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....................................................................................................................................................................................
A(a, 0)
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k = −A′′C
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a=
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� ¬ º ±F¬ � ©o²�©+° · ª© µ f(x)
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z − e2kπi2n+1
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0
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n=0
(−1)n
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x3
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1
x3
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x3
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e−w sec w dw
= e−w sec w tan w −∫
tan w d(e−w) −∫
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i+j=0
(−1)0(
2i
i
)(
2j
j
)
+∑
i+j=1
(−1)1(
2i
i
)(
2j
j
)
+ · · ·
+∑
i+j=2n
(−1)2n
(
2i
i
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2j
j
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x0 l x1 l x2 l . . .lx2n [@e=b�gPS�VP`dWPwFoDjdb?WY^ ∞
∑
k=0
(−1)k
(
2kk
)
xk �&[cb�g�[cbMq{S)X@^d�J|([%e�j,S
1√
1 + 4x· 1
√1 + 4x
=1
1 + 4x=
∞∑
k=0
(−1)k(4x)k l��S�j,WFe�j,X@oDwPSfb�gFU(b
Bn = 1 − 4 + 42 − · · · + 42n = 15
(
42n+1 + 1) �
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[D0D1D2] + [E0E1E2] + [F0F1F2]
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µ °�²V « ©�ª�¬;©3» º ²��
B ·Lº °Lª$ ·L¶DD1
µ °�²DD2
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∠D1DD2 = ∠UDV = Ax
�+º » º F(µ`¸ F ´ � ∠EID = 180◦ − C�
· ª�¬ µ ª∠UID = C
x � ª ¶C·/F F8· � ª�¬ µ ª∠DID1 = 2C
� µ °�² ª�¬ µ ª∠DD2D1 = C
x � · °;�©`®L¯'©<°Lª F ´ �ª ¸$º(µ °�� F ©<ABC
µ °�²DD1D2
µ`¸ © º » º F�µ<¸ x� ¬;© ¸,µ ª º8· ·L¶ º » º F(µ`¸,º ª ´ º @ª�¬;©¸,µ ª º�·3·L¶ ª�¬;© º�¸ ± º�¸ ±¯�» ¸,µ ² º(º�� · ª�¬ µ ª
[DD1D2] =(
r
R
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[ABC]x
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A
B CD
D0
D1
D2
E
F
IU
V
�)���
�*ePU�X%WPZ(WYo�qGX ~�l���S�W ° bdU([%e
[EE1E2] = [FF1F2] =(
r
R
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[ABC]�
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1
2
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4r2 cos B cos C)
sin A =1
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[E0E1E2]U6e�w
[F0F1F2]� � bI^_WYXcX3W6��qIb�gFU(b
[D0D1D2] + [E0E1E2] + [F0F1F2]
=1
2(sin 2B + sin 2C + sin 2A) = 2r2 sin A sin B sin C
= 2r2(
a
2R
) (
b
2R
) (
c
2R
)
=r2
R2[ABC]
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�¯'±F¬¦ª�¬ µ ªf(0) = 1
� �¬ · � ª�¬ µ ª
2
∫ 1
0
x2f(x) dx +1
12≤[∫ 1
0
f(x) dx
]2 x
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x � ©o²�©�³'°�©
Mp =
∫ 1
0
xpf(x) dxµ °�²
M0 =
∫ 1
0
f(x) dx�
µ °�²3�¬ · � ª�¬ µ ªp + 2
2Mp +
2pf(0) − (p + 1)
4(p + 1)≤ M2
0
�
�)�Fu
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(
x − 12
)
+ 12
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�� eYb,S�Z)`MU(bd[@e�Z ° ~EVPU6`5bdq!l6��S�Z(S!b
Mp =
[
xp
∫ x
0
f(t) dt
]1
0
−∫ 1
0
[
pxp−1
∫ x
0
f(t) dt
]
dx
= M0 − p
∫ 1
0
∫ x
0
xp−1f(t) dt dx�
|([%e�j,Sf[%qTj�WFe�jdU)sFS WFe
[0l
1]lA��S gFU)sFS
f(t) ≥ f(x) − f(0)
x − 0t + f(0)
^_WF`0 ≤ t ≤ x ≤ 1
�4��S(e�j�SYl
Mp ≤ M0 − p
∫ 1
0
∫ x
0
(
xp−2[f(x) − f(0)] t + f(0)xp−1)
dt dx
= M0 − p
2
∫ 1
0
xp [f(x) + f(0)] dx
= M0 − p
2Mp − pf(0)
2(p + 1)
�±�gPS(`dS(^_WF`dSYlp + 2
2Mp +
2pf(0) − (p + 1)
4(p + 1)≤ M0 − 1
4= M2
0 −(
M0 − 1
2
)2
≤ M20
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f[@q�U
X@[@e�S�U�`x^%o�e�jdbd[%WFe¤U�e�wM0 = 1
2
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f(x) = m(
x − 12
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+ 12
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0
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0
f(x) dx − 1
3
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ln 2− 1
x H�¸$·GD ©ª�¬ µ ª
(
2a
b + c
)p
+
(
2b
c + a
)p
+
(
2c
a + b
)p
≥ 3x
�)�F�
£F���5KP0%">��/T�62E0_<9�����G�(�+�6$!�)�dª+� ����-�/�7���7=$��@"@Q6<�0%�_2;��2;0_<�����76"50>���®8¯ S)b
x =2a
b + c
ly =
2b
c + a
lPU6e�wz =
2c
a + b
�*±9gPS�ex ≥ 0
ly ≥ 0
lz ≥ 0
lU6e�w=b�gPS�Z![]sFS(eE[@e�S6¡Yo�U�Xc[cb>~ ° S6j,WFayS(q
xp + yp + zp ≥ 3 À �®Áo�e�w�S(`Ib�gPSf^_WYX@X%W6�&[%e�ZAU�wPwF[]bd[3WFePU(X9j�WFePq,b,`MU�[@eYb®�
1
x + 2+
1
y + 2+
1
z + 2= 1
� À �®Á
¯ S)bq =
ln 3
ln 2− 1 ≈ 0.585
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xp + yp + zp
3
)1p
≥(
xq + yq + zq
3
)1q �
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� À �®ÁR�[cb�gPWYoFb�X%WFqrq;WY^�Z(S�e�S(`MU�Xc[cb>~�lL��S�a.U!~�U6qGqGo�a.S�b�gFU)b
a = min{al
bl
c} �±�gPS(ex =
2a
b + c≤ 1
U6e�wyz =
4bc
(c + a)(a + b)≥ 4bc
(2c)(2b)= 1
�?²AW�bdSfb�gFU)b À �®Áj,U�e ° S�W ° bdU([%e�S�w ° ~EU6w�wY[%e�Zxb�gPS&^_WYXcX3W6�&[@e�Zxb5��W�[%e�S�¡Fo�U(X@[]bd[3S(q(�
xq + 2
(
2
x + 1
)q
≥ 3l À ��Á
yq + zq ≥ 2
(
2
x + 1
)q � À �®ÁRTS=�&[@XcXLVP`dW�sFS À ��ÁCo�e�wPS�`xb�gPSyj,WFePq�b,`MU([%eYb 0 ≤ x ≤ 1
U6e�w À �®ÁCo�e�wPS�`xb�gPSj�WFePq,b,`MU�[@eYbdqyz ≥ 1
U�e�w À �®Ár�+±�gY[@q*�&[cX@XDq�o���j,SfbdW;V�`dW(sFS À �!Ár�±�W;VP`dW�sFS À �6Á�lPj,WFePqG[%w�S(`Lb�gPSf^%o�e�jdbd[3WFe
f(x) = xq + 2
(
2
x + 1
)q l^_WF`
0 ≤ x ≤ 1�J±9gPS�e
f ′(x) = qxq−1 − q
(
2
x + 1
)q+1 �²AW6�;lY^_WF`
0 < x ≤ 1lPwPS(��e�S
g(x) = (q − 1) ln x − (q + 1) ln
(
2
x + 1
) �
� ��t
±�gPS(ef ′(x)
U�e�wg(x)
gFU)sFSfb�gPS&qrU�ayS&q�[3Z)e.WFe(0, 1]
�L|([%e�j,S
g′(x) =q − 1
x+ (q + 1)
(
1
x + 1
)
=2qx + q − 1
x(x + 1)
l��S�gFU)sFS
g′(x) = 0^_WF`
x = x0 = (1 − q)/(2q) < 1�*p6o�`Mb�gPS(`,a.WF`dSFl
g′(x) < 0^_WF`x ∈ (0, x0)
l6U�e�wg′(x) > 0
^_WF`x ∈ (x0, 1)
�H��S(e�j�SYlg[%qLq�b,`M[3jdbdX ~�wPS6j�`dS(U6q�[%e�Z
WFe(0, x0]
U�e�wEq,b,`d[%jdbdX]~=[@e�j�`dS�U�qG[@e�Z&WFe[x0, 1]
�|([%e�j,S
g(1) = 0U6e�w
limx→0+
g(x) = +∞ l�[]b�^_WYX@X%W6��q=b�gFU(b�b�gPS�`dSiS��6[@q�bMqx1 ∈ (0, x0)
qGoDjGg.b�gFU)bg(x1) = 0
��p�o�`5b�gPS�`dayWF`dSYlg(x) > 0
^_WF`x ∈ (0, x1)
lU6e�w
g(x) < 0^_WF`
x ∈ (x1, 1)�4��S(e�j�SYl
f ′(x1) = 0lf ′(x) > 0
^_WF`x ∈ (0, x1)
lU6e�w
f ′(x) < 0^_WF`
x ∈ (x1, 1)�f±9gPS�`dS)^_WF`dSFl
f[%qxq�b,`M[3jdbdX ~y[@e�j�`dS�U�qG[@e�Z=WFe
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f(1) = 3l?��S
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WFe[0, 1]
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x + 1 =yz + 2y + 2z + 4
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p = ql+b�gPS�e
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À U!Á 1
2 − cos A+
1
2 − cos B+
1
2 − cos C≥ 2
v
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5 − cos A+
1
5 − cos B+
1
5 − cos C≤ 2
3
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R
l À �®Á
∑
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4R2
l À �®Á
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16Rr − 5r2 ≤ s2 ≤ 4R2 + 4Rr + 3r2 � À ��Á
À U!ÁfhioPX]bd[%VFX]~Y[%e�Z ° W�b�gTq�[3wPS�q&WY^Jb�gPS.Z![]sFS(eT[@e�S6¡Yo�U�Xc[cb>~ ° ~ ∏
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l��S�W ° bdU([%e=b�gPS�S6¡YoP[]sYU(X3S(eYb*[@e�S6¡Yo�U�Xc[cb>~
2∏
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s2 ≤ 4R2 + 8Rr − 5r2 �� e;b�gPSfX@[%Z(g�b?WY^9[%e�S�¡Fo�U(X@[]b�~ À ��Á�lY[cbIqGo��yj�S(q?bdW;q{gPW6� b�gFU(b
4R2 + 4Rr + 3r2 ≤ 4R2 + 8Rr − 5r2 �k�oFbLb�gY[%q?[%qCS�¡FoP[ sYU�X%S�eYbLbdW
2r ≤ Rl���gY[3jGg=[%q*b�gPSx��S(XcX3ÃF��e�W6��ey�YoPX%S�` � e�S6¡Yo�U�Xc[cb>~n�
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72Rr − 9r2 ≤ 5s2 �
� �6�
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2r ≤ R�
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� X � 1
λ − cos A+
1
λ − cos B+
1
λ − cos C≥ µ 1 � & 2 ≤ µ < 6
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µ + 6
2µ-
� 0 � 1
λ − cos A+
1
λ − cos B+
1
λ − cos C≤ µ 1 � & 0 < µ ≤ 2
3
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µ + 6
2µ-
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