24
? ( ) 4M 1 M 2 M 3 O 1 O 2 O 3 O 4G 1 G 2 G 3 O P (4G 1 G 2 G 3 ) P (4M 1 M 2 M 3 )+ P (4O 1 O 2 O 3 ) P 4O 1 O 2 O 3 . . . . q . . . G 1 G 2 G 3 M 1 M 2 M 3 O O 1 O 2 O 3 r O r i O i i =1 2 3 r 2 = r 3 =2 r 1 r 1 = r> 0 g = P (4G 1 G 2 G 3 ) m = P (4M 1 M 2 M 3 ) o = P (4O 1 O 2 O 3 ) o =8+2r r 4 g o m g<m + o

M O G O P G P 4M )+ P 4O P 4O - Canadian Mathematical Society · b gPS&^%o e jdbd[3WFe f(x) = xlnx [%qCj,WFesFSG Pl6 rSePqrSe q

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G1

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0

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x3

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0

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n − 1 ≤√

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3

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� À �®Á|([%e�j,S

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3

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(1 + x + y)(1 + x)(1 + y)

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2xy(2 + x + y)

(1 + x + y)(1 + x)(1 + y)

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1 − x

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(1 + x + y)(1 + x)(1 + y)

=xy(1 + x + y) − (1 − x − y)xy

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2xy(x + y)

(1 + x + y)(1 + x)(1 + y)

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2(x + y)

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2

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1

9

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n ≥ 3l9��S.gFU!sFS

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3<

4

5

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Sn =n∑

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1 + a1 + a2

+n∑

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2

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i

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j

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i

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j

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x0 l x1 l x2 l . . .lx2n [@e=b�gPS�VP`dWPwFoDjdb?WY^ ∞

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(−1)k

(

2kk

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xk �&[cb�g�[cbMq{S)X@^d�J|([%e�j,S

1√

1 + 4x· 1

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∞∑

k=0

(−1)k(4x)k l��S�j,WFe�j,X@oDwPSfb�gFU(b

Bn = 1 − 4 + 42 − · · · + 42n = 15

(

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[D0D1D2] + [E0E1E2] + [F0F1F2]

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� µ °�² ª�¬ µ ª∠DD2D1 = C

x � · °;­�©`®L¯'©<°Lª F ´ �ª ¸$º(µ °�� F ©<­ABC

µ °�²DD1D2

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r

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B CD

D0

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D2

E

F

IU

V

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�*ePU�X%WPZ(WYo�qGX ~�l���S�W ° bdU([%e

[EE1E2] = [FF1F2] =(

r

R

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[ABC]�

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1

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�|([%a;[@X@U6`I`dS(X@U(bd[%WFePqCgPWYX3w=^_WF`

[E0E1E2]U6e�w

[F0F1F2]� � bI^_WYXcX3W6��qIb�gFU(b

[D0D1D2] + [E0E1E2] + [F0F1F2]

=1

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= 2r2(

a

2R

) (

b

2R

) (

c

2R

)

=r2

R2[ABC]

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R ≥ 2rV�`dW(sF[%w�S(q?b�gPS�wPS�q�[%`dS�w;[@e�S6¡Yo�U�Xc[cb>~n�K � %,�N%,��5����� 6 Dk� K���2F.�!�p b K(' - �<m1 � �8% 7 � � ��,���D�94���%C� 7 9 1 = ��% 6 � 7 !(M 1�K>7 %,"5��� �$G�n !<22! p

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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

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+ 12

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�� eYb,S�Z)`MU(bd[@e�Z ° ~EVPU6`5bdq!l6��S�Z(S!b

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[

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)

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4= M2

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M0 − 1

2

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≤ M20

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f[@q�U

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2

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f(x) = m(

x − 12

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+ 12

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2

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0

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0

f(x) dx − 1

3

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¨<©�ªa�b�c « ©o° · °��F°�© � µ ª º D © ¸ © µ6F °#¯�» « © ¸ ­ ��µ °�² F © ª p ≥ ln 3

ln 2− 1

x H�¸$·GD ©ª�¬ µ ª

(

2a

b + c

)p

+

(

2b

c + a

)p

+

(

2c

a + b

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≥ 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

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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

[0, x1]U6e�wEq,b,`d[%jdbdX]~EwPS6j�`dS(U6q�[%e�Z&WFe[x1, 1]

�|([%e�j,S

f(0) = 2q+1 = 2ln 3/ ln 2 = 2log2 3 = 3U6e�w

f(1) = 3l?��S

j�WFe�j,XcoDw�Sfb�gFU)bf(x) ≥ 3

WFe[0, 1]

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2

)1q ≥

(√

y +√

z

2

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yz ≥ 1l

(√y +

√z

2

)2

≥ 2

x + 1

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1

x + 2= 1 − y + z + 4

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yz + y + z

yz + 2y + 2z + 4

v��gPS�e�j,SFl

x + 1 =yz + 2y + 2z + 4

yz + y + z− 1 =

y + z + 4

yz + y + z

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(y + z + 2√

yz)(y + z + 4) ≥ 8(yz + y + z)l

(y + z)2 + 2(y + z)(√

yz − 2) + 8√

yz − 8yz ≥ 0l

(y + z − 2√

yz)(y + z + 4√

yz − 4) ≥ 0l

(√

y −√

z)2(y + z + 4√

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a = b = c� � e�U�wPwF[]bd[3WFe�lJ[c^

p = ql+b�gPS�e

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c[@q*\GS�`dW=U6e�w=b�gPS�W�b�gPS�`Jb5��W;U�`dS�S�¡Fo�U(XM�

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À U!Á 1

2 − cos A+

1

2 − cos B+

1

2 − cos C≥ 2

v

À3° Á 1

5 − cos A+

1

5 − cos B+

1

5 − cos C≤ 2

3

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��� � cos A =

s2 − 4R2 − 4Rr − r2

4R2

l À �®Á

��� � cos A =

R + r

R

l À �®Á

��� � cos B cos C =

r2 + s2 − 4R2

4R2

l À �®Á

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16Rr − 5r2 ≤ s2 ≤ 4R2 + 4Rr + 3r2 � À ��Á

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��� � (2 − cos A)

l��S�W ° bdU([%e=b�gPS�S6¡YoP[]sYU(X3S(eYb*[@e�S6¡Yo�U�Xc[cb>~

2∏

��� � cos A + 4

��� � cos A ≥ 4 + 3

��� � cos B cos C

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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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l��gY[3jGg=[%q�U6Z)U�[@e.S6¡YoP[]sYU(X3S(eYbLb,W�b�gPS��YoPX%S�` � e�S6¡Yo�U�Xc[cb>~

2r ≤ R�

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� X � 1

λ − cos A+

1

λ − cos B+

1

λ − cos C≥ µ 1 � & 2 ≤ µ < 6

�+�λ =

µ + 6

2µ-

� 0 � 1

λ − cos A+

1

λ − cos B+

1

λ − cos C≤ µ 1 � & 0 < µ ≤ 2

3

����λ =

µ + 6

2µ-

��� � � ������������������������

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