÊE�3*ÉB�d ?@:2:<3 d EGF��BFDE{=

k3�$@A<561K7>3�;<�63�� d ?@:2:<3 d EGF �@F�E�=

lF�: 5 d ?@:2:<3 d EG38;

3 ٠CG38�\A21�58C���C�58I<H Ga� H@CG3�38�*CG38�\A21�58C�FDE{=MFKLMI21DF4367�E*H@56E

l ≤ 3a�Ó A<CÊE*H23�C9¡

k ≤ lJ<3 d 56A<7,3�F�ft5�7,3\E�?Bf�38;2�63�7;@F�7 d ?@:2:<3 d E�7NE*H23M��C�58I<H¢¡tE*H23�:~E*H23�CG3!FK7S5M7,3\EN?Bf��@3�CÊEGF d 3�7 d H2?@7,36:�?@:<3�f4CG?@L 3�5 d H38;2�63ME*H@5�E�;BFK7 d ?@:2:<3 d EG7�?@C>1K3�5\�@3�7£?@:<3!�@3�CÊE�3*É a � H23�CG3�fp?@CG3@¡t}�3 H@5\�@3~3�$@A<561�F�E�=�FDf

k = 358:<;�:<383�;�?@:@1�= d ?@:27*F4;23�C

k ∈ {1¡2} a�6A<I<I<?@7,3

k = 1a�� H23�:ME*H236CG3�F�7#5��@3�CÊE�3*É

v}�H2?@793�CG36L ?6�B5�1m14365��@367�EÊ}�?�?@CE*H@CG383 d ?@LMIm?@:<36:BEG7 a Î 3 d 56A<7,3

vH@5�7N;238��CG3�3

3¡2E*H23�CG3QLMA<7�E#J<3Q5�:�3�;<�63�F�: d F4;23�:BE}�FDE*H

v}�H2?@793�CG3�L�?��B561t;BFK7 d ?@:2:<3 d EG7uE*H23���C�5�ImH a� f

k = 2¡#7*A<I<I<?@7,3�E*H23�CG3�L�?��B561�?Bf��@36C�EGF d 367

v5�:<;

w;@F�7 d ?@:2:<3 d E�7�E*H23��C�58I<H a�� f

vFK7#58;�Öp5 d 36:BEuEG?

w¡BE*H236:�E*H23�CG3�5�CG3£fp?BA<C�38;2�63�7uÖ{?BFK:@F�:<� {v

¡w} E�?!E*H23CG3�7�ES?Bf�E*H23���C�5�ImH a �6FK: d 3

k = 2¡G\{v

¡w} H@587>EÊ}�? d ?@LMIm?@:<36:BEG7\¡�3�5 d H }�F�E*HEÊ}�?M3�;<�6367�E�? {v

¡w} a �BF�E*H23�C�I<56FKCu?Bf�3�;<�6367�;BFK7 d ?@:2:<3 d EG7uE*H23���C�5�ImH a

Page 27: WholeIssue 33 3 - CMS-SMC

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� fvFK7�:<?�ES5�;8Öp5 d 3�:BE>EG?

w¡/E*H236:�E*H23�CG3�58CG3�EÊ}�?�?@C>E*H@CG3�3 d ?@L�I<?@:<3�:BE�7£F�:

G\{v¡w} a � f�E*H@CG383B¡<E*H236CG3Q58CG3Q7*F É�38;2�63�7�Ö{?BF�:@FK:<� {v

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2¡2¡�58:<;

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A58:<;

B¡BE*H236:�}�3£L�5\=M587)7)A<L�3Q?@:<338;2�63�56E

vd ?@:2:<3 d EG7NEG?

A¡¢}�HBF�1K3!E*H23 ?�E*H236C�EÊ}�? d ?@:2:<3 d E�E�?

Ba �6F�LMFD1K5�CG1 =t¡tE*H2338;2�63�7!FK: d FK;<36:BE�}�FDE*H

w5�CG3È;BF��BF4;238; ?@:<3�58:<;�EÊ}�? ai� H23�CG36L ?6�B5�1#?Bf�E*H23�EÊ}�?7)F�:<�\143\E�?@:�3�;<�6367�;BFK7 d ?@:2:<3 d EG7uE*H23���C�5�ImH a

à a�Ø 3\EABCD

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5�:<;BD

L 3�3�EU5�EU5�I<?BFK:BEMatb I<?BFK:BE

O?@:�E*H23N1�F�:<3S7938��L�3�:BE

MCFK7u79361K3 d EG38;Q7)A d HE*H@56E

O 6= M58:<;

OB < OCa�� H23 d FKC d 1K3 d 3�:BE�CG3�;M5�E

OI2587)7)F�:<��E*H@CG?BAm�6H�Im?BF�:BEG7

B58:<;

DL 3�3�E�7�E*H23S1�F�:<3

AB5�EuIm?BF�:BE

B5�:<;M5�I<?BFK:BE

XÌ }�H23�CG3

X = B}�H23�:E*H23N1�F�:<3

ABFK7UE�58:<�636:BEOE�?£E*H23 d F�C d 143 Í 5�:<;�L�383\EG7UE*H23N1DFK:<3

BC56EUI<?BFK:BE

B58:<;!5Im?BF�:BE

Ya�Ø 3\EUE*H23�1DFK:<367

DX5�:<;

DYL 3�3�EUE*H23S1�F�:<3£7938��L�3�:BE

AC56E

P5�:<;

CG3�7)Im3 d EGF��@3�1�= a ��É8I<CG36797uE*H23N�B561�Am3£?Bf OQ

OP

F�:!E�36C�LM7�?Bft}�H23�: MA

MO= t

aµ ����� �!�"��# �� · ��� � � � � � � ��� � � ��� � µ � � ��� � � ���HÄ ��� � ��# � � ½�6FK: d 3�E*H23'$@A<58;2C�F�1�56EG3�C�561

DXBYFK7 d = d 1�F d ¡8}�3£H@5��@3∠AXD = ∠BY D

= ∠BOA = ∠BOP �H23�: d 3@¡�$BA<5�;@CGFD1K5�E�36C�5�1

BOPXF�7 d = d 1�F d�a� ?@:27*F4;23�CtE*H23#F�:��@3�C�7*F4?@:

I?Bf@I<?B143

O58:<;Im?8}�36C

OB2 ar� H236:�E*H23 d F�C d 143 d FKC d A<L��7 d C�F4JBFK:<�BOPX

LM58I27UEG?�E*H23>1�F�:<3AB

a� HBA<7\¡I(P ) = A

¡/}�HBF d H�FKLMI21DF4367�E*H@56EOP · OA = OB2 a

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A

B C

D

MO

P

QX

Y

�6FKL�F�1�58C�1�=¢¡8}�3£?2JBE�5�F�:OQ · OC = OB2

a�� H236CG36fp?@CG3B¡OQ

OP=

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OC=

MA + MO

MA − MO=

MA

MO+ 1

MA

MO− 1

=t + 1

t − 1

a

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BC

PQ+

CA

QR+

AB

RP≥ 6

aµ ���I¹ � � �� · ��� � � � � �*� � � � � �!� � µ � � � � � � ���yÄ � � � ��# � ��� ��#�� · ��Ä � � ��� � � #�Ä ��� �� ��¹� � ��� � �!� � �wÄ � � � � �p· ��� �"��� ��� � µ � � �!#4½ � � � � � � � # � � � � ������� �!�"��# ��º � � � #�Ä ��� �� ��¹� �#½Ò 3�A<793#E*H23�7*E�58:<;@58CG;�:<?�EG5�EGF4?@:

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Page 28: WholeIssue 33 3 - CMS-SMC

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Ò 3�H@5\�@3PQ = 2(s − b) sin

B

2= 2(s − b)

(s − c)(s − a)

ca

¡

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2= 2(s − c)

(s − a)(s − b)

ab

¡

RP = 2(s − a) sinA

2= 2(s − a)

(s − b)(s − c)

bc

aÒ 3#5�I<I@1�=>E*H23 b � Ù M�� � :<3�$BA<5�1DFDE{=�58:<;SA<793�E*H23#58J<?6�@3�3*É�I2CG3�7)7)FK?@:27/fp?@C

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¡58:<;RP

EG?M�63\EBC

P Q+

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RP≥ 3 · 3

BC · CA · AB

P Q · QR · RP= 3 · 3

(abc)2

8[(s − a)(s − b)(s − c)]2a

Ó F�:25�1D1�=¢¡�}�3MA<7,3�E*H23�}�361D14�)��:<?8}£:�FK:<3�$@A<561�F�E�=abc ≥ 8(s − a)(s − b)(s − c)

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P Q+

CA

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RP≥ 3

3√

8 = 6a

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Page 29: WholeIssue 33 3 - CMS-SMC

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DE F

Ø 3�EB′ 58:<; C′ J<3�E*H23�CG3 �¢3 d EGF4?@:27>?Bf B 58:<;

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Page 30: WholeIssue 33 3 - CMS-SMC

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Page 31: WholeIssue 33 3 - CMS-SMC

�����

�6FK: d 3B¡S¡P¡�5�:<;

R58CG3 d ?@: d = d 1DF d ¡<}�3MH@5��@3

∠BSR = ∠BPR �7)F�: d 3

A¡R¡S¡@58:<;

C5�CG3 d ?@: d = d 1DF d ¡

∠RCA = ∠RSAa�� H23�:

∠BSR = ∠BPR = ∠QPC = ∠PCQ = ∠RCA = RSAa

� HBA<7\¡SR

JBFK793 d E�7∠BSA

a �6FK: d 3BR = RA

¡�E*H23>58:<�\1K3SJBFK793 d EG?@C�E*H238?@CG36LË�\F �@3�7A<7BS = SA

a�� ?@:27,3�$@Am36:BEG1�=¢¡∠BRS = 90◦ aS� H23�:

∠BPS = 90◦ Ì J23 d 5�A<793B¡S¡P¡B58:<;

R58CG3 d ?@: d = d 1�F d�Í9aÒ 3�H@5\�@3

∠CPS = 90◦ − ∠QPC = 90◦ − ∠BSR = ∠RBS = ∠ABS58:<;∠SCP = ∠SCR = ∠SAR = ∠SAB

a� H236CG36fp?@CG3B¡/E�CGF�58:<�\1K3�7

ABS58:<;

CPS5�CG3�7)F�LMFD1K5�C a �6FK: d 3 4ABS

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PS

BS=

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AB=

1

2

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k7*A d H E*H@56E,¡¢fp?@C�58:�=�I<?@7)F�EGF��@3

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Ì � Í

Page 32: WholeIssue 33 3 - CMS-SMC

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fp?@Cu79?@L 3�I<?@7)F�EGF��@3a¡b¡c7*A d H!E*H@5�E

a2 > bc¡@58:<;�;236�<:<3

f(x) = (b2 − ca)x2 + (a2 − bc)x + (c2 − ab)a

Ó CG?@L Ì � Í ¡m3�FDE*H236Cf(x) ≥ 0

fp?@C�561�1¢CG3�561x?@C

f(x) ≤ 0fp?@C�5�1D1tCG365�1

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f(1) = a2 + b2 + c2 − ab − bc − ca > 0Ì :<?�EG3£E*H@5�E

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2(A1A2 + B1B2 + C1C2) = AB + BC + CAa

Page 33: WholeIssue 33 3 - CMS-SMC

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∠BAA1 =π

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AIa 0@EG?B1436L!= � 7 � H23�?@CG3�L �\F��@367

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Page 34: WholeIssue 33 3 - CMS-SMC

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CnÍ9a,� :�I258CÊEGF d A21�58C9¡

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Page 45: WholeIssue 33 3 - CMS-SMC

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aÌ 5 Í �8H2?8}�E*H@5�E

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ma≤ 12R

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� �% ! b2 + c2

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?Ï 3\E�36C�L�FK:<3NE*H23£C�5�:<�63�?Bf

1

R

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wa

a

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BC5�E�F�EG7SL�F4;2�)I<?BFK:BE

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b2 + c2 = 2(AD2 + BD2)

= 2(AD2 + BD · DC)

= 2(AD2 + AD · DK)

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ma≤ 4R �

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ma≤ 3(4R) = 12R

a

Page 48: WholeIssue 33 3 - CMS-SMC

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Ì J Í �6FK: d 3ha =

2[ABC]

a

5�:<;R =

abc

4[ABC]

¡8}�3£?2JBE�5�F�:b2 + c2

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(

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2bc

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≥ 4R¡

}�HBF d H!FKLMI21DF4367uE*H@5�E ∑ � �% ! b2 + c2

ha≥ 3(4R) = 12R

aÌ d�Í�Ø 3�E

sw =1

R

� �% ! b2 + c2

wa

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(4, ∞)a

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b2 + c2

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2

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B − C

2

cosA

2

= 4 cosB − C

2

a�£36: d 3B¡

sw ≥ 4∑

� �% ! cosB − C

2

a�6FK: d 3

cos x ≥ 1 − 2x/πfp?@C

0 ≤ x ≤ π/2¡8}�3�H@5\�@3

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2=

� �% ! cos|B − C|

2≥

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1 − 1

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58:<;b = c = x + 1

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2x

(x + 1)52

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x?@:

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Page 51: WholeIssue 33 3 - CMS-SMC

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Page 52: WholeIssue 33 3 - CMS-SMC

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

A

B C

E

F F ′

H

K..............................................

ω...........................................

µ.......................................

θ

.

..

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β

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

α

Ó CG?@Lsin θ = 5/13

5�:<;cos ω = 3/5

¡O}�3�?2J�EG56FK:tan θ = 5/12

58:<;tan ω = 4/3

a!Ò 3�H@5\�@3α = θ + ω

a!� HBA<7\¡tan α = tan(θ + ω) = 63/16

a�6FK: d 3tan α > 0

¡<}�3!79383£E*H@56EαF�7N5�:�5 d A@EG3!58:<�\1K3 aS� HBA<7�¡

E1DF4367NJ<3\EÊ}�3836:

F ′58:<;CÌ 7*FK: d 3 4BEF ′ FK7uFK79?@7 d 361K3�7 Í9a ��3�: d 3@¡ µ = ∠FCB = ∠F ′BC = β + θ

a� 3*É�E,¡�7)F�: d 3β + 2α = π

Ì 7*A<L�L�FK:<��E*H23N58:<�\1K3�7uF�: 4BEF ′ Í ¡6}�3 d 5�1 d A21K5�E�3tan β = tan(π − 2α) = − tan(2α) =

2016

3713

aÒ 3SE*H236: ;238;BA d 3SE*H@5�E

tan µ = tan(β + θ) =253

204

a�6FK: d 3

tan θ =HK

BK

¡tan µ =

HK

KC

¡@58:<;BK + KC = BC = 6

¡8}�3�?2J�EG56FK:HK =

6 tan θ tan µ

tan θ + tan µ=

1265

676

a'D� ��� ��������� 8 � �)�� ��0&�59�'E3:)���'(+(' /26 �F�4 ?5��<1�$��� �N<7�1� � ��)�"$� " N� �$��4 )�"$� " N� �$��4 �:�$����< " "���

%���� �*�)������< ��"�=�� 6 F#%��+� ��'E9�',6������$4J32��������4 �>� "��5MO� = F#%�6�$ F&?23(9�67)�4 � < �������$� <1)������1##���O� )�� "�� �? ��<����� �N<L� ��4 � � !�� < ��4��.C24�? )�'2=E3&6OF#%('E3&- 6OK�%1��)$)�4E35"��5M�#5��$�" �L�$�JB5���&����N4PF$'&4�?:)*'&= E'���9�%1��3�N'(+�C ? )�4 ?5�O� ��� < �#���0���$GJ��"$�N<L�$GA4�67������8>�*��M� $4/'/�����7� < " = /2��C �0��3&� � '/K/0>',+1-:'�����4 %P"�G �����4JF59 4?:)*'&=/B �'/F,�*'�B.02CD+1� �F�+ ���4.��< ��35" !�<1>�N4.� 6 4�? )�'2=19�',6OF#%16 ��'E��0�'� /'&419*"� $"�� � �� $</F�<7�1� 4 C&��" >" 4�*"�!#"$�>= � 6OF#%('E�+� $�'E3 � ��+A9���354 � ��G �$� < "��E? ��<����� �N<L� � � �D+1� � � ������$� "$�,$4D)��OK�����#���� �N4�+ � = �$C�� �)�F#% ��C&)0�:��3�/#4 �#" ���N<��*4 +(��4�?:)*'&= �2C � )���3&02�,��4 ���$�$��< " 4:+ ��4(?:)*'&= - K �>K�) � ����+1054�@ " �L� 8��$GJG �*� 4�1##�E+1�$�1##��� � "���>�O=�$ �59���3��2K JC(C24���<7��� "J? ��<����� �N<L� ��4,��" � < � "0>" 4:F$'&4D?:)*'&=�"�����1##�D!#��� !:�$������KDC,������������� � � �������GJ"�� � $< �2�������5� �,"��$�/�L� ���$�2� �A� #5�E����� ��� <7�$�:KE9 �P�1� ��8>GA< ���N<L�$�����P�����,< �#MN�$�O���$MN�NK

02�$�#� �M ���� MN�$GJG ���#� �� �1#�"���#5� ����!:�$M��L���"�� ��<7MN������"$�����P����"$�, "$�& ������#�"�� � #5���*"��L��� � �tan θ

�P��� � #�"0��� � ��8��P� �.���$�HK = 2

K,9 #�< �,��$<L�L�$�&"��& �� ��##�$�1##���(� #5�����,"����E"�� �/!#" < � �,� ���O��<LMO���� ��G 8���� � � �$�

HK"$�,

tan θ4#�7��"0�*< ���A� #5�EG ��"$��< ���A� ��O��<LMO���E�L�A���$�D����"0���� ��K

����� [�l�������T����������������*i,!.0/+12/�3�4�5È6�7 HXW(YMW s W s D 3Q<r.b7 G�ZBD�J�C�D t;/�.$4 G�[uJ 5�: D quU

4ABC�+'$]^&[S�( �rS�"$&+U�'�YZU�'b&+%bU^S+d�'b%lYM_>&�"�'�#�%l_m#�`�'$]^&[S�U�c�de&+_

BS�U�(

Cj�&�&�'S+'�'$]^&ÈW�#�YMU�'

P�2S�U�(X'$]^&ÈS�( �rS�"$&+U�'[YMU�'�&�%�U^S�d2'b%lYM_>&�"�'�#�%l_#�`�'$]^&ÈS�U�c�de&+_

AS�U�(

CjÈ&�&Q'mSQ'I'$]^&�W�#�YMU�'Q

� ]�S�%lS�"�'�&�%lY��0&['$]^#�_O&['b%�YZS�U�c�dM&�_pYZU n ]�Ye"0] AQ + BP = AB

Page 53: WholeIssue 33 3 - CMS-SMC

�� ��

µ ����� �!�"��#���uà ��u´���� ����� � � ¸ # �A¹ � � � ��� ��ºK´ � � � � � � ´ � � � � � � À � ����# ��# � � ��� ��� � � ��� � ��� � ���"���Wa

� H23#E�C�FK5�:<�\14367OFK:S}�HBF d HAQ+BP = AB

58CG3�I<CG3 d FK79361 =�E*H2?@793�}�FDE*H£5>C�F4�6H�E58:<�\1K3£56ECaÏ 3�:<?�EG3SJ�=

3α¡3β¡�5�:<;

3γE*H23N5�:<�\14367u5�E

A¡B¡�58:<;

C¡8CG3�7)Im3 d EGF��@3�1�=¢¡�5�:<;J�=

RE*H23 d F�C d A<L�C�58;@FDA<7�?Bf 4ABC

a�� 3 d 5�1D1<E*H@5�E�fp?@Cu5�:�=MCG365�1m:@A<L J23�Cθ¡

sin 3θ = sin θ(

3 − 4 sin2 θ)

= sin θ(1 + 2 cos 2θ)

= 4 sin θ sin(

π

3− θ

)

sin(

π

3+ θ

) a� H2367,3SF4;23�:BEGF�EGF4367�}�F�1D1¢J<3SA<7,3�;Q}�F�E*H2?BA@E d ?@LML 36:BE aÒ 3�H@5\�@3

AB

sin γ=

2R sin 3γ

sin γ= 2R(1 + 2 cos 2γ)

a Ì � Í� H23 Ø 5\} ?Bf��6F�:<3�7�FK: 4AQC

�\F��@367AQ

sin γ=

AC

sin(α + γ)=

AC

sin(

π3

− β) =

2R sin 3β

sin(

π3

− β)

= 8R sin β sin(

π3

+ β)

= 4R(

cos π3

− cos(

π3

+ 2β)

)

= 2R(

1 − 2 cos(

π3

+ 2β)

) a Ì � Í�6FKL�F�1�58C�1�=¢¡

BP

sin γ= 2R

(

1 − 2 cos(

π3

+ 2α)

) a Ì � Í� 7)F�:<� Ì � Í ¡ Ì � Í ¡@58:<; Ì � Í ¡8}�3�79383NE*H@56E

AQ + BP = ABF�f¢58:<;�?@:@1 =�FDf

1 − 2 cos(

π3

+ 2β)

+ 1 − 2 cos(

π3

+ 2α)

= 1 + 2 cos 2γ �E*H@56E�F�7�¡cos 2γ − 1

2= − cos

(

π3

+ 2α)

− cos(

π3

+ 2β) a Ì � Í

� ?8} }�3�CG3W}£CGF�E�3£J<?�E*H�7)FK;<367#?Bf Ì � Í �cos 2γ − 1

2= cos 2γ + cos 2π

3= 2 cos

(

π3

− γ)

cos(

π3

+ γ) ¡

− cos(

π3

+ 2α)

− cos(

π3

+ 2β)

= −2 cos(α − β) cos(

π3

+ α + β)

= −2 cos(α − β) cos(

2π3

− γ)

= 2 cos(α − β) cos(

π3

+ γ) a

� 7)F�:<�>E*H23�793�3*É�I2CG3�7)7)FK?@:27�FK: Ì � Í ¡8}�3�7,3�3NE*H@56EAQ + BP = AB

F�ft5�:<;�?@:@1 =�FDfcos(

π3

− γ)

cos(

π3

+ γ)

= cos(α − β) cos(

π3

+ γ) a Ì � Í

� f�E*H236CG3�F�7!5�C�F4�6H�E£58:<�\1K3�56ECÌ E*H@56E£F�7�¡UFDf

γ = π6

Í ¡OE*H23�: J2?�E*Hr7*F4;23�7!?Bf3�$BA<56EGFK?@: Ì � Í 58CG3 �)3�CG? a � fCFK7>:<?�E�5!CGFK�6H�E#58:<�\1K3 Ì

γ 6= π6

Í ¡<E*H236: Ì � Í CG38;BA d 367�EG?

Page 54: WholeIssue 33 3 - CMS-SMC

�� ��

cos |α − β| = cos(

π3

− γ) a � :�E*HBF�7 d 58793 Ì � Í F�7�:<3,�@36CS795�EGFK7*�¢3�;t¡OJ<3 d 56A<7,3ME*H23d ?@7*FK:<3NfKA<: d EGFK?@:!FK7�FK:�Ö{3 d EGF��@3�?@:

[0, π]58:<;

0 ≤ |α − β| < α + β = π3

− γ < πa

'D� ��� ��������� 8 � � 6 F#%��+����',9�'E6��0���>4E32��������4 �>� "��5MO� =1F#%16�$ F&?&3(9>6L) 4�� < ��������� <�)������ #5���O�)�� "��L� ?5��<1�$��� �N<7�1� 4 ��� !�� < ��4(�.C24J?:)*'&=�'�$:CD)�9>C ��6L) 0�K,-/�+� 67)�41B�"��&��" ��*<7� %/< ��# )�M�#5����� 4/',�1##�����N4/5���$�$MN� =� E'���9�%1��3 �O',+�CP?:) 4P? �O� � � < �5����� ��GJ��"��N< ��GA4,6 ������8>�*��M� >4:'E�$��� � < "�=�$ � 9>��3%�2K JC(C24 ��<7��� "? ��<����� �N<L� ��4 ��"�� < � "0>" 4&F$'D4(?:)*'&=#"$�, � #5�(!#��� !:�$������K

������� [�l�������T����������������* ,!.0/+12/�3�4�5È6�7��³:���=B4�t ZBD < D :Zt{t;4 G�Y / @ 4 J^G�? . D�J �$4� P &Q'

A�B�C R &['$]^&�S�U�c�de&+_�#�`BSÃ'b%lYMS�U�c�de&^ � ]^# n '$]�S+'

(

cos 12A + cos 1

2B + cos 1

2C) (

csc 12A + csc 1

2B + csc 1

2C)

−(

cot 12A + cot 1

2B + cot 1

2C)

≥ 6√

3

� 303Q4 J <M: D t{tr7 <r=�4©3 D yX4�1�.$/^/�LÈ6�7��s 4�L A 4�< � .l3$t D�J�D+C :� G ~aJ : } 4�.l3$:z<e7 /�L s D . D�� 4 } / Gs D . D � 4 } / G Z /�3 J : D D�J 5 84�.���4 C / } : J^D���� :u/ J�J 4 ZBD :Ztu4�7 G�� tM3$:;4 x^D yi1�6^4�t{t G D�J 5x = D .lt;4�3 YMW�� :ey9: J�J :;4 G � J�C 4Ot;/ s < D <u4 ~hJ : } 4Q.z30:;<r7 G s D�J � J^C 4�tu/ G���mGm~ s�� ��|�D tz<r=B4Q.NOD�J / @ 3 GI~ .�3 @ tZ: J 4 J^C 7�y J^D 3$: @ y G [rJ�J 3+6�. @ � A�G � @ 3Q<r.l: D���� W � W ��:{t;/ �3Q4 } :� G ,!. D�J��lD�J : G�!@�C /�30t D�} : D���D�J 5 � W N W s yX4�4 J�A�G w D tz<l6^/�y�yX4Ot G <r=B4���4�<r=�4�.lt D�J 5�3+

� YMU�"b&A + B + C = π

��S�U�(0 < A

�B�C < π

� n &�]�SQf�&0 = cos π

2= cos

(

12(A + B + C)

)

= cos 12A cos 1

2B cos 1

2C − cos 1

2A sin 1

2B sin 1

2C

− sin 12A cos 1

2B sin 1

2C − sin 1

2A sin 1

2B cos 1

2C

� &(�Ygf�Ye(^& R V sin 12A sin 1

2B sin 1

2CS�U�(T%�& �$S�%�%lS�U�c+&Ç'$]^&['b&+%bjT_p'�#T# R 'lS�YZU

cot 12A cot 1

2B cot 1

2C = cot 1

2A + cot 1

2B + cot 1

2C

� YMU�"b&cot 1

2A�cot 1

2B�BS�U�(

cot 12CS�%�&9&+S�"0]ÈW!#�_$Y{'�Y f�&�� n &iS�W�W�dgV�'$]^&"!�#%$'&(#quU�& ) � S�d{Y{'uVi'b#Tc+&�'

cot 12A cot 1

2B cot 1

2C ≥ 3 3

cot 12A cot 1

2B cot 1

2C�

n ]�Ye"0]iYMjTW^d{Ye&+_m'$]�SQ'3

cot 12A cot 1

2B cot 1

2C ≥

√3 *u�,+

- VTS�U�#�'$]^&�%ºS�W^W^d{Ye"�S+'�YM#�U�#�`!'$]^&.!�#%$'&(# quU�&/) � S+dZYg';VB� n &�]�S�f�&(

cos 12A + cos 1

2B + cos 1

2C) (

csc 12A + csc 1

2B + csc 1

2C)

−(

cot 12A + cot 1

2B + cot 1

2C)

=cos 1

2A + cos 1

2C

sin 1

2B

+cos 1

2B + cos 1

2C

sin 1

2A

+cos 1

2A + cos 1

2B

sin 1

2C

≥ 63

(

cos 1

2A cos 1

2B cos 1

2C

sin 1

2A sin 1

2B sin 1

2C

)2

= 6 3

cot 12A cot 1

2B cot 1

2C ≥ 6

√3�

Page 55: WholeIssue 33 3 - CMS-SMC

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}�H23�CG3�}�3�H@5��@3!A<7938; Ì � Í EG?�?2JBE�5�F�: E*H23!1K5�7*E�FK:<3�$@A<561�F�E�= a � $BA<5�1DFDE{=�H2?B1K;27>F�fO5�:<;?@:@1�=!F�fcot 1

2A = cot 1

2B = cot 1

2Ca �6F�: d 3

cot tFK7�F�:�Ö{3 d EGF �@3�?@: (

0, π2

) ¡8}�3�79383E*H@56E�3�$@A<561�F�E�=�H2?B1K;27�FDft58:<;M?@:@1�=!F�f 4ABCFK7#3�$@A2FD1K5�E�36C�5�1 a

'D� ��� ��������� 8 ��'E3&0>'E-�� '���9�4()�"$� ���$���*4/F$'D4�? )�'2=()�F5C,9�9 �:3DC� �+E42'E��8����O� ?5��<1�$��� �N<7�1� 4� �$�5� ���$G ���!� 4 '�� 4P?:)*'&=(F#%16�$ F&?&3(9>6L) 4 � < �������$� <:)������1##���O� )�� "�� � ?5��<1�$��� �N<7�1� 4,��� !�� < ��4 �.C:4P?:)*'&='�$:C&)#9>C ��67) 0�K(-/�+� 67)�4 B�"��&��" ��*<7� %/< ��# )�M�#5����� 4,'E� #5�����N4(/5���$�$MN� =���C(% + /DK(%��:?DB ��354 /5� "$�,��$�� "�< � <L�*4#'���=D0&���,'� E'E6 ��' ?#4D%��$��� 02�$����4(F0#�< ��" =,B ��-�? ��'H+�K ��?23(9 ��4�-/������� 4 $�'&4P? )�'2=�$ � 9>��3�2K AC,C:4.��<7��� " ?5��<1�$��� �N<7�1� 4 ��"�� < � "�>"�41F$'D4�?:)*'&= ��6(@>%�CP?54 $#��� F#�$GJG ����<7�1� F#��� �7�)���*4 P< �#� ���%P"��$����4 �+� 4,?:)*'&=#"$�,J�1##�,!#��� !:�$������K

�*"��5�����&�&##���P��1�1#�"��5� #5�&!#��� !:�$����E< �5�$;���"�� <L� �,< �2"E�O!:�$ML< " ��ML"$���E� ��G �$���������#��� "��$< �5�$;���"�� <L� <L���� �$� � �$�#M�� <L�$���

f4

g : (0, π) → � +��"�� < � �1��< �0� ∏

� ��� � � � f(A) ≥ λ"$�, �*<L� #5��� ∏

� ��� � � � g(A) ≥ µ�$�

� ��� � � � f(A)g(A) ≥ νKD67� � #5� ��� ���2M "����*4

(

� ��� � � � f(A)

)(

� ��� � � � g(A)

)

−∑

� ��� � � � f(A)g(A) ≥ 6(λµ)1/3 �"$�,�< � � #5�E���$MN�$�,JM "����*4

(

� ��� � � � f(A)

)(

� ��� � � � g(A)

)

−∑

� ��� � � � f(A)g(A) ≥ 6ν1/3 !67� 8���� # M "������N4:�$;���"�� <L� �.##���>� � �$���$;�� < � "��L��� "��#� � < "��0� �L����K � "$�#���$�D"�� ��� �#��� ����1#�"��:� #5�,"�!�!�� <7M "�� <7�$� � �< �#�$;��$" � <7� <7��� ��K �1� 4 ��K �� 4 ��K ;.��4 ��K �1L/< �>� 7�: � <L�*�1>�(G �$���������5��� " ��< �#�$;��$" � <7� <7���E� �2�1##��� �$�OG

(

� ��� � � �(

cos(A/2))p

)(

� ��� � � �(

csc(A/2))p

)

−∑

� ��� � � �(

cot(A/2))p ≥ 6 · 3p/2

� �$�D"$�+�P!:�$�N<L� <���/����" ��� ��G 8����pK

� %� %��%��)$%�� 7�: ?"!BA$* &D&���� � �C& �.��! � /2�$�$G �$� � <LM�6 �5�$;���"�� <L� <L��� �E *�� &(�.'D�6FCG *�* ' 5 *�H 3 0 �1��! ��I ' *1� � ��J � � ��7�8�K8 !

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jÈ&�&Q'l_Ã'$]^&�`r#^"�S�dIS�k�YM_�S+'N

P &Q'H

S�U�(K R &�W�#�YMU�'l_�#�U MF

_ � "0]\'$]�S+'HN ⊥ MF

S�U�(KN ⊥ MN

qr` 1

HN=

1

2

(

1

a+

1

b

) S�U�(NK =

√ab

* n ]^&�%�& a > b > 0+b�F_O]^# n '$]�SQ'

KI = (a + b)/2`r#�%m_>#�jÈ&Ã_$YecQU�Y��B"�S�U�'ºW!#�YZU�'

I#�U

MN

� s /�t @ <M:;/ J 6�7 � W N W s yX4�4 J�A�G w D tz<l6^/�y�yX4Ot G <r=B4���4�<r=�4�.lt D�J 5�3+ � U�&W!#�_0_0Y R dM&W�#�_0Yg'�Ye#�UT`r#�% I

YM_p'$]^&W!#�YZU�' n ]^&+%�&�'$]^&�d{YMU�& MNYZU�'b&+%l_O&�"�'�_'$]^&�dZYZU�&�W!&+%bW�&�U�(�Ye" � dZS�%h'b#

MKSQ'

K��'$]^&�%�YMc+]�'h'b%�YZS�U�c�dM&�_

HNKS�U�(

NKIS�%�&_0YZjTY{dMS�%O n ]^&+U�"$&�� IK/KN = KN/NH

��S�U�(

KI =

√ab ·

√ab

2ab/(a + b)=

a + b

2

� � 5 W�� � &�_O]�S+dZd�_O&�& R &+dM# n '$]�SQ'a'$]^&�W�#�YMU�'

IYM_IjÈ#�%�&��O_0YMcQU�Y��!"bS�U�'��m'$]�S�Ui"�S�U

R &_>&�&+UT`e%�#�j � jÈ&�&�U���� _�_>#�d � '�YM#�Ua Fqe'mYZ_Q��YZUT`eS�"�'O��'$]^&"$&+U�'b%�&¡#�`a" � %uf�SQ' � %�&i#�`B'$]^&"$#�U�YM"�S+'º'$]^&�W!#�YZU�'M

*{'$]�S+'mYM_���'$]^&¡"b&�U�'b%�&¡#�`!'$]^&¡"bYM%�"bdM& n ]^#�_O&" � %uf�S+' � %�&�YZ_�'$]^&_>S�jÈ&ÃS�_m'$]^&�"$#�U�YM"�� _�S+'I'$]^&QYM%º_O]�S�%�&�(9W!#�YZU�'M

+0 *

Page 56: WholeIssue 33 3 - CMS-SMC

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nÊn a µ ��� � �!� ��#�º���� � � � ����� � ��º���# � � � � � � � ���³ � � � ���4½ � � � � ´ �"�����_¸ # �A¹ � � � ��� � À��· � � ��� � ��� � � ¸ µ �NaÒ 3�5�797*A<L 3SE*H@56E�E*H23 d ?@:@F d FK7�E*H23Q361D1�F�I<793

x2/α2 + y2/β2 = 1 �E*H23£fp? d A<7

FF�7>E*H23!Im?BF�:BE

(c, 0)¡<}�H23�CG3

c795�EGFK7*�¢367

c2 + β2 = α2 a!Ó ?@C#7,?@L�3£�B5�1DAm3M?Bf t¡

0 < t ≤ π/2¡<1K3�E

MJ<3�E*H23�Im?BF�:BE

(α cos t, β sin t)a#� 7)F�:<�£7*E�58:<;@58CG; d 561 d A21DA<7�¡?@:<3�;23�EG3�CGLMF�:<3�7�E*H23�C�5�;BF�A<7#?Bf d A<C{�B56EGA<CG3£5�E

ME�?MJ23

k3/(αβ)¡8}�H236CG3

k2 = α2 sin2 t + β2 cos2 ta

� H23�7*14?@I<3�?BfOE*H23!EG5�:<�63�:BEN56EM

FK7 −β cos t/α sin t¡/7,? E*H@5�E�E*H23�7*14?@I<3�?BfOE*H23:<?@C�LM5�1>E*H23�CG3�LMA<7�E~J23

α sin t/β cos ta¾� H236CG36fp?@CG3B¡

MNH@587 E*H23�3�$@A<5�EGF4?@:

y − β sin t =α sin t

β cos t(x − α cos t)

¡2?@C(β cos t)y − (α sin t)x + c2 sin t cos t = 0

a� HBF�7�1DFK:<3�L 3�3�E�7uE*H23

xÙ 5WÉ8F�7�56E N

(

c2 cos t

α, 0

)

�}�H23�: d 3@¡

MN2 = β2 sin2 t +

(

β2 cos t

α

)2

=β2k2

α2

a Ì � Í�6FK: d 3SE*H23�3�$@A<5�EGF4?@:�?Bf<E*H23�1DFK:<3

MFF�7

y = −(x − c)β sin t/(c − α cos t)¡�E*H23;@F�7*E�58: d 3�f4CG?@L

NEG?

MFF�7

NH =

βc2 sin t cos t

a− βc sin t

β2 sin2 t + (c − α cos t)2

a� H23£7 $@A<5�CG38�*CG?2?�EUE�36C�LËFK:!E*H23£;<36:<?@LMF�:256EG?@Cu7*FKLMI21DF��m3�7�5�7�fp?B1�1K?8}£7 �β2 sin2 t + (c − α cos t)2 = β2 sin2 t + c2 − 2αc cos t + α2 cos2 t

= β2 sin2 t + c2 − 2αc cos t + (β2 + c2) cos2 t

= α2 − 2αc cos t + c2 cos2 t = (α − c cos t)2 �E*HBA<7�¡

NH = βc sin t/α¡25�:<;

MH2 = MN2 − NH2 =β2k2

α2− β2c2 sin2 t

α2

=β2(

α2 sin2 t + (α2 − c2) cos2 t)

− β2c2 sin2 t

α2

=β2(α2 − c2 cos2 t − c2 sin2 t)

α2=

β4

α2

a�£36: d 3B¡

MH = β2/αÌ � Í

Ì }�HBF d H¢¡@F�: d F4;23�:BE�5�1D1�=¢¡BF�7#J2?�E*HQE*H23�1�56EGA<7�CG3 d EGA<LÕ?Bf<E*H23�3�1�1DFKI27,3S58:<;QE*H23�C�5�;BF�A<7#?Bfd A<C{�B56EGA<CG3£56E(α, 0)

Í)a

Page 57: WholeIssue 33 3 - CMS-SMC

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Ó CG?@L E*H23�7*FKL�F�1�58C�C�F4�6H�EÈE�CGF�58:<�\1K3�7MNK

5�:<;MHN

¡�}�3�7,3�3cE*H@5�EMK/MN = MN/MH

aO� ?@L�J@F�:@FK:<�#E*HBFK7uI2CG?@Im?@CÊEGF4?@:£}�F�E*H!3�$@A<5�EGF4?@:27 Ì � Í 5�:<;Ì � Í ¡�}�3£79383NE*H@5�EMK =

MN2

MH=

k2

α

aÓ F�:25�1D1�=¢¡Bf4CG?@LËE*H23�7)F�LMFD1K5�C�C�F4�6H�EUE�CGF�58:<�\1K3�7

IMK58:<;

NMH¡

IM =MN · MK

MH=

k3

αβ

¡7,?�E*H@56E

IMFK7uE*H23�C�5�;BF�A<7#?Bf d A<C{�B56EGA<CG3£5�E

M¡B587 d 1�5�F�L 3�; a

F#�$G�!��L�$�L�1����� ��� <7�$���1M "�G � � ���$G '�$:C&)#9>C ��67) 0�K:-/�+� 67)�4(B�"$�&�*"� ��*<L� %E< � # )�M�#5����� 4:',�1##�����N4/5���$�$MN� =#"���J�1##�,!#��� !:�$������K

9 ##������� ��� <7�$����� ���#"�� "�< � �7��"����-/��GA< �A������ " �����#��� "�� � �$�OG � �$�/" MO�$��<7M � #�"��&!#���0���MN�����1##�MN���#�7��� � �2M ���&��"��O������� �$�D"��J"�� 8$<L� � "$�!��MO�$��<7M�� �#"�� "�< � �7�O� ��< �/8>"�����J�$� � #5�E� "�� �$�(���$MN�O��G ��#�< �7�1-/��GA< �$�< �/8>"$����J�$� � #5�1�$MOMN���#�7� <7M <7�1��K A� #�"��$�,!#���������#� �� A���Q� �,����� ��� <7�$� � �$��� � #5��� ��#A<L�5< � ��" � <1J�$��� � � �$�"$� �*� � < !#���*4 8��$M "��$���,<L�,#�"��/�1##� �*< � � ��� � �2�$�N< �0��� "�GA< � < "�� � �$�OG � � "���KE9 #5� MN�$�O�����O!:�$�,$< ����� �$�OG ��� "$� � �$��1##� # ��!:��� 8���� " "$�,J�1##�,!#"$� "�8���� "1GJ"��A8��1��8�� "�< �5��J��"��N< � ��K

9 #�< �2!#����8$�L��G�!#������<1����D"P�N< G�!��7�PMN�$�����7�*��M�� <L�$� � �$���1##� MO���5� ���P� �:M��$�&�*"�� �$���P� �#"�MN�$��<LM:"��#"$�+�!:��< �#�

M�1#�"���< �&�5���:�$�J�1##�,GJ" N�$�&" �$< ���0����< �

M�L���1##� � ��ML<

F"���

F ′ = ML"�� �N

�1##��!:��< �#� ��##�����/�1##�8$< ���$MN� �$�2� �

∠FMF ′ G �$�$� �FF ′ K � 9 #�< � < �D�1##���#�$�OGJ"���� � � #5�,MN�$��<LM5"��

M=�< ���1##�EML"$���E� ��"(!#"�� "�8���� "�4

MF ′ < �P� " ���� �L� 8���� #5�/� < �5���1#������ � #M

�1#�"�� < ��!#"$� " � �7�*�2�L� �1##� " �$< ��K �/-/�����5�K

� �J8���� #5�/!:��< �5���#5�����/�1##��!:��� !:���,$<LM���� "$�D� �

MN"��

NG �$�$� �

MF= � #5���

I< �,� #5��!:��< �5� ��#5�����/� #5��!:��� !:���,$<LM���� "$�

� �MF

"��K

G �$�$� �(� #5�,�#�$�OGJ"��MN

KDF��L��"�� � � 4 �1##�,�5�$G 8���� �a

"$�,b

�1#�"���" !�!:��"��:< �J�1##��!#����8$�L��G"$���(" � ���5MN� <7�$� � �

M= � #5�"�E"$���/�$��� �/< ��$< ���$M�� � �E���*� "�� ��1� ��� #5���L������� #��(� � � #5�(����GA<�' GJ" N�$�:"$�, ����GA<�'

GA< �#�$��" �����P� #�"�� "$������� �$" � � � ����5���L��J8 �A� #5�����E�7�$� �L��� ��KE9 #5�/�#��� "�� <7�$� ���$��GJ�E� � #�"��$�18��$��� M�##�$�����8 � �#"�� " < � �L� � �$�&<L� �(��<LMO�/���*� "�� <L�$���&#�< !��L�A�1##�1MN�$�����7�*��M�� <L�$� � �2� #5�(!:��< �#�

IK

����[Ç [�l�������T����������������* ,!.0/+12/�3�4�5È6�7��³:e= D tr7 Z 4 J ����4 G�Z . D 3�/ }�G�Y /�y D�J : D P &Q'

F (n) R &�'$]^&�U � j R &+%�#�`^#�U�&+_�YZUÃ'$]^& R YZU^S�%uV&$k�W^%�&�_0_0YM#�U#�`�'$]^&�W!#�_$Y{'�Y f�&YMU�'�&�c+&+%n \ #�%m&0k�S�j�W�de&��

F (5) = F(

101(2)

)

= 2�

F (15) = F(

1111(2)

)

= 4

P &Q'Sk =

∞∑

n=1

F k(n)

n(n + 1)

� n ]^&+%�& F k(n)YM_F(�& �!U�&�(�%�&�" � %l_0Y f�&+d V R V F 1 = F

S�U�(

F k = F ◦ F k−1`r#�%

k ≥ 2

*MS�+ � %�#�f�&['$]�SQ'S1 = 2 ln 2

* R + � %�#�f�&['$]�SQ' 18

5ln 2 − 1

15≤ S2 ≤ 4 ln 2

*r" + � %�#�f�&['$]�SQ' 218

25ln 2 − 7

25≤ S3 ≤ 11 ln 2

*r( +

? � #�jTW � '�&Sk

� � 5 � quU�'$]�YZ_aW^%�# R dM&�jX��'$]^&�&$k�W^%�&�_0_0YM#�U F k(n)

j�&�S�U^_(

F (n))k�Q%lSQ'$]^&�%�'$]�S�U

'$]^&k �

� Y{'�&�%lSQ'b&�#�`F��S�_�_$'lS+'�&�(9S R #+f�&� *

Page 58: WholeIssue 33 3 - CMS-SMC

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µ ����� �!�"��# �"� � ��� � Ì 5 Í �� � ����� � � �¢¾ ��# � � � � ¸ � ����� �!# � #!Ä� � # ��� � � � �~Å # # � ��� � � � �� ����� � ���2aÓ ?@CS5�:�=ÈIm?@7*FDEGF �@3MFK:BEG38�636C

n¡U;<36:<?�E�3ME*H23~JBFK:25�C{=�CG36I<CG367,36:BEG5�EGF4?@:r?Bf

nJ�=

∞∑

j=0

zj(n)2j a�� H23�: F (n) =∞∑

j=0

zj(n)a�� ?8}

S1 =∞∑

n=1

F (n)

n(n + 1)=

∞∑

n=1

∞∑

j=0

zj(n)

n(n + 1)=

∞∑

j=0

∞∑

n=1

zj(n)

n(n + 1)

=∞∑

j=0

∞∑

ν=0

2j−1∑

u=0

1

(ν2j+1 + 2j + u)(ν2j+1 + 2j + u + 1)

=∞∑

j=0

∞∑

ν=0

(

1

ν2j+1 + 2j− 1

ν2j+1 + 2j + 2j

)

=∞∑

j=0

∞∑

ν=0

(

1

(2ν + 1)2j− 1

(2ν + 2)2j

)

=

∞∑

j=0

1

2j

∞∑

ν=0

(

1

2ν + 1− 1

2ν + 2

)

=

∞∑

j=0

1

2j

∞∑

n=1

(−1)n−1

n= 2 ln 2

aµ ����� �!�"��#��"� � ��� � � Ì J Í ��#�� Ì d�Í �� � � � � � � � � � � �Wa

Ò 3£H@5��@3F (2m) = F (m)

58:<;F (2m + 1) = 1 + F (2m) = 1 + F (m)

aØ 3�Eak ≤ Sk ≤ bk

a�� H236:Sk =

∞∑

n=1

F k(n)

n(n + 1)=

∞∑

m=0

(

1 + F (m))k

(2m + 1)(2m + 2)+

∞∑

m=1

F k(m)

2m(2m + 1)

=

∞∑

m=0

1

(2m + 1)(2m + 2)+

∞∑

m=0

k−1∑

p=1

(

kp

)

F p(m)

(2m + 1)(2m + 2)

+

∞∑

m=0

F k(m)

(2m + 1)(2m + 2)+

∞∑

m=1

F k(2m)

2m(2m + 1)

= ln 2 +∞∑

m=1

F k(m)

2(2m + 1)

(

1

m + 1+

1

m

)

+

k−1∑

p=1

(

kp

)

F p(m)

(2m + 1)(2m + 2)

= ln 2 +1

2Sk +

∞∑

m=1

k−1∑

p=1

(

kp

)

F p(m)

(2m + 1)(2m + 2)

a�£36: d 3B¡

Sk = 2 ln 2 + 2

k−1∑

p=1

(

k

p

) ∞∑

m=1

F p(m)

(2m + 1)(2m + 2)

a Ì � Í

Page 59: WholeIssue 33 3 - CMS-SMC

�� �

�6FK: d 3 1

(2m + 1)(2m + 2)≤ 1

4m(m + 1)

¡Bf4CG?@L Ì � Í }�3£H@5��@3

Sk ≤ 2 ln 2 +1

2

k−1∑

p=1

(

k

p

) ∞∑

m=1

F p(m)

m(m + 1)≤ 2 ln 2 +

1

2

k−1∑

p=1

(

k

p

)

bpa

Ó CG?@L�I<5�C�E Ì 5 Í ¡�}�3£H@5��@3b1 = 2 ln 2

58:<;¢¡@fp?@Ck = 2

¡�}�3 d ?@: d 1�Am;23SE*H@56ES2 ≤ 2 ln 2 +

1

2

(

2

1

)

b1 = 4 ln 2 �

E*HBA<7�¡b2 = 4 ln 2

auÓ ?@Ck = 3

¡8}�3£H@5\�@3S3 ≤ 2 ln 2 +

1

2

[(

3

1

)

+

(

3

2

)]

= 11 ln 2¡

}�HBF d H�L 3658:27uE*H@5�Eb3 = 11 ln 2

a� :�E*H23Q?�E*H236C�H@5�:<;t¡ 1

(2m + 1)(2m + 2)≥ 1

5m(m + 1)

fp?@Cm ≥ 2

¡25�:<;f4CG?@L Ì � Í }�3£H@5��@3

Sk ≥ 2 ln 2 +2

5

k−1∑

p=1

(

k

p

) ∞∑

m=2

F p(m)

m(m + 1)+

1

6

k−1∑

p=1

(

k

p

)

≥ 2 ln 2 +2

5

(

k−1∑

p=1

(

k

p

) ∞∑

m=2

F p(m)

m(m + 1)− 1

2

k−1∑

p=1

(

k

p

)

)

+2(2k − 2)

12

≥ 2 ln 2 − 2k − 2

30+

2

5

k−1∑

p=1

(

k

p

)

apa

Ó ?@Ck = 2

¡8}�3 d ?@: d 1DAm;<3Sf4CG?@L I258CÊE Ì 5 Í E*H@56ES2 ≥ 2 ln 2 − 1

15+

2

5

(

2

1

)

a1 =18

5ln 2 − 1

15

a�6FK: d 3

a2 = 185

ln 2 − 115

¡8}�3£H@5��@3S3 ≥ 2 ln 2 − 1

5+

2

5

[(

3

1

)

a1 +

(

3

2

)

a2

]

≥ 118

25ln 2 − 7

25

a$�"$� � � ���D"�� ���J�����1�$�� 8 �.�$C�� �A)�F#%(��CD)��:��3�/#4��#" ���N<��*4D+(��4/?:)*'&=:"$�, �1##�P!#��� !:�$������K($�"$� �

� ��2����GJ"�< ���E� !:���:K

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ABC R &ÃS�U�V¡'b%�YZS�U�c�dM&� � ]^# n '$]�S+'∑

� ����� � �

1 + 2 sin A − cos 2A

8 + 3 cos(

A

2

)

cos(

B − C

2

)

+ cos(

3A

2

)

cos

(

3(B − C)

2

) ≤ 1

Page 60: WholeIssue 33 3 - CMS-SMC

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� � � � � � � � � ��º � � � � � � � � � # �!�"� ��� ������� �!�"��# ���� · � � � � �¤´�� ����� � � � � Ã�� � � # � � � ��# � ���� # � � � � � � � � � � � �WaÒ 3��<C�7*E�L�58�83SE*H23Sfp?B1�1K?8}�FK:<�S?2J2793�C{�B5�EGF4?@:27��

1 + 2 sin A − cos 2A = 2 sin A + 2 sin2 A¡ Ì � Í

2 cos(

A

2

)

cos(

B − C

2

)

= cos(

A + B − C

2

)

+ cos(

A + C − B

2

)

= cos(

π − 2C

2

)

+ cos(

π − 2B

2

)

= sin C + sin B¡ Ì � Í

2 cos(

3A

2

)

cos

(

3(B − C)

2

)

= cos

(

3(A + B − C)

2

)

+ cos

(

3(A + C − B)

2

)

= cos

(

3(π − 2C)

2

)

+ cos

(

3(π − 2B)

2

)

= − sin 3C − sin 3Ba Ì � Í

� 7)F�:<� Ì � Í ¡ Ì � Í ¡B58:<;!E*H23Sfp?@C�L�A21K5sin 3x = 3 sin x − 4 sin3 x

¡�}�3£�63\E8 + 3 cos

(

A

2

)

cos(

B − C

2

)

+ cos(

3A

2

)

cos

(

3(B − C)

2

)

= 8 + 32(sin B + sin C) − 1

2(sin 3B + sin 3C)

= 2(4 + sin3 B + sin3 C) ≥ 2(3 + sin3 A + sin3 B + sin3 C)a Ì � Í

Ø 3�EL;23�:<?�EG3£E*H23�143�f�E�7)FK;<3Q?Bf�E*H23�F�:<3�$BA<5�1DFDE{=�F�:�E*H23QI2CG?2J@1K3�L 7�EG5�E�36L 36:BE a� 7)F�:<� Ì � Í 58:<; Ì � Í ¡8}�3£?2JBE�5�F�:

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� �% ! 2 sin A + 2 sin2 A

2(3 + sin3 A + sin3 B + sin3 C)=

� �% ! (sin A + sin2 A)

∑ � �% ! (1 + sin3 A)

a

�£36: d 3B¡8F�E�7*A � d 3�7�E�?£79H2?8}iE*H@5�E ∑ � �% ! (sin A+sin2 A) ≤ ∑ � �% ! (1+sin3 A)¡�}�HBF d H

FK7#3�$@A2F �B5�1K3�:BE�FK:M7*A d*d 3�7)7)FK?@:!EG?∑

� �% ! (1 + sin3 A − sin A − sin2 A) ≥ 0¡

� �% ! (1 + sin A)(1 − sin A)2 ≥ 0a

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λ = (90 + 69√

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A = B = C = π3

K

Page 61: WholeIssue 33 3 - CMS-SMC

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¡�CG3679I<3 d EGF �@361 =�143\E

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OA¡OB

¡¢58:<;OC

¡¢CG3679I<3 d EGF �@361 = a Ï 3�:<?�EG3�E*H23 d FKC d A<LMC�5�;BF�A<7\¡¢F�:2C�5�;BF�A<7\¡t5�:<;7,36LMF�Im36CGF�L 3\E�36C�?Bf 4ABCJ�=

R¡r¡@58:<;

s¡@CG3�7)Im3 d EGF��@3�1�= a 0<CG?��@3NE*H@56E

2(KD + ME + NF ) ≥ R + 3r +s2 + r2

2R

aµ ����� �!�"��# �� · � � � � �p´�� ����� � � � � à � � � # ��� � � #�� � � � ��� � � � ���� � � ��� � � � ���Wa

Ó F�C�7*E,¡�}�3£I2CG?��@3SE*H@5�EKD2

R2=

1

4+ cos2 A + cos A cos(B − C)

a Ì � ÍÒ 3 d 58:M5�797*A<L 3NE*H@56E

BFK7 d 1K?@7,36COE�?

AE*H@5�:!E�?

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� f∠A

FK7�5�C�F4�6H�E!5�:<�\143B¡�E*H236:�E*H23ÈI<?BFK:BE�7D5�:<;

Od ?BFK: d FK;<3�5�:<; Ì � Í F�73�$BA2F��B561436:BE�EG?

R = 2KO¡8}�HBF d H!F�7 d 143658C�1�=QE�C�Am3 a

� f∠A

F�7�5 d A@EG3@¡�E*H236:∠KOD = ∠AOB + ∠BOD

= 2C + A

= π − (B − C)

58:<;OD = R cos A

ai� H23 Ø 5\} ?Bf � ?@7)F�:<3�758I2I21DF43�;�FK: 4KOD=@FK361K;27 Ì � Í9a

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A

B CDK

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Ó CG?@L Ì � Í ¡8}�3£�63�EKD2

R2=

[

cos A +1

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]2

+1

4sin2(B − C)

≥[

cos A +1

2cos(B − C)

]2 ¡58:<;!E*H236CG36fp?@CG3B¡

KD ≥ R∣

∣cos A + 12cos(B − C)

∣ ≥ R[

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] a

Page 62: WholeIssue 33 3 - CMS-SMC

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�6FKL�F�1�58C�F�:<3�$BA<5�1DFDEGFK3�7uH2?B14;£fp?@CME

5�:<;NF

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� �% ! cos A = 1 +r

R

¡∑

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4R2

¡58:<; ∑

� �% ! sin B sin C =r2 + s2 + 4Rr

4R2

¡}�3�?2J�EG56FK:

2(KD + ME + NF )

≥ 2R

[

� �% ! cos A + 12

� �% ! cos(B − C)

]

= R

[

2

(

1 +r

R

)

+r2 + s2 − 4R2

4R2+

r2 + s2 + 4Rr

4R2

]

= R + 3r +s2 + r2

2R

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x ∈�

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H(a, b, c)YM_�U�#�U �>&+j�W�';VÈYZ`2S�U�(X#�U�dgVÈY{`

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Page 63: WholeIssue 33 3 - CMS-SMC

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b < 158:<;

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pJ<3>58:�=QCG365�1@:@A<L J23�C7)A d HQE*H@56E

ap2 − (1 − b)p + c ≤ 0a�Ó ?@Cu3)É�5�L�I@143B¡ d H2?2?@793

p = (1 − b)/(2a)F�f

a > 058:<;

p = c/(1 − b)F�f

a = 0a/� H236:QE*H23NfKA<: d EGFK?@:

h(x) = px795�EGFK7*�¢367UE*H23fKA<: d EGF4?@:2561mFK:<3�$@A<561�F�E�=!FK:!E*H23�I<CG?2JB1436L a�� HBA<7\¡

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x ≥ 0a�� H23�:

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h(

h(ax))

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YM_C′A · C′B = C′F ·

C′C = C′N2 n ]^&+U�"$&��

C′N

C′A=

C′B

C′N=

C′B − C′N

C′N − C′A

Page 64: WholeIssue 33 3 - CMS-SMC

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� HBA<7\¡(C′N/C′A) · (C′B/C′N) = (C′B − C′N)2/(C′N − C′A)2

¡�?@C Ì FK:E�36C�LM7�?Bf�?@C�F436:BE�3�;�1DFK:<3�7,3���L 36:BEG7 ÍBC′

C′A= −BN2

NA2

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CA′

A′B= −CL2

LB2

5�:<; AB′

B′C= −AM2

MC2

a� H236CG36fp?@CG3B¡

AB′

B′C· CA′

A′B· BC′

C′A= −AM2

MC2· CL2

LB2· BN2

NA2= −1

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