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A
BBN
B
CD
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A1
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� ?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¡
����
}�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
��� �
� 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
¡w} EG?�E*H23�E*H@CG3�3 d ?@LMIm?2�:<3�:BE�7U5�:<;SE*H23,=�LMA<7�E�J<3�;BF��BF4;238;
2¡2¡�58:<;
2a �@5 d H�7*A d H�I<56FKC/?Bf23�;<�6367O7,36I<5�C�56EG3�7E*H23Q��C�58I<H a�� f¢E*H23�CG3�5�CG3�EÊ}�? d ?@LMIm?@:<36:BEG7
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
J<3S5�C�H2?@L JBA<7�}�F�E*H∠A < 90◦ a�Ø 3�EOF�EG7UEÊ}�?�;BFK58�6?@:25�1�7 AC
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�:<;
Q¡
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@¡�$BA<5�;@CGFD1K5�E�36C�5�1
BOPXF�7 d = d 1�F d�a� ?@:27*F4;23�CtE*H23#F�:��@3�C�7*F4?@:
I?Bf@I<?B143
O58:<;Im?8}�36C
OB2 ar� H236:�E*H23 d F�C d 143 d FKC d A<L��7 d C�F4JBFK:<�BOPX
LM58I27UEG?�E*H23>1�F�:<3AB
a� HBA<7\¡I(P ) = A
¡/}�HBF d H�FKLMI21DF4367�E*H@56EOP · OA = OB2 a
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A
B C
D
MO
P
QX
Y
�6FKL�F�1�58C�1�=¢¡8}�3£?2JBE�5�F�:OQ · OC = OB2
a�� H236CG36fp?@CG3B¡OQ
OP=
OA
OC=
MA + MO
MA − MO=
MA
MO+ 1
MA
MO− 1
=t + 1
t − 1
a
Ð a �6A<I2Im?@793>E*H@5�E�E*H23NF�: d FKC d 1K3�?Bf 4ABCF�7UEG5�:<�63�:BE�E�?£E*H23N7)FK;<367
AB¡BC
¡CA56EuI<?BFK:BE�7
P¡Q¡R¡BCG3�7)Im3 d EGF��@3�1�= a 0<CG?��@3NE*H23�fp?B1�1K?8}�FK:<�>F�:<3�$BA<5�1DFDE{= �
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?@:
a¡b¡cfp?@C�E*H23>7*F4;23�7�?BfBE�C�FK5�:<�\143
ABC¡85�:<;
sfp?@CUE*H23�7,36LMF�Im36CGF�L 3\E�36C a
��� �
Ò 3�H@5\�@3PQ = 2(s − b) sin
B
2= 2(s − b)
√
(s − c)(s − a)
ca
¡
QR = 2(s − c) sinC
2= 2(s − c)
√
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ab
¡
RP = 2(s − a) sinA
2= 2(s − a)
√
(s − b)(s − c)
bc
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A
B
B′
C
C′
DE F
Ø 3�EB′ 58:<; C′ J<3�E*H23�CG3 �¢3 d EGF4?@:27>?Bf B 58:<;
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BC′ 5�:<;B′C
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Ø 3�EOJ<3�E*H23 d FKC d A<L d 36:BE�CG3N?Bf 4ABC
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J<3D1
5�:<;E1¡@CG3679I<3 d EGF �@361 = a
���#
Ò 3�H@5\�@3OD1 = O2I2 = 1
2(c − a) = 1
2AD58:<;
OE1 = O3I3 = 12(b − a) = 1
2AE
ab 1K79?¢¡
∠D1OE1 = ∠DAEa �£36: d 3B¡
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A
B C
D
D1
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∠BRCa
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SJ23uE*H23�7,3 d ?@:<;NI<?BFK:BEt?Bf8F�:BE�36C�7,3 d EGF4?@:�?Bf6E*H23 d FKC d A<L d FKC d 1K3�7�?Bf6E�CGF�58:<�\1K3�7
BPR58:<;
RCAa
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A
B C
PQR
S
�����
�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
FK7SFK79?@7 d 361K3�7}�FDE*HBS = SA
¡BFDEUfp?B1�1K?8}£7uE*H@5�E 4CPSF�7�FK79?@7 d 361K3�7U}�FDE*H
PS = SCa�b 1K79?¢¡
PS
BS=
CP
AB=
12AB
AB=
1
2
¡L�58�8F�:<�
∠SBP = 30◦ F�:MCGFK�6H�EUE�CGF�58:<�\1K3 PBSa�� H236CG36fp?@CG3B¡
∠BRC = ∠BRS + ∠SRC = ∠BRS + ∠SRP
= ∠BRS + ∠SBP = 90◦ + 30◦ = 120◦ aÚ aMÓ FK:<; E*H23M��CG3656EG3�7�E>CG365�1�:@A<L�J<36C
k7*A d H E*H@56E,¡¢fp?@C�58:�=�I<?@7)F�EGF��@3
a¡b¡c}�F�E*H
a2 > bc¡
(a2 − bc)2 > k(b2 − ca)(c2 − ab)a
µ ���I¹ � � �� · ��� � � � � � � ��� � � ��� � µ � � ��� � � ���HÄ �J� � ��# � ��� · �"� � � �}´�� �!� � � � � � à � � � # �� � � #�� � � � # ��³ � � � � � ´ ��� # � �� � ��# �G· � � ����# �(¼�À���� � � � � � � ��# � � ½ � � Ä �I¹ � ´�� ����� � � � ¿ �������� �!�"��#4½
� H23���CG3656EG3�7�EkF�7
4aÓ F�C�7*E�7*A<I<I<?@7,3eE*H@5�E
(a2 − bc)2 > k(b2 − ca)(c2 − ab)}�H23�:<3,�@36C
a¡b¡c > 0
5�:<;a2 > bc
a�Ø 3�Et ∈ (0, 1)
a �6FK: d 312 > t · t
¡8}�3£H@5\�@3(1 − t2)2 > k(t2 − t)(t2 − t)
¡f4CG?@L }�HBF d H�}�3£;238;BA d 3SE*H@56E
(
1 + t
t
)2
> ka
� Eufp?B1D14?8}£7�E*H@5�Ek ≤ lim
t→1
(
1 + t
t
)2
= 4a
� ?8}À}�3Q}�FD1�1�7,H2?8} E*H@5�E(a2 − bc)2 > 4(b2 − ca)(c2 − ab)
}�H23�:<3,�@36Ca¡b¡c > 0
5�:<;a2 > bc
a�b 797*A<L 3£?@:!E*H23 d ?@:BE�C�58C{=QE*H@56E(a2 − bc)2 ≤ 4(b2 − ca)(c2 − ab)
Ì � Í
� � �
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
xa�b�d EGA<561�1 =t¡E*H23�fp?@CGL 36C#H2?B1K;27�7*FK: d 3
f(1) = a2 + b2 + c2 − ab − bc − ca > 0Ì :<?�EG3£E*H@5�E
a = b = cFK7#3*É d 1�Am;238;�J�=
a2 > bc¡@5�:<;M7,?
a2 + b2 + c2 > ab + bc + caÍ)a
� E�fp?B1�1K?8}£7uE*H@5�Eb2 − ca
FK7�I<?@7)F�EGF��@3 a�� ?8}�¡8}£C�FDEG3f(x) = (bx − c)2 − ag(x) − x(a2 − bc)
¡}�H23�CG3
g(x) = cx2 − 2ax + ba �6FK: d 3
a2 − bc > 05�:<;
g
(
c
b
)
=c(c2 − ab) + b(b2 − ac)
b2> 0
Ì 7*FK: d 3c2 − ab
H@5�7£E*H23~7)58L�3~7*F4��:�5�7b2 − ca
J�= Ì � ÍpÍ ¡/}�3�H@5\�@3f(
c
b
)
< 0¡
5 d ?@:BE�C�5�;BF d EGFK?@: a�� HBF�7 d ?@L�I@143\E�367�E*H23�I<CG?2?Bf a
� 3*É�E�}�3S1K?2?2��5�Eu79?B1�A@EGFK?@:27�E�?!I2CG?2J@1K3�LM7#?BfmE*H23Q�NA<:<��58C�FK5�:��r56E*H236L�5�EGF d 5�1� 1�=@L�I@FK58; ������� Ù ����� � Ó FKC�7�E � ?BA<:<;��\F �@3�:M56E � ����� � � ����� aÇ a�b CG3 d EG5�:<�\A21K5�C�J2C�F d �QH@587��@?B1�A<L�3
V = xd L 3 ¡@5�:<;M7*A<CGf45 d 3£5�CG3�5 S = y
d L 2 aÓ F�:<;!E*H23£L�FK:@F�L�561@�@?B1�A<L�3�fp?@C�}�HBF d Hx = 10y
aµ ����� �!�"��# �� � ��� ��� #�� � # � ´ ��� � � � � � ��� � � #�� � ½Ø 3�E
a¡
b¡u58:<;
cJ<3 E*H23�;@F�L 36:27)FK?@:27�?Bf�E*H23�J@CGF d � a�� H236:
x = abc5�:<;
y = 2a2 + 2b2 + 2c2 a Î =!E*H23 b � Ù M�� � :<3�$BA<5�1DFDE{=t¡8}�3£H@5��@3
x2/3 = (a2b2c2)1/3 ≤ a2 + b2 + c2
3=
y
6
aÒ H23�:
x = 10y¡2}�3��63\E
x2/3 ≤ x
60�E*H@56E�FK7\¡
x ≥ 603 = 216000a ��?@CG3�?��@36C,¡
}�H23�:a = b = c = 60
}�3MH@5��@3x = 10y = 216000
a �£36: d 3B¡tE*H23ML�FK:@F�L�561�@?B1�A<L�3�fp?@C�}�HBF d Hx = 10y
FK7216000
aÚ aNØ 3�E
ABCJ<3�5�E�CGF�58:<�\1K3 auÒ 3�;2CG?@I 5QI<3�CGIm36:<;@F d A21K5�C�f4CG?@L
AE�?!E*H23£F�:BE�36C�:2561J@F�7,3 d E�?@C�7�7�EG5�C�EGF�:<�!f4CG?@L
B58:<;
C¡¢E*H236F�C>fp3�3�ENJ236F�:<�
A15�:<;
A2a � :�E*H23M795�L 3}£5�=�}�3£;<3��m:<3
B1¡B2
5�:<;C1¡C2a 0<CG?��@3NE*H@56E
2(A1A2 + B1B2 + C1C2) = AB + BC + CAa
� � �
µ ����� �!�"��# �� · � � � � �p´�� ����� � � � � à � � � # ��� � � #�� � ½Ø 3�EIJ<3SE*H23SFK: d 3�:BE�CG3�?Bf
∆ABCa�� H23�:t¡
∠BAA1 =π
2− B
2=
A + C
2>
A
2= ∠BAI
¡∠CAA2 =
π
2− C
2=
A + B
2>
A
2= ∠CAI
a� HBA<7\¡
A15�:<;
A25�CG3�?@:�?@I2Im?@7*FDEG3�7)FK;<367#?BfmE*H23�JBFK793 d EG?@C
AIa ��?@CG38?6�@3�C9¡
∠IAA1 =π
2− ∠AIA1 =
π
2−(
A
2+
B
2
)
=C
2
Ì 7*FK: d 3∠AIA1 = π − ∠AIB
Í 58:<;M7*FKL�F�1�58C�1�=¢¡∠IAA2 =
B
2
a � E�fp?B1D14?8}£7�E*H@5�E
IA1 = AI sin
(
C
2
) 58:<;IA2 = AI sin
(
B
2
) a Ì � ÍÒ FDE*HQE*H23Sf458L�F�1DFK5�C�:<?�EG5�EGF4?@:27
a = BC¡b = CA
¡B58:<;c = AB
¡8}�3�561K79?MH@5\�@3AA1 = c sin
(
B
2
) 58:<;AA2 = b sin
(
C
2
) a Ì � Í� ?8}�¡U?2J2793�C{�@3ME*H@56E
A¡
A2¡I¡O5�:<;
A15�CG3 Ì F�:�E*HBFK7Q?@CG;23�C Í ?@:�E*H23 d F�C d 143�}�F�E*H;@F�58L�3�EG3�C
AIa 0@EG?B1436L!= � 7 � H23�?@CG3�L �\F��@367
A1A2 ·AI = IA2 ·AA1+IA1 ·AA2¡f4CG?@L }�HBF d H¢¡BA<7*FK:<� Ì � Í 5�:<; Ì � Í ¡8}�3£�63�E
A1A2 = c sin2
(
B
2
)
+ b sin2
(
C
2
) a� H236CG3£58CG3�58:25614?2�6?BA<7�CG3�7*A21DE�7�fp?@C
B1B258:<;
C1C2a�� ?8} }�3£H@5��@3
2(A1A2 + B1B2 + C1C2)
= 2(a + b) sin2
(
C
2
)
+ 2(b + c) sin2
(
A
2
)
+ 2(c + a) sin2
(
B
2
)
= (a + b)(1 − cos C) + (b + c)(1 − cos A) + (c + a)(1 cos B)
= 2(a + b + c) −[
(c cos B + b cos C)
+ (a cos C + c cos A) + (a cos B + b cos A)]
= 2(a + b + c) − (a + b + c) = a + b + c = AB + BC + CAa
� 3*É�E�}�3�14?2?2��56E>7,?B1DA@EGF4?@:27Nfp?@C#E*H23~�SA<:<��5�CGF�58:��r56E*H236L�5�EGF d 5�1 � 1 =2LMI2F�5�;�����B� Ù ������� ¡ Ó F�:25�1 � ?BA<:<;��\F��@36:M56E � ����� ��� ���8� aà a&Ò 3 d ?B14?BA<C�E*H23��@36C�EGF d 367 ?Bf>5
2003�)�6?@:�}�FDE*H CG38;¢¡>JB1�Am3B¡�5�:<;i��CG3�3�:�7)A d HE*H@56E£:<3�F4�6H2J2?BA<C�7 d 58:2:<?�E�H@5\�@3 E*H23~7)58L�3 d ?B1K?BA<C a � :�H2?8} LM58:�=�}£5�=@7 d 5�:�}�35 d�d ?@LMI21DFK79HQE*HBFK7 ×
� � �
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2003 Ù �6?@: a � HBFK7£��C�58I<H�F�7SE*H23 d = d 1K3 C2003¡58:<;!E*H23�I<CG?2JB1436L�FK7uEG?M;23�EG3�CGLMF�:<3SE*H23£:@A<L�J<36C�?BfmFDE�7�I<CG?@I<3�C d ?B14?BA<C�FK:<��7 a
� E£F�7Q5 d 1�587)7)F d 561�3*É83�C d F�7,3 Ì � � 8¡ � � Í EG?�I<CG?6�@3 E*H@5�E,¡uL�?@CG3��63�:<36C�5�1D1�=¢¡UE*H23:@A<L J23�C�?BftI<CG?@I<3�C d ?B1K?BA<CGF�:<��7�?Bf Cn}�FDE*H
qd ?B14?BA<C�7�F�7
PCn(q) = (q − 1)n + (−1)n(q − 1)Ì d 5�1D143�;NE*H23 d H@CG?@LM56EGF d Im?B1 =2:<?@L�FK561�?Bf
CnÍ9a,� :�I258CÊEGF d A21�58C9¡
PC2003(3) = 22003−2
a����'H��$���%!���(7� � Ø2aÈØ ?����587 �W¡ � � � ����# � �"��� ����� � � � ��� � � � ��#�� � � � � ��� � � � ¡ � ?@CÊE*H2�W��?B1�1�58:<;¢¡3*É83�C d F�7,3 � Ù �8� a� � �*a�� ?@L�3�7 d A�¡ ³ � � ��� � � � �!# � � � ����# � �"��� � � � � # � Ä�� � � � � � � ��� ¡ Ò F�1K3,=t¡3*É83�C d F�7,3 ��� Ù � � aÚ a�Ø 3\E
tJ<3S5N�BÉ83�;!Im?@7*FDEGF �@3>FK:BEG38�636C a�Ø 3\E
ft(n);23�:<?�EG3>E*H23S:@A<L J23�C�?Bf<FK:BEG38�636C�7
k7)A d H�E*H@5�E
1 ≤ k ≤ n5�:<; (k
t
) FK7u?2;2; a Ì � f1 ≤ k < t
¡6E*H23�: (kt
)
= 0a Í 0<CG?��@3
E*H@56E�FDfnF�7S5M7)A � d F436:BEG1�=���CG3�5�E>Im?8}�36C>?Bf
2¡¢E*H236: ft(n)
n=
1
2r
¡m}�H236CG3rFK7S5�:
FK:BEG38�636C�;<3\E�36C�L�FK:<3�;�J�=t58:<;�F�:<;<36Im36:<;<36:BE�?Bf
na
µ ����� �!�"��# ��~³ � � � � � ´ ��� # � �� � ��# ��· ���!���# ��¼�À�� � � � � ��� � � #�� � ½Ø 3�E
n = 2p }�FDE*H 2p > 2ta�Ø 3\E
t =p∑
i=0
ti2i ¡\}�H23�CG3 ti ∈ {0
¡1} fp?@CO365 d H i
a� ?�EG3�E*H@5�E�E*HBF�7UFK7OE*H23SJBFK:25�C{=!3)É8I<5�:27)FK?@:!?Bf
t3)É d 3�IBE�E*H@5�E
tid 58:!J<3 �)3�CG?£fp?@C�7,?@L�3
i = p¡p − 1
¡. . .
a£Ø 3\ErJ23£E*H23Q:@A<L�J<36C�?Bf
i7)A d H�E*H@5�E
ti = 1a#� H23�:
rF�7#E*H23:@A<L J23�C�?Bf
17�FK:SE*H23>JBFK:25�C{=�3)É8I<5�:27)FK?@:�?Bf
t �E*HBA<7\¡
rFK7 d 1K3�5�CG1 =�F�:<;<36Im36:<;<36:BEO?Bf
na
Ó ?@C1 ≤ k ≤ n
¡<143\Ek =
p∑
i=0
ki2i ¡@}�H236CG3 ki ∈ {0
¡1} fp?@C�3�5 d H i
a£� 3 d 561�1Ø A d 5�7#� � H238?@CG36L Ì 7,3�3 � � Í ¡8}�HBF d H�7*E�56EG3�7uE*H@5�E
(
k
t
)
≡p∏
i=0
(
ki
ti
)
(mod 2)a
� HBA<7\¡ (kt
) FK7N?2;2; FDf�58:<;~?@:@1�=�F�fki ≥ ti
fp?@C�365 d Hia>� H236CG36fp?@CG3B¡ (k
t
) FK7N?2;2; FDf58:<;�?@:@1�=~F�f
ki = 1}�H236:
ti = 1¡�58:<;
kid 58:�J<3M36F�E*H23�C
0?@C
1?�E*H23�C�}�FK793 a�� Efp?B1�1K?8}£7Nf4CG?@L¾E*H23�;236�<:@FDEGFK?@:È?Bf
rE*H@5�E#E*H23�CG3M58CG3M3)�5 d EG1�=
2p−r F�:BE�3��63�C�7 k7)A d H
E*H@56E1 ≤ k ≤ n
5�:<; (kt
) FK7#?2;<; Ì :<?�EG3SE*H@56Er ≥ 1
7)F�: d 3t > 0
¡�}�HBF d H�36:27)A<CG367E*H@56Eu5�E�1436587�E�?@:<3£?BfmE*H23
ki7�FK7�:<?@:<� �)3�CG?¢¡B7,?�E*H@56E
k ≥ 1Í9a�� H236:
ft(n)
n=
2p−r
2p=
1
2r
a
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A
B CDE F
G
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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
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02�$�#� �M ���� MN�$GJG ���#� �� �1#�"���#5� ����!:�$M��L���"�� ��<7MN������"$�����P����"$�, "$�& ������#�"�� � #5���*"��L��� � �tan θ
�P��� � #�"0��� � ��8��P� �.���$�HK = 2
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� ]�S�%lS�"�'�&�%lY��0&['$]^#�_O&['b%�YZS�U�c�dM&�_pYZU n ]�Ye"0] AQ + BP = AB
�� ��
µ ����� �!�"��#���uà ��u´���� ����� � � ¸ # �A¹ � � � ��� ��ºK´ � � � � � � ´ � � � � � � À � ����# ��# � � ��� ��� � � ��� � ��� � ���"���Wa
� H23#E�C�FK5�:<�\14367OFK:S}�HBF d HAQ+BP = AB
58CG3�I<CG3 d FK79361 =�E*H2?@793�}�FDE*H£5>C�F4�6H�E58:<�\1K3£56ECaÏ 3�:<?�EG3SJ�=
3α¡3β¡�5�:<;
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A¡B¡�58:<;
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RE*H23 d F�C d A<L�C�58;@FDA<7�?Bf 4ABC
a�� 3 d 5�1D1<E*H@5�E�fp?@Cu5�:�=MCG365�1m:@A<L J23�Cθ¡
sin 3θ = sin θ(
3 − 4 sin2 θ)
= sin θ(1 + 2 cos 2θ)
= 4 sin θ sin(
π
3− θ
)
sin(
π
3+ θ
) a� H2367,3SF4;23�:BEGF�EGF4367�}�F�1D1¢J<3SA<7,3�;Q}�F�E*H2?BA@E d ?@LML 36:BE aÒ 3�H@5\�@3
AB
sin γ=
2R sin 3γ
sin γ= 2R(1 + 2 cos 2γ)
a Ì � Í� H23 Ø 5\} ?Bf��6F�:<3�7�FK: 4AQC
�\F��@367AQ
sin γ=
AC
sin(α + γ)=
AC
sin(
π3
− β) =
2R sin 3β
sin(
π3
− β)
= 8R sin β sin(
π3
+ β)
= 4R(
cos π3
− cos(
π3
+ 2β)
)
= 2R(
1 − 2 cos(
π3
+ 2β)
) a Ì � Í�6FKL�F�1�58C�1�=¢¡
BP
sin γ= 2R
(
1 − 2 cos(
π3
+ 2α)
) a Ì � Í� 7)F�:<� Ì � Í ¡ Ì � Í ¡@58:<; Ì � Í ¡8}�3�79383NE*H@56E
AQ + BP = ABF�f¢58:<;�?@:@1 =�FDf
1 − 2 cos(
π3
+ 2β)
+ 1 − 2 cos(
π3
+ 2α)
= 1 + 2 cos 2γ �E*H@56E�F�7�¡cos 2γ − 1
2= − cos
(
π3
+ 2α)
− cos(
π3
+ 2β) a Ì � Í
� ?8} }�3�CG3W}£CGF�E�3£J<?�E*H�7)FK;<367#?Bf Ì � Í �cos 2γ − 1
2= cos 2γ + cos 2π
3= 2 cos
(
π3
− γ)
cos(
π3
+ γ) ¡
− cos(
π3
+ 2α)
− cos(
π3
+ 2β)
= −2 cos(α − β) cos(
π3
+ α + β)
= −2 cos(α − β) cos(
2π3
− γ)
= 2 cos(α − β) cos(
π3
+ γ) a
� 7)F�:<�>E*H23�793�3*É�I2CG3�7)7)FK?@:27�FK: Ì � Í ¡8}�3�7,3�3NE*H@56EAQ + BP = AB
F�ft5�:<;�?@:@1 =�FDfcos(
π3
− γ)
cos(
π3
+ γ)
= cos(α − β) cos(
π3
+ γ) a Ì � Í
� f�E*H236CG3�F�7!5�C�F4�6H�E£58:<�\1K3�56ECÌ E*H@56E£F�7�¡UFDf
γ = π6
Í ¡OE*H23�: J2?�E*Hr7*F4;23�7!?Bf3�$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?
�� ��
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
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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
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� 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�#�'$]^&�%ºS�W^W^d{Ye"�S+'�YM#�U�#�`!'$]^&.!�#%$'&(# quU�&/) � S+dZYg';VB� n &�]�S�f�&(
cos 12A + cos 1
2B + cos 1
2C) (
csc 12A + csc 1
2B + csc 1
2C)
−(
cot 12A + cot 1
2B + cot 1
2C)
=cos 1
2A + cos 1
2C
sin 1
2B
+cos 1
2B + cos 1
2C
sin 1
2A
+cos 1
2A + cos 1
2B
sin 1
2C
≥ 63
√
(
cos 1
2A cos 1
2B cos 1
2C
sin 1
2A sin 1
2B sin 1
2C
)2
= 6 3
√
cot 12A cot 1
2B cot 1
2C ≥ 6
√3�
�� ��
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2A = cot 1
2B = cot 1
2Ca �6F�: d 3
cot tFK7�F�:�Ö{3 d EGF �@3�?@: (
0, π2
) ¡8}�3�79383E*H@56E�3�$@A<561�F�E�=�H2?B1K;27�FDft58:<;M?@:@1�=!F�f 4ABCFK7#3�$@A2FD1K5�E�36C�5�1 a
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f4
g : (0, π) → � +��"�� < � �1��< �0� ∏
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∏
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∑
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)(
∑
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(
∑
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)(
∑
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)
−∑
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(
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)(
∑
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csc(A/2))p
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−∑
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qr` 1
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MKSQ'
K��'$]^&�%�YMc+]�'h'b%�YZS�U�c�dM&�_
HNKS�U�(
NKIS�%�&_0YZjTY{dMS�%O n ]^&+U�"$&�� IK/KN = KN/NH
��S�U�(
KI =
√ab ·
√ab
2ab/(a + b)=
a + b
2
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IYM_IjÈ#�%�&��O_0YMcQU�Y��!"bS�U�'��m'$]�S�Ui"�S�U
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+0 *
�� ��
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 $@A<5�CG38�*CG?2?�EUE�36C�LËFK:!E*H23£;<36:<?@LMF�:256EG?@Cu7*FKLMI21DF��m3�7�5�7�fp?B1�1K?8}£7 �β2 sin2 t + (c − α cos t)2 = β2 sin2 t + c2 − 2αc cos t + α2 cos2 t
= β2 sin2 t + c2 − 2αc cos t + (β2 + c2) cos2 t
= α2 − 2αc cos t + c2 cos2 t = (α − c cos t)2 �E*HBA<7�¡
NH = βc sin t/α¡25�:<;
MH2 = MN2 − NH2 =β2k2
α2− β2c2 sin2 t
α2
=β2(
α2 sin2 t + (α2 − c2) cos2 t)
− β2c2 sin2 t
α2
=β2(α2 − c2 cos2 t − c2 sin2 t)
α2=
β4
α2
a�£36: d 3B¡
MH = β2/αÌ � Í
Ì }�HBF d H¢¡@F�: d F4;23�:BE�5�1D1�=¢¡BF�7#J2?�E*HQE*H23�1�56EGA<7�CG3 d EGA<LÕ?Bf<E*H23�3�1�1DFKI27,3S58:<;QE*H23�C�5�;BF�A<7#?Bfd A<C{�B56EGA<CG3£56E(α, 0)
Í)a
�� ��
Ó CG?@L E*H23�7*FKL�F�1�58C�C�F4�6H�EÈE�CGF�58:<�\1K3�7MNK
5�:<;MHN
¡�}�3�7,3�3cE*H@5�EMK/MN = MN/MH
aO� ?@L�J@F�:@FK:<�#E*HBFK7uI2CG?@Im?@CÊEGF4?@:£}�F�E*H!3�$@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� �
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�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<�'
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IK
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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�&� *
�� �
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
�� �
�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¡
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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
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� ��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
�� �
� � � � � � � � � ��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
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∑
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∑ � �% ! (1 + sin3 A)
a
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∑
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A = B = C = π3
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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
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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
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A
B CD
K
O
� f∠A
F�7#?2JBEGA<793@¡8E*H23�:∠KOD = ∠BOD − ∠AOB
= (π − A) − 2C
= B − C
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atb ��5�F�:E*H23 Ø 5�}Â?Bf � ?@7*FK:<367�F�: 4KOD=@FK361K;27 Ì � Í9a
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............................
A
B CDK
O
Ó CG?@L Ì � Í ¡8}�3£�63�EKD2
R2=
[
cos A +1
2cos(B − C)
]2
+1
4sin2(B − C)
≥[
cos A +1
2cos(B − C)
]2 ¡58:<;!E*H236CG36fp?@CG3B¡
KD ≥ R∣
∣cos A + 12cos(B − C)
∣
∣ ≥ R[
cos A + 12cos(B − C)
] a
�8���
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5�:<;NF
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� �% ! cos A = 1 +r
R
¡∑
� �% ! cos B cos C =r2 + s2 − 4R2
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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C′N
C′A=
C′B
C′N=
C′B − C′N
C′N − C′A
�8���
� 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
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AB′
B′C· CA′
A′B· BC′
C′A= −AM2
MC2· CL2
LB2· BN2
NA2= −1
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K = AP ∩ BQ¡R = PB′ ∩ QA′ ¡@5�:<;
C = B′A∩A′B5�CG3 d ?B1�1DFK:<3658C a6� HBA<7\¡
KFK7/E*H23 d ?@L�L�?@:�Im?BF�:BEO?Bf�E*H23�1DFK:<367
CR¡
BQ¡@58:<;
AP¡@587#;23�7*FKCG3�; a
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BQ ∩ ACaNb 1�7,?¢¡
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