550 Bdt Chon Loc

550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng

1

S GD & T VNH PHC -------------**--------------

550 BI BT NG THC CHN LC

H v tn : Trn Mnh Cng T : Khoa hc t nhin

n v : Trng THCS Kim X Vnh Tng Vnh Phc

VNH PHC, XUN NHM THN 2012 .

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2

550 BI BT NG THC CHN LC -----------**-----------

1.Cho a, b, c l cc s thc khng m tha mn iu kin ab + bc + ca = 1.Chng minh rng

25111

+

++

++ accbba

.

2.Cho a,b,c l cc s thc dng .Chng minh rng

223

)1()1()1( 222222 -++-++-+ accbba

3.Cho a,b,c )1;0( .Chng minh rng 1)1)(1)(1( anRxxx n sao cho

1

...,...2

222

2121 -

+++=+++na

xxxaxxx nn

Chng minh rng :

na

xi2

;0 , i = 1,2,...,n.

14.Cho a,b,c (0;1) .Chng minh rng 1

444

-+

-+

- cbbaca

baac

bc

accb

ab

15.Cho a,b,c l cc s thc dng tho mn iu kin abc 1.Chng minh rng cba

ac

cb

ba

++++

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3

16. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng 3 61 .

a b c ab bc ca+

+ + + +

17. Cho a,b,c l cc s thc dng.Chng minh rng

ac

cb

ba

ac

cb

ba 222

2

3

2

3

2

3

++++

18. Cho x,y,z l cc s thc dng tho mn iu kin x2 + y2 + z2 + 2xyz = 1.Chng minh rng a) xyz

81

b) x + y + z 23

c) xy + yz + zx 22243

zyx ++ d) xy + yz + zx xyz221+

19. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = xyz .Chng minh rng xy + yz + zx 1113 222 ++++++ zyx 20. Cho x,y,z l cc s thc dng tho mn iu kin x,y,z > -1 .Chng minh rng 2

11

11

11

2

2

2

2

2

2

++

++

+++

+++

+yx

zxz

yzy

x

21. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1.Chng minh rng 2

222

++

+++

+++

baac

accb

cbba

22.Cho a,b,c 0 tho mn iu kin a4 + b4 + c4 2(a2b2 + b2c2 + c2a2) .Chng minh rng a2 + b2 + c2 2(ab + bc + ca) 23. Cho x,y,z l cc s thc dng tho mn iu kin x2 + y2 + z2 = xyz.Chng minh rng a) xyz 27 b) xy + yz + zx 27 c) x + y + z 9 d) xy + yz + zx 2(x + y + z) + 9 24. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = 3.Chng minh rng zxyzxyzyx ++++ 25. Cho a,b,c l cc s thc dng. Chng minh rng

43

2.

2.

2.

++++

++++

++

++++

bacc

baac

acbb

accb

cbaa

cbba

26. Cho a,b,c l cc s thc dng. Chng minh rng

abcb

caba

bcac

ac

cb

ba

++

+++

+++

++


cbacabcab

babac

acacb

cbcba

++++

+-

++-

++-

)(322

3

22

3

22

3

28.Cho cc s thc dng a,b,c,x,y,z tho mn iu kin a + x = b + y = c + z = 1.Chng minh rng 3

111)(

+++

cxbzayxyzabc

29. Cho a,b,c l cc s thc dng. Chng minh rng )(

41

222cba

bacca

acbbc

cbaab

++++

+++

+++

30.Cho a,b,c,d l cc s thc tho mn iu kin 12222 =+++ dcba .Tm gi tr nh nht ca biu thc Q = a3(b + c + d) + b3(c + d + a) + c3(d + b + a) + d3(a + b + c) 31. Cho x,y,z l cc s thc dng .Chng minh rng 1

))(())(())((

++++

++++

+++ yzxzzz

xyzyy

y

zxyxx

x


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4

++

++

+

++

++

+ba

cac

bcb

ac

bab

aca

cb4

33. Cho x,y,z l cc s thc dng .Chng minh rng 3(x2y + y2z + z2x)(xy2 + yz2 + zx2) xyz(x + y + z)3 34. Cho a,b,c l cc s thc dng tho mn iu kin { } { } 1,,min,,max - cbacba Chng minh rng 1 + a3 + b3 + c3 + 6abc 3a2b + 3b2c + 3c2a 35. Cho x,y,z l cc s thc dng tho mn iu kin x2 + y2 + z2 + 2xyz = 1.Chng minh rng 8(x + y + z)3 10(x3 + y3 + z3) + 11(x + y + z)(1 + 4xyz) 12xyz . 36. Cho a,b,c l cc s thc dng tho mn iu kin a2 + b2 + c2 = 3. Tm gi tr nh nht ca biu thc P =

3332

52

52

5

+++

+++

++

bac

acb

cba

37. Cho a,b,c l cc s thc dng. Chng minh rng 27 +

++++

+

+

+

cbacba

abc

cab

bca 111

)(6222222

38. Cho a,b,c (0;1) tho mn iu kin ab + bc + ca = 1 .Chng minh rng

-+

-+

-

-+

-+

- cc

bb

aa

c

c

b

b

a

a 222222

11143

111

39.Cho x,y,z 1 tho mn iu kin x + y + z = 1.Chng minh rng

1027

11

11

11

222

++

++

+ zyx

40.Cho 1=++ zyx .Chng minh rng (1 x)2(1 y)2(1 z)2 215xyz(x + y)(y + z)(z + x) 41. Cho x,y,z l cc s thc dng tho mn iu kin xyz = x + y + z + 2. Chng minh rng a) xy + yz + zx 2(x + y + z)

b) xyzzyx23

++

42. Cho x,y,z l cc s thc tho mn iu kin x2 + y2 + z2 = 2.Chng minh rng 2+++ xyzzyx 43. Cho a,b,c,d l cc s thc dng. Chng minh rng 0

+-

++-

++-

++-

baad

addc

dccb

cbba

.

44. Cho x,y l cc s thc dng . Chng minh rng xy + yx > 1. 45. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng (a + b)(b + c)(c + a) 4(a + b + c 1) 46. Cho a,b,c l cc s thc dng. Chng minh rng (a 2 + b2 + c2)(a + b c) (b + c a) (c + a b) abc(ab + bc + ca) 47. Cho a,b,c l cc s thc dng. Chng minh rng 3 + a + b + c +

abccba

ac

cb

ba

cba ++++

+++++1

)1)(1)(1(3

111

48. Cho nxxx ,...,, 21 l cc s thc dng tho mn iu kin 1...21 =nxxx .Chng minh rng

n

n

n

in

n

i

ni

n

i

n

xxxn

++P===

1)1(.

111

49. Cho a,b,c,d l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng a3 + b3 + c3 + abcd min

+

2791

,41 d

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5

50. Cho a,b,c l cc s thc dng. Chng minh rng (1+ a2)2(1 + b2)2(a c)2(b c)2 (1+ a2)(1 + b2)(1 + c2)(a b)2(b c)2(c a)2

51. Cho nxxx ,...,, 21 , Ryyy n ,...,, 21 tho mn iu kin 1...... 222

21

222

21 =+++=+++ nn yyyxxx

Chng minh rng

--

=

n

iii yxyxyx

1

21221 12)(

52.Cho naaa ,...,, 21 l cc s nguyn dng khc nhau tng i mt . Chng minh rng )...(

312

... 2122

22

1 nn aaan

aaa ++++

+++

53. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1.Chng minh rng

433

)3()3()3(

++

++

+ cabcba

bcab

ac

abca

cb

54. Cho a,b,c,d l cc s thc tho mn iu kin (1+ a2)(1 + b2)(1 + c2)(1 + d2) = 16.Chng minh rng 53 -+++++- abcdbdacdacdbcab 55. Cho a,b,c l cc s thc dng. Chng minh rng (a2 + 2)(b2 + 2)(c2 + 2) 9(ab + bc + ca) 56. Cho x,y,z l cc s thc tho mn cc iu kin zyx 0 , n > 2 tho mn iu kin

11 211

+=

==

nx

xn

k k

n

kk

Chng minh rng

)1(

24

1 21

21

2

-++>

== nn

nn

kk

n

kk

xx

62. Cho a,b,c l cc s thc dng. Chng minh rng a2 + b2 + c2 + 2abc + 3 (1 + a)(1 + b)(1 + c) . 63. Cho a,b,c l cc s thc dng. Chng minh rng .8

)(2)2(

)(2)2(

)(2)2(

22

2

22

2

22

2

++++

+++++

+++++

bacbac

cabcab

cbacba

64.Cho x,y l cc s thc dng v m,n l cc s nguyn dng .Chng minh rng (n 1)(m 1)(xm+n + ym+n) + (m + n 1)(xmyn + xnym) mn(xm + n 1y + ym + n 1x) .

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6

65. Cho a,b,c,d,e l cc s thc dng tho mn iu kin abcde = 1. Chng minh rng

310

11111

+++

++++

+++

++

+++

++++

eabceaeabe

deabdedead

cdeacdcdec

bcdebcbcdb

abcdababca

.

66.Cho )2

;0(,,p

cba .Chng minh rng

0)sin(

)sin().sin(.sin)sin(

)sin().sin(.sin)sin(

)sin().sin(.sin

+--

++

--+

+--

abacbcc

cacbabb

cbcabaa

67. Cho a,b,c l cc s thc dng. Chng minh rng .333333222222444 cabcabaccbbaaccbbacba ++++++++++ 68. Cho a,b,c,x,y,z l cc s thc dng. Chng minh rng ax + by + cz + ).)((

32

))(( 222222 zyxcbazyxcba ++++++++

69.Cho a,b,c l di ba cnh ca mt tam gic.Chng minh rng .213

++

-++

ca

bc

ab

ac

cb

ba

70. Cho nxxx ,...,, 21 > 0 , n > 2 tho mn iu kin 1...21 =+++ nxxx .Chng minh rng

.1

11

11==

--

+

n

i i

in

i n x

xn

x

71. Cho a,b,c l cc s thc khng m tho mn iu kin a2 + b2 + c2 + abc = 4.Chng minh rng 20 -++ abccabcab 72. Cho a,b,c l cc s thc dng. Chng minh rng { }.)(,)(,)(max

32223 accbbaabc

cba----

++

73. Cho a,b,c l cc s thc dng. Chng minh rng .

3.

2.

33

3 cbabaa

abcaba +++

++

74. Cho x,y,z l cc s thc dng tho mn iu kin (x + y + z)3 = 32xyz.Tm gi tr ln nht v gi tr nh nht ca biu thc P =

4

444

)( zyxzyx

++++

75. Cho a,b,c,d l cc s thc dng. Chng minh rng (a + b)3(b + c)3(c + d)3(d + a)3 16 42222 )( dcbadcba +++ 76. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 1. Tm gi tr ln nht ca biu thc P =

caca

bcbc

abab

-+

-+

- 1)(

1)(

1)( 444


)1(

3)1(

1)1(

1)1(

133 abcabcaccbba +

+

++

++

78. Cho a,b,c l cc s thc dng tho mn iu kin a2 + b2 + c2 = 9. Chng minh rng 2(a + b + c) abc 10 79. Cho x,y,z l cc s thc dng. Chng minh rng

2222222 )(9111

zyxxzxzzyzyyxyx ++

+++

+++

++

80. Cho ba s thc dng a ,b ,c tho mn iu kin ab + bc + ca 1 .Chng minh rng :

( )22 2 2 2 2 21 1 1 9

a b ca ab b b bc c c ca a+ +

+ ++ + + + + +

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7

81. Cho a,b,c,d l cc s thc dng. Chng minh rng 2(a3 + 1)(b3 + 1)(c3 + 1)(d3 + 1) (1 + abcd)(1+ a2)(1 + b2)(1 + c2)(1+ d2) . 82. Cho a,b,c l cc s thc dng. Chng minh rng (a + b)4 + (b + c)4 + (c + a)4

74

(a4 + b4 + c4) .

83. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng .

21

21

21

11

11

11

cbaaccbba ++

++

+

+++

+++

++

84. Cho a,b,c l cc s thc dng tho mn iu kin 21ab + 2bc + 8ca 12.Tm gi tr nh nht ca biu thc P = .

321cba

++

85. Cho a,b,c,x,y,z l cc s thc dng tho mn iu kin xy + yz + zx = 3. Chng minh rng 3)()()( +

+++

+++

+yx

bac

xzac

bzy

cba

86. Cho a,b,c l cc s thc dng. Chng minh rng .

53

)()(

)()(

)()(

22

2

22

2

22

2

++-+

+++-+

+++-+

cbacba

bacbac

acbacb

87. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng (a2 + b2) (b2 + c2)( c2 + a2) 8(a2b2 + b2c2 + c2a2)2 88. Cho a,b,c,d l cc s thc dng tho mn iu kin abcd = 1. Chng minh rng 1

)1(1

)1(1

)1(1

)1(1

2222

++

++

++

+ dcba


bac

acb

cba

bac

acb

cba

++

++

+

++

++

+ 222

22

2

22

2

90. Cho a,b,c l cc s thc dng tho mn iu kin abc + a + c = b. Tm gi tr ln nht ca biu thc sau

P = 222 1

31

21

2cba +

++

-+

91.Cho n s thc naaa ,...,, 21 .Chng minh rng

21

2

)...(*

jnji

ii

i aaa ++

N

92. Cho a,b,c l cc s thc dng. Chng minh rng 3222

++

++

+ acc

cbb

baa

93. Cho x,y,z l cc s thc dng. Chng minh rng (xy + yz + zx)

49

)(

1

)(

1

)(

1222

+

++

++ xzzyyx

94.Cho nxxx ,...,, 21 l cc s thc dng tho mn iu kin

=

+n

i

nix

1

2)13( .

Chng minh rng .

3161

1

nx

n

i i

+=

95.Cho naaa ,...,, 21 l cc s thc dng. Chng minh rng

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8

(n 1)(a1n + a2

n + + ann ) + n a1a2an (a1 + a2 + + an)(a1n -1 + a2n -1 + + ann - 1)

96. Cho nxxx ,...,, 21 > 0 tho mn iu kin 1...21 =nxxx . Chng minh rng nxxx

n

iij

njii --

= 1

22

1

)( .

97. Cho naaa ,...,, 21 11-

2. Tm gi tr nh nht ca biu thc

.)1(1

...

1

21= --

n

i i

n

an

aaa

98. Cho naaa ,...,, 21 [ )1;0 tho mn iu kin

a = 33... 222

21

+++n

aaa n

Chng minh rng

2222

22

1

1

11...

11 ana

a

a

a

a

a

a

n

n

-

-++

-+

-

99. Cho a,b,c,x,y,z l cc s thc dng tho mn iu kin (a + b + c)(x + y + z) = (a2 + b2 + c2)(x2 + y2 + z2) = 4 Chng minh rng abcxyz

361

< .

100. Cho nxxx ,...,, 21 > 0 , n > 2 tho mn iu kin 1...21 =nxxx .Tm hng s kn nh nht sao cho 1

1

1...

1

1

1

1

21

-+

+++

++

nxkxkxk nnnn

.

101. Cho nxxx ,...,, 21 > 0 , n > 2 tho mn iu kin 1... 22221 =+++ nxxx .Tm hng s kn ln nht sao cho (1 1x ) (1 2x ) (1 xn) nn xxxk ...21 102. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng

23

)(1

)(1

)(1

333

++

++

+ bacacbcba

103. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng 1

555555

+++

+++

++ caacca

bccbbc

abbaab


3 3 3 3 3 3

1 1 1 1a b abc b c abc c a abc abc

+ + + + + + + +

105. Cho x,y,z l cc s thc dng v tch xyz = 1.Chng minh rng: 1 1 1

11 1 1x y y z x z+ +

+ + + + + +.

106. Cho x,y,z l cc s thc dng tho mn iu kin 4111 =++zyx

. Chng minh rng

12

121

21

++

+++

+++ zyxzyxzyx

.

107. Cho cc s thc khng m x, y thay i v tha mn x + y = 1. Tm gi tr ln nht v gi tr nh nht ca biu thc ( )( )

2 2S 4x 3y 4y 3x 25xy= + + +

108. Cho x,y,z l cc s thc dng thay i v tho mn iu kin xyz = 1.Tm gi tr nh nht ca biu thc:

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9

2 2 2( ) ( ) ( )

2 2 2

x y z y z x z x yP

y y z z z z x x x x y y

+ + += + +

+ + + 109.Cho x,y,z l ba s dng v tho mn iu kin 1++ zyx .Chng minh rng

.821112

22

22

2 +++++z

zy

yx

x

110.Cho cc s dng x,y,z tho mn iu kin xyz = 1.Chng minh rng

33111 333333

++

+++

+++

zxxz

yz

zy

xy

yx.

Khi no ng thc xy ra ? 111.Cho x ,y l cc s thc thay i .Tm gi tr nh nht ca biu thc 2 2 2 2( 1) ( 1) 2 .A x y x y y= - + + + + + -

112. Cho hai s thc x0, y0 thay i v tho mn iu kin ( ) 2 2x y xy x y xy+ = + - Tm gi tr ln nht ca biu thc A =

33

11yx

+ .

113. Cho ab> 0. Chng minh rng

a

bb

b

aa

+

+

21

221

2

114. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng

3333

2

3

2

3

2 18111cbab

caabc

cab

++

++

++

+


21

1)1(1

1)1(1

1)1(1

222222

++++

++++

+++ accbba

116. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng 1

11

11

11

+-

+-

+-

ac

cb

ba


43

)1)(1()1)(1()1)(1(

333

++

+++

+++ ba

cca

bcb

a

118. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng

41

111

++

++

+ bca

abc

cab

119. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng 132222 +++ abccba 120.Cho a,b,c l cc s thc dng tho mn iu kin a2 + b2 + c2 = 1. Chng minh rng 34

1+++

abccba

121. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng cabcabbcaabccab ++++++++ 1 . 122. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng (1 + a)(1 + b)(1 + c) 8(1 a)(1 b)(1 c) . 123. Cho a,b l cc s thc dng tho mn iu kin a + b = 1. Chng minh rng

31

11

22

+

++ b

baa

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10

124.Cho cc s thc x,y .Chng minh rng 3(x + y + 1)2 + 1 3xy . 125. Cho a,b,c l cc s thc dng. Chng minh rng 333

11)(2

ab

ba

baba +

++ .

126.Cho a,b,c 1. Chng minh rng )1(111 +-+-+- abccba . 127. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = xyz .Chng minh rng

23

1

1

1

1

1

1222

++

++

+ zyx.


888 222

++

++

+ abc

c

cab

b

bca

a .

129. Cho a,b,c,d l cc s thc dng. Chng minh rng

32

32323232

+++

+++

+++

++ cbad

badc

adcb

dcba

.

130. Cho a,b,c,d l cc s thc dng tho mn iu kin ab + bc + cd + da = 1. Chng minh rng

313333

++

+++

+++

+++ cba

dbad

cadc

bdcb

a .


abcab

cabca

bcabc

222 222 ++

++

+

abcc

cabb

bcaa

2221

2

2

2

2

2

2

++

++

+ .

132. Cho a,b,c l cc s thc dng. Chng minh rng cba

abc

cab

bca

++++333

.

133. Cho a,b,c l cc s thc dng tho mn iu kin a2 + b2 + c2 = 3. Chng minh rng

23

11

11

11

+

++

++ cabcab

.

134. Cho a,b,c l cc s thc dng. Chng minh rng 1+

++

+++

++bacb

cbba

ac

cb

ba

.

135.Cho a,b,c 43

- v a + b + c = 1. Chng minh rng

109

111 222

++

++

+ cc

bb

aa

.

136. Cho x,y,z l cc s thc dng tho mn iu kin xyz = 1 .Chng minh rng 2

6336

99

6336

99

6336

99

++

++

+++

+++

+xxzz

xzzzyy

zyyyxx

yx.

137.Cho 0> zyx .Chng minh rng 222

222

zyxyxz

xzy

zyx

++++ .

138. Cho 0> cba .Chng minh rng cba

bca

abc

cba

+--

+-

+-

43222222

.

139. Cho x,y,z l cc s thc dng. Chng minh rng

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11

9

33))((

)(222

222 +

+++++++++zxyzxyzyx

zyxzyxxyz.

140. Cho naaa ,...,, 21 > 0 v 1...21 0 . Chng minh rng

nn

n

n aaaa

a

a

a

aa

aa

+++++++ - ...... 211

221

3

22

2

21 .

143. Cho x,y,z l cc s thc dng tho mn iu kin xyz = 1 .Chng minh rng )(2222 zxyzxyzyxzyx +++++++ . 144. Cho x,y,z l cc s thc dng tho mn iu kin xyz xy + yz + zx .Chng minh rng xyz 3(x + y + z)

145.Cho x,y,z > 1 v 2111=++

zyx. Chng minh rng

111 -+-+-++ zyxzyx . 146. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca = 1. Chng minh rng

abca

cc

bb

a1

61

61

61

333 +++++ .

147.Cho 2,2,2 zyx . Chng minh rng (x3 + y)(y3 + z)(z3 + x) 125xyz . 148. Cho a,b,c,d l cc s thc dng tho mn iu kin 32222 )( badc +=+ . Chng minh rng 1

33

+db

ca

.


222

++

++

+ bac

acb

cba

.


bcc

abb

aca

cbbca

baabc

accab

++

++

+

++

++

+ )()()(.

151. Cho a,b,c,d l cc s thc dng. Chng minh rng 4

++

+++

+++

+++

adbd

dcac

cbdb

baca

.

152. Cho x,y,u,v l cc s thc dng. Chng minh rng

vuuv

yxxy

vuyxuvuyxuxy

++

+

++++++

.

153. Cho x,y,z l ba s thc thuc on [ ]4;1 v zxyx , . Tm gi tr nh nht ca biu thc P =

xzz

zyy

yxx

++

++

+ 32 .

154. Cho x,y,z l ba s thc thuc on [ ]9;1 v zxyx , . Tm gi tr ln nht v nh nht ca biu thc P =

xzz

zyy

yxx

++

++

+ 2 .

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12

155. Cho x,y,z l ba s thc thuc on

3;31

v zxyx , .Chng minh rng

57

+

++

++ xz

zzy

yyx

x .


+++

+

+

+

312111

abc

cbaac

cb

ba

.

157. Cho x,y,z l cc s thc khng m tho mn iu kin x + y + z = 1. Chng minh rng x2y + y2z + z2x

274

.

158. Cho a,b,c,d,e,f l cc s thc dng tho mn iu kin a + b + c + d + e + f = 1 v ace + bdf

1081

.

Chng minh rng abc + bcd + cde + def + efa + fab

361

.

159.Ch a,b,c [ ]1;0 .Chng minh rng a2 + b2 + c2 a2b + b2c + c2a + 1 . 160.Cho a,b,c 0 v a + b + c abc .Chng minh rng a2 + b2 + c2 abc . 161. Cho a,b,c 0 v a + b + c abc .Chng minh rng a2 + b2 + c2 3 abc . 162. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = xyz .Chng minh rng xy + yz + zx 9(x + y + z) . 163. Cho 4321 ,,, xxxx l cc s thc dng tho mn iu kin 14321 =xxxx . Chng minh rng

+++++++++4321

432134

33

32

31

1111,max

xxxxxxxxxxxx .

164. Cho a,b,c,x,y,z l cc s thc dng. Chng minh rng

)(3)( 3333

zyxcba

zc

yb

xa

++++

++ .

165. Cho a,b,c,d l cc s thc dng tho mn iu kin 1

11

11

11

11

4444=

++

++

++

+ dcba.

Chng minh rng abcd 3 . 166.Cho x,y,z > 1. Chng minh rng ( ) zxyzxyxyzzxyyzx xyzzyx +++++ 222 222 . 167.Cho 0 abc .Chng minh rng (a + 3b)(b + 4c)(c + 2a) 60abc . 168. Cho x,y,z l cc s thc dng. Chng minh rng )(2222 yzxyzyx +++ . 169.Cho cc s thc a,b,c tho mn iu kin a2 + b2 + c2 = 1. Chng minh rng

53

212121

222

+

++

++ ab

cca

bbc

a.

170. Cho a,b,c l cc s thc dng tho mn iu kin abc 1. Chng minh rng

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13

1111444444

++

+++

+++ cbacbacba

.

171.Cho a > b > c > 0 , x > y > z > 0 . Chng minh rng

43

))(())(())((

222222

++

+++

+++ bxaybyax

zcazcxaxcz

ybcybzczby

xa.

172. Cho 3 s thc khng m , ,a b c . Chng minh rng: ( )

2 2 2 2 1 2a b c abc ab bc ca+ + + + + + Du ng thc xy ra khi no ?

173. Cho ba s thc , ,a b c i mt phn bit. Chng minh rng :

2 2 2

2 2 2 2( ) ( ) ( )a b c

b c c a a b+ +

- - - 174. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 3. Chng minh rng

23

111 222

++

++

+ ac

cb

ba

.


222

++

++

+ acc

cbb

baa

.

176.Cho 1...,0,...,,,, 2121 =+++> nn xxxxxxba . Chng minh rng

)(

1...

1

3

32

32

21

31

babababa +

+++

++

+ nxx

xx

x

xx

x

n

n .

177.Cho [ ]1;0, yx . Chng minh rng

xyyx +

++

+ 12

1

1

1

122

.

178.Cho x,y,z > 0 , xyz = 1 , *,111

Nkzyxzyx

++>++ . Chng minh rng

kkkkkk

zyxzyx

++>++111

.

179. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng a. 11.1.1 333 -++-++-+ bacacbcb . 180. Cho x, y, z l cc s thc dng tho mn iu kin x + y + z = 1. Chng minh rng

827

11

11

11

-

+-

+- xzyzxy

.


++++

++

cbacba

ac

cb

ba 111

)(2

.

182. Cho hai s ,x y lin h vi nhau bi ng thc 2 22 7( ) 2 10 0x xy x y y+ + + + + = . Tm gi tr ln nht v gi tr nh nht ca biu thc 1S x y= + + .

183. Vi a,b l cc s thc tho mn ng thc 49

)1)(1( =++ ba , hy tm gi tr nh nht ca biu thc

44 11 baP +++= .

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14


abcd

dcbadcba

++++++

3333

1111 .

185.Cho [ ]1;0,, cba . Chng minh rng

2111

++

++

+ abc

cab

bca

.


3

1++ accbba .


cc

bb

aa

ca

ac

ab

-+

+-+

+-+

++

11

11

11

2 .


cba

bacba

ac

cb

ba

++-

+++++2222 )(4

.


34

)1)(1()1)(1()1)(1( 33

2

33

2

33

2

++

+++

+++ ac

c

cb

b

ba

a.


1222 222

++

++

+ cc

bb

aa

.

191. Cho x,y,z l cc s thc dng tho mn iu kin 1xyz . Chng minh rng

0225

25

225

25

225

25

++

-+

++-

+++

-yxz

zzxzy

yyzyx

xx .

192. Cho a,b l cc s thc dng. Chng minh rng

+

+

++

++

21

221

243

43 22 baabba .

193. Cho a,b,c l cc s thc dng tho mn iu kin 1111 =++cba

. Chng minh rng

(a 1)(b 1)(c 1) 8 .

194.Cho x l s thc dng v n l s nguyn dng. Chng minh rng

1 + xn + 1 1)1(

)2(-+

n

n

xx

.

195. Cho x l s thc dng v n l s nguyn dng. Chng minh bt ng thc sau

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15

121

21

1)1(

++

+

++ n

n

nn xxxx

Khi no ng thc xy ra ? 196. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng

43

)1)(1()1)(1()1)(1(

+++

+++

++ acc

cbb

baa

.

197. Cho a,b,c l cc s thc khng m tho mn iu kin ab + bc + ca = 31

. Chng minh rng

31

11

11

1222

+-

++-

++- abccabbca

.


32

)1()1()1( -+-+- bcaabccab .


)(23

cbaba

c

ac

b

cb

a++

++

++

+ .


14

14

14

1

-+

-+

- cabcab.

201. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca = 1. Chng minh rng

abc

cbaabc

3

33

)(61

3 +++ .

202. Cho a,b,c l cc s thc dng tho mn iu kin a,b,c 1. Chng minh rng

9111

)2(

+++

cbaabc .

203. Cho x,y,z l cc s thc dng tho mn iu kin 1=++ zxzxyzyzxyxy . Chng minh rng

21

33

6

33

6

33

6

+

++

++ xz

zzy

yyx

x.

204. Cho a,b,c l cc s thc dng tho mn iu kin a,b,c [ ]2;1 . Chng minh rng

10111

)(

++++

cbacba .


42

2

2

2

2

2

2

2

abcd

dcbaad

dc

cb

ba +++

+++ .

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16

206.Cho [ ]2;0x . Chng minh rng

433 334 ++- xxxx .


21

321

321

321

222222

+++

+++

++ accbba.

208. Cho x,y l cc s thc dng tho mn iu kin x2 + y2 = 1. Chng minh rng

2341

1)1(1

1)1( +

+++

++

xy

yx .

209.Cho a,b,c ( ]1;0 . Chng minh rng

)1)(1)(1(311

cbacba

---+++

.

210. Cho x,y,z l cc s thc dng tho mn iu kin

21

21

41

3=

++

++

+ zz

yy

xx

Chng minh rng

9

243

81

zyx .


++

++

+

++++

cba

bac

acb

cbacba

23111

)(333

333 .

212.Cho x l mt s thc bt k . Chng minh rng

17)1(16)1(

81

42

48

+++

x

xx.


cbacba

acac

cbcb

baba

++++

++

+++

+++ 222222222

.3 .


2>+

++

++ ba

cac

bcb

a.

215. Cho x,y,z l cc s thc khng m tho mn iu kin x + y + z = 1, 2n .Chng minh rng

xny + ynz + znx 1)1( ++

n

n

nn

.

216. Cho x,y,z l cc s thc dng. Chng minh rng 16xyz(x + y + z) 3 444 )()()(3 xzzyyx +++ .

217. Cho x,y,z l cc s thc dng tho mn iu kin xyz = 1. Chng minh rng

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17

33

2

3

2

3

2

++

+++

+++ yxxz

zxzzy

yzyyx

x .

218. Cho x,y,z l cc s thc dng tho mn iu kin xyz = 1. Chng minh rng

17722

22

7722

22

7722

22

++

+++

+++ xzxz

xzzyzy

zyyxyx

yx.


4

331+

++

++

+ abcabc

cabb

bcaa

.


cbaacaca

cbcbc

babab

+++-

++-

++-

2

33

2

33

2

33

35

35

35

.

221. Cho x,y,z l cc s thc tho mn iu kin x,y,z 1- v x3 + y3 + z3 x2 + y2 + z2 Chng minh rng x5 + y5 + z5 x2 + y2 + z2 .


2173111 222

+++

+++

++

bac

acb

cba .

223. Cho x,y,z,t l cc s thc dng tho mn iu kin xyzt = 1. Chng minh rng

34

)(1

)(1

)(1

)(1

3333

+++

+++

+++

++ zxyzxytyxtyxtztxztxzytyztyzx.

224. Cho )1,(...,0,...,, 2121 +++> nkkaaaaaa kk . Chng minh rng

1...

...11

21

1

21 ++++++

+++ nk

nn

nk

nn

aaa

aaa.

225. Cho a, b, c l di 3 cnh ca mt tam gic c chu vi bng 2. Tm gi tr nh nht ca biu thc 3 3 34( ) 15P a b c abc= + + + .


++

++

+

++

++

+ca

bcb

aba

cb

aca

cbc

ba2 .

227. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca = 1. Chng minh rng a + b + c + abc

9310

.

228. Cho a,b,c l cc s thc dng tho mn iu kin a2b2 + b2c2 + c2a2 a2b2c2 Chng minh rng

23

)()()( 22322

223

22

223

22

+

++

++ acb

accba

cbbac

ba .


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18

5127291

11

11

1333

+

+

+

cba.

230. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 1. Chng minh rng

27

11

11

11

2

2

2

2

2

2

++

+++

+++

ac

cb

ba

.

231. Cho x,y,z l cc s thc khng m tho mn iu kin x2 + y2 + z2 = 1.Tm gi tr ln nht ca biu thc P = 6(y + z x) + 27xyz .

232. Tm gi tr ln nht, gi tr nh nht ca biu thc

P = 61245

145

+++-+--aa

aa trong a l tham s thc v

45

1 - a .

233. Cho x,y,z l cc s thc dng tho mn iu kin xy + yz + zx = 1.Tm gi tr nh nht ca biu thc

3 3 3

= + ++ + +

x y zP

y yz z xz x xy.

234.Cho a, b, c l cc s dng, chng minh bt ng thc

2

2333

2

3

2

3

2

++

++

+ bac

acb

cba

235.Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 3. Chng minh rng .

abcabc

cac

bcb

aba

5312

111 +

++

++

+.

236. Cho a, b, c l cc s thc dng, chng minh bt ng thc

83

333

++

++

+ bac

acb

cba

.

237. Cho x,y l cc s thc dng tho mn iu kin x + y = 1. Chng minh rng

3241133

+++ xyyx

.

238. Cho a,b,c,d l cc s thc tho mn iu kin a2 + b2 = c + d = 4. Chng minh rng ac + bd + cd 4 + 4 2 .

239.Cho x,y,z vi x = max{ }zyx ,, .Chng minh rng

33 22111 ++++++xz

xy

yx

.

240.Choa l s thc dng v x,y,z l cc s thc tho mn iu kin xy + yz + zx = 1. Chng minh rng

2

811)( 222

aa ++-++ zyx .

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19

241.Cho a,b,c > 1. Chng minh rng

bacb cba

ac logloglog ++ 33 abc .

242.Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng

.181

33

6

33

6

33

6

+

++

++ ba

cac

bcb

a

243. Cho x, y, z l cc s thc dng tho mn iu kin x + y + z = 1. Chng minh rng

23

+

++

++ zxy

zxyzx

yzxyz

xy .

244.Cho x l s thc khng m. Chng minh rng

91

22++

+xx

x.

245. Cho a,b l cc s thc tho mn iu kin a > b 0 . Chng minh rng

5)32)((

322

2

+-+

bbaa .

246. Cho a,b l cc s thc dng tho mn iu kin a + b = 4. Chng minh rng

2a + 3b + 18106

+ba

.

247. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 3. Chng minh rng 5555 33222 +++++ accbba .

248.Cho x,y,z l cc s thc dng . Chng minh rng (x + y + z)6 432xy2z3 .

249.Cho [ ]1;0x . Chng minh rng

16913 4242 ++- xxxx .

250. Cho a,b,c,d l cc s thc dng ,chng minh bt ng thc

625

2856153

253

253

253

2

+

+

+

+

ad

dc

cb

ba

.

251. Cho a,b,c,d l cc s thc dng tho mn iu kin a + b + c + d 1 . Chng minh rng

4911

111

111

111

1

++

++

++

++

addccbba.

252. Cho a,b,c,d l cc s thc dng tho mn iu kin 16abcd . Chng minh rng

16

240112121212

++

++

++

++

bad

adc

dcb

cba .

253. Cho a,b l cc s thc dng tho mn iu kin a + b 1 . Chng minh rng

20111

2233++

+ abbaba .

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20

254. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c 1 . Chng minh rng

281111111

222222+++

++

++

+ cabcabaccbba.

255. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 3.Chng minh rng 5555 63))(2())(2())(2( ++++++++ cbcacbabcbacaba .

256. Cho a,b,c l cc s thc dng tho mn iu kin (a2 + a + 2)(b + 1)2(c2 + 3c) = 64. Chng minh rng a3b4c5 1 .

257. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c 23

.Chng minh rng

34311

311

311

3

++

++

++

accbba.

258. Cho a,b,c,m,n,p l cc s thc dng tho mn iu kin a + b + c 1 v m + n + p23

.Chng minh

rng

3912

112

112

1

++

++

++

pcnbma.

259.Cho x,y,z l cc s thc. Chng minh rng 2222 )333(4)3)(3)(3(27 zxyzxyzyx +++++ .

260. Cho x,y,z l cc s thc tho mn iu kin x + y + z < 0 v 4xz > y2. Chng minh rng 2x2 + y2 + 5z2 + 6xy + 7xz + 2yz > 0 .

261. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca = abc.Chng minh rng (a + b c 1)(b + c a 1)(c + a b 1) 8 .

262. Cho a,b,c l cc s thc dng . Chng minh rng

.29

2 222

2

22

2

22333

++

+++

+++

+++

acbac

bcacb

abcba

abccba

263. Cho a,b l cc s thc dng tho mn iu kin ab = 1. Chng minh rng

.111

33

+

++ a

bb

a

264. Cho x,y,z l cc s thc khng m tho mn iu kin x + y + z = 2. Chng minh rng .2)(2 444333 zyxzyx +++++

265. Cho a,b,c l cc s thc dng . Chng minh rng

.24111

32

32

32

++

++

+

cac

bcb

aba

266. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1.Chng minh rng ).1(2))()(( cbaaccbba ++++++

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21

267. Cho x,y,z l cc s thc dng . Chng minh rng

6111

)1( ++++++

+++ zyx

xy

yz

zx

zyxxyz .

268.Cho [ ]1;0,,, dcba . Chng minh rng

.31111

++

++

++

+ abcd

dabc

cdab

bcda

269. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca = 1.Chng minh rng

121

)()()( 2228

222

8

222

8

+

++

++ ac

ccb

bba

a.

270. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c 3 .Chng minh rng

.84111

272

3

2

3

2

3

+++++

cabcabac

cb

ba

271.Cho a,b,c l cc s thc dng tho mn iu kin

cbacba111

1111

6222

+++

++

Chng minh rng

121

101

101

101

++

+++

+++ cbacbacba

.

272. Cho a,b,c,d,e,f l cc s thc tho mn iu kin ab + bc + cd + de + ef = 1.Chng minh rng

7cos2

1222222p

+++++ fedcba .


.3127

111 232323

+++

+++

++ ccc

bbb

aaa

274. Cho x,y,z l cc s thc dng . Chng minh rng

33222222

xyx

xzxzzyzyyxyx

zxyzxy ++

++++++++

++.

275. Cho a,b l cc s thc dng .Chng minh rng

3444

13.

413

.33++

++++abab

baba .

276. Cho a,b,c l cc s thc dng .Chng minh rng

.1

3)1(

1)1(

1)1(

1+

+

++

++ abcaccbba

277. Cho x,y,z l cc s thc khng m .Chng minh rng ))()((9)(8 2222333 xyzzxyyzxzyx +++++ .

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22

278. Cho x,y,z l cc s thc dng .Chng minh rng

0222222

+-

++-

++-

yxyz

xzxy

zyzx

.

279.Cho x,y l cc s thc dng tho mn iu kin x + y = 1.Tm gi tr nh nht ca biu thc P = xy + yx .

280. Cho a,b,c l cc s thc khng m .Chng minh rng

222 +

-+

+-

++-

++a

acc

cbb

bacba .

281. Cho x,y,z l cc s thc khng m tho mn iu kin x + 2y + 3z = 41

.Tm gi tr ln nht ca biu

thc

P = 2

33

2

33

2

33

632729

5468783

242232

xxzzx

zyzyz

yxyxy

+-

++-

++-

.

282.Cho

21

;0,,, dcba .Chng minh rng

)1)(1)(1)(1(4

4

dcbaabcd

dcbadcba

----

----+++

.

283. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 1. Tm gi tr nh nht ca biu thc P = 222222222 2)(3)(3 cbacabcabaccbba ++++++++ .

284. Tm gi tr nh nht ca biu thc P = 103214 22 ++--++- xxxx .

285. Cho x,y l hai s thc khng m thay i.Tm gi tr ln nht v nh nht ca biu thc

P = )1)(1()1)((

22 yxxyyx

++--

.

286.Cho a,b,c l di ba cnh ca mt tam gic .Chng minh rng

9))()((3111

)( ---

+

++++

abcaccbba

cbacba .

287.Cho 10 s thc khng m )5,...,2,1(, =iba ii tho mn iu kin )5,...,2,1(122 ==+ iba ii v 125

24

23

22

21 =++++ aaaaa .Hy tm gi tr nh nht ca biu thc

P = 54321

54321

aaaaa

bbbbb

++++++++

.

288. Cho x,y,z l cc s thc khng m.Chng minh rng [ ] )2)(2)(2())()(( 2 yxzxzyzyxxyzxzzyyx +++++++++ . 289. Cho a,b,c l cc s thc dng .Chng minh rng

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23

3222 )()()(

41

3 abcaccbbacba +++

++

.

290.Cho 0,...,, 21 >nxxx v nxxx n 2...21 =+++ vi n 3 .Chng minh rng

3

)1(2

11 1 3-

+

= =

nn

x

xn

j

n

i i

j

ji

.

291.Cho hm s [ ) +=+x

ttdt

xfRf1 20022002

)(,,1: .Chng minh vi cc s thc x1,x2,...,xn 1 ,ta c

....

ln)(...)()( 2121

n

xxx

n

xfxfxf nn ++++++

292.Cho cc s thc a,b,c tho mn iu kin 30 cba .Chng minh rng 36)9)(()9)(()9)(( 222 --+--+-- ccbbcaaba .

293.Cho cc s thc naaa ,...,, 21 .Chng minh rng

22221

3 332

31 ...... nn aaaaaa ++++++ .

294. Chng minh rng vi mi s dng a,b,c ta lun c

3

2222

3

++++

++cabcabcba

ac

cb

ba

.

295. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 1. Tm gi tr nh nht ca biu thc

.24

)(4

)(4

)( 222

-++

-++

-+

bac

acb

cba


1777 33

3

33

3

33

3

++

+++

+++ aabcc

ccabcb

bbabca

a.

297. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 3.Chng minh rng

3111

++

+++

+++ aba

ccac

bbcb

a.


..193

999999 222333222

cbacba

bacc

acbb

cbaa

++++

++

+++

+++

299. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1.Chng minh rng

1121212 333

++

++

+ cc

bb

aa

.

300.Cho cc s thc x1,x2,...,xn vi 2n tho mn iu kin x1 + x2 +...+ xn 0 v 1... 222

21 =+++ nxxx .t M = max{ }nxxx ,...,, 21 .Chng minh rng

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24

)1(

1

-

nnM .

301. Cho naaa ,...,, 21 )1( n l cc s thc dng. Chng minh rng

++++

+

+++

++

+++

nnn aaan

aaaaaan

1...

111

1...

11

111

...11

212121

ng thc xy ra khi no ? 302. Cho x,y,z l cc s thc dng tho mn iu kin 3333 =++ zyx . Tm gi tr ln nht ca biu thc P = xyzzxyzxy -++ )(3 .

303. Chng minh rng

))()((

)(

2

1111 23

3 accbbaabccba

abcaccbba ++++++

++

++

++

vi mi a,b,c > 0 .


275

)(814

+++

abccabcab

.

305.Cho a,b,c l cc s thc khng m phn bit .Chng minh rng

2

5511

)(

1

)(

1

)(

1)(

222222 +

-

+-

+-

++accbba

cba .

306. Cho x,y l cc s thc dng tho mn iu kin x + y + 1 = 3xy. Tm gi tr ln nht ca biu thc

P = 22

11)1(

3)1(

3yxyx

yxy

x--

++

+.

307.Cho cc s thc x,y tha mn iu kin 322 ++ yxyx .Chng minh rng 3343334 22 ----- yxyx . 308.Cho a,b,c l di ba cnh mt tam gic.Chng minh rng )()()(3 222222333 bacacbcbaabccba ++++++++ . 309.Cho n l s nguyn vi n >3.Gi naaa ,...,, 21 l cc s thc tho mn min njiaa ji =- 1,1

Tm gi tr nh nht ca biu thc

P = .3

1=

n

kka

310. Cho a,b,c l cc s thc dng tho mn iu kin a3 + b3 = c3.Chng minh rng a2 + b2 c2 > 6(c a)(c b) .

311.Cho x,y,z l cc s thc tho mn iu kin x3 + y3 + z3 3xyz = 1.Tm gi tr nh nht ca biu thc P = x2 + y2 + z2 .

312. Cho 1,...,, 21 >naaa .Chng minh rng

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25

nx

xx

x

xx

x

xx n 4...2

1

4

32

3

21 +++ .

313. Cho 30021 ,...,, aaa khng m tho mn iu kin 1300

1

==i

ia .Tm gi tr ln nht ca biu thc

P = jiji

ji aa|,

.

314. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = xy + yz + zx. Chng minh rng

11

11

11

1222

++

+++

+++ xzzyyx

.

315. Cho x,y,z l cc s thc dng .Chng minh rng [ ]222222322232223 )()()()()()( xzzxzyyzyxxyxyzyxzzxyzyx ++++++++++ .

316.Cho x,y,z l cc s thc dng tho mn iu kin 21

11

11

11

222=

++

++

+ zyx.Chng minh rng

31

21

21

21

333+ .

330. Cho a,b,c l cc s thc khng m .Chng minh rng )(3)( 2 abccabbcacba ++++ .

331. Tm gi tr nh nht ca biu thc P = 444221222212222 222222 ++++++-+++++-+ yxyxyxyxyxyx .

332. Cho a,b,c l cc s thc tho mn iu kin a + b + c = 1 v ab + bc + ca > 0. Tm gi tr nh nht ca biu thc

P = cabcabaccbba ++

+-

+-

+-

5222 .

333. Cho a,b,c,d l cc s thc dng tho mn iu kin 3(a + b + c + d) + 4(abc + bcd + cda + dab) = 8

Chng minh rng ab + ac + bc + ad + bd + cd 2 .


2

211

1

211

1

211

1

+++

+++

++c

ab

ca

b

.


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27

)(9)3)(3)(3( 201020122010201220102012 cabcabccbbaa +++-+-+- .

336. Chng minh rng nu phng trnh sau c nghim 01234 =++++ cxbxaxx

th 34222 ++ cba .Du bng xy ra khi no ?

337.Tm cc gi tr ca a,b phng trnh 01234 =++++ axbxaxx

c nghim v tng 22 ba + t gi tr nh nht . 338.Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = 1. Chng minh rng xyzxzzyyxzxyzxy 5)(4 222222 +++++ .

339.Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = 1. Chng minh rng

49

222

++

++

+ yxz

xzy

zyx

.

340. Cho a,b,c,d,e,f l cc s thc dng .Chng minh rng

fedcbafdbeca

feef

dccd

baab

+++++++++

+

++

++

))(( .

341.Cho 1,1 ba .Chng minh rng

88

322222 ba

baabba +

+

+

-.

342.Cho a,b,c l cc s thc dng .Chng minh rng

.4

)(2)(2

)( 2222

ab

baab

baba

ba --

+

+-

343. Cho a,b,c l cc s thc tho mn iu kin abc = 1.

Chng minh rng t nht hai trong ba s a

cc

bb

a1

2,1

2,1

2 --- u ln hn 1 .

344. Cho a,b,c,d l cc s thc dng .Chng minh rng

2+

++

++

++ ba

dad

cdc

bcb

a .

345.Cho a > b > 0 . Chng minh rng

.8

)(28

)( 22

bba

abba

aba -


28


611

11

11

11

11

11

--+--+--accbba

.


1327

11

11

11

+

++

++ cabcab

.

349.Cho a,b,c l di ba cnh ca mt tam gic tho mn iu kin a + b + c = 1.Chng minh rng

21

4222 +++ abccba .


.111111

29

cbaaccbbacba++

++

++

+

++

351. Cho x,y,z l cc s thc .Chng minh rng

0121212 222

2

22

2

22

+

-+

+-

++

-z

xzy

zyx

yx.

352. Cho x ,y l cc s thc thay i tho mn iu kin 122 =++ yxyx .Tm gi tr nh nht v gi tr ln nht ca biu thc A = x3y + xy3 .

353. Cho a,b,c l cc s thc khng m tho mn iu kin a2 + b2 + c2 = 1. Chng minh rng

)(43

111 222ccbbaa

ac

cb

ba

+++

++

++

.

354. Cho x,y,z l cc s thc dng tho mn iu kin x2 + y2 + z2 = 1. Chng minh rng

31222 ++ xyzzxyyzx .

355.Cho [ ]1;0,, zyx Tm gi tr ln nht ca cc biu thc S = x2y y2x v P = x2y + y2z + z2x x2z y2x z2y .

356. Cho a,b,c l cc s thc tho mn iu kin a < b < c , a + b + c = 6 , ab + bc + ca = 9 Chng minh rng 0 < a < 1 < b < 3 < c < 4 .

357. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca + 2abc = 1.Chng minh rng 2(a + b + c) + 1 32abc .

358. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = 3 v a,b,c l cc s khc 0 Chng minh rng

222222 111

11123

c

z

b

y

a

x

cba ++

++

+++ .

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29

359.Cho

23

;1, yx . Chng minh rng

222323 yxyxxy +-+- .

360. Cho a,b,c l cc s thc tho mn iu kin a + b + c = 1.Chng minh rng )(41222 cabcabcba +++++ .


2

)(22

6

66

6

22

6 cbaabcba

cac

bcb

a ++

++

++

+ .

362.Cho 3

1,,0 1 tho mn iu kin a + b + c = abc .Chng minh rng (a2 1)(b2 1)(c2 1) 8 .


))(())(())((

222

bcacabc

cbabcab

cababca

bac

acb

cba

+++

+++

++

+++

+

++

++

.


bacacbcbaba

cac

bcb

a 34343444

444

444

4

222222+++++++++++++ .

369. Cho a,b,c l ba s thc dng tho mn a + b + c + 1 = 4abc.Chng minh rng

cbacabcab

1113

111++++ .


3

.6abc

cbac

bab

aca

cb ++

++

++

+ .

371. Cho a,b,c,d l cc s thc khng m tho mn a2 + b2 + c2 + d2 = 4. Chng minh rng

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30

2 (4 ab bc cd da) )12( + (4 a b c d) .


cabcab

cba

abc

c

cab

b

bca

a

++++

+

++

++ 222 222

.


222333

cbaba

abccacabcb

cbabca

++++

+++

+++

.

374. Cho a,b,c l cc s thc khng m nhng khng c hai s no trong ba s ng thi bng 0 Chng minh rng

cba

abcbaabc

accab

cbbca

++

++

+++

+++ 3

322

2

322

2

322

2 9 .

375. Cho a,b,c l cc s thc .Chng minh rng 333333222222 ))()((3 accbbaacaccbcbbaba +++-+-+- .

376. Cho x,y,z l cc s thc khng m .Chng minh rng

5444 )(121

)()()( zyxyxzxzyzyx +++++++ .

377. Cho a,b,c l ba s thc dng tho mn abc = 1. Chng minh rng 0)()()( 222 -+-+- aacccbbba .

378. Cho a,b,c l cc s thc dng .Chng minh rng ))((3)( 2222223 cabcabaccbbacabcab ++++++ .

379. Cho a,b,c l cc s thc dng phn bit .Chng minh rng

2222

222222

)(16

cbaabc

cabcabcbabcaccbabcaba

++

---+++++++

.

380.Cho a,b l hai s thc phn bit tho mn iu kin 1111 ++-=+=++- bababa

Tm gi tr nh nht ca biu thc A = ba +

381.Cho 10,10


31

)(2))((3 22222222 dbabcdcadcdcbaba +-+-+- .

384. Cho a,b,c l cc s thc dng .Chng minh rng a) ))(()( 4442333 cabcabcbacba ++++++ .

b) 35552444 ))(()(9 cbacbacba ++++++ .

385. Cho a,b l hai s thc tho mn iu kin 6789 22 ++ baba . Chng minh rng 91257 ++ abba .


)(2

111111222 cbacabcabaccbbacba ++

+++

++

++

+++ .


cabcabaccbba

cabcabbacacbcba

+++++

+++

++

++

+ ))()(()(4

)(1

)(1

)(1

333 .


1444444 22

2

22

2

22

2

++

+++

+++ acac

ccbcb

bbaba

a .

389. Cho a,b,c l cc s thc dng tho mn iu kin abc 8 .Chng minh rng

11

11

11

1222

+-

++-

++- ccbbaa

.

390. Cho x,y,z l cc s thc dng tho mn iu kin 4111)( =

-+-+

zyxzyx . Tm gi tr nh nht

ca biu thc

A =

++++

444444 111)(

zyxzyx .

391. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca = 1 .Chng minh rng

25

)(1

)(1

)(1

2

22

2

22

2

22

++

+++

+++

acac

cbcb

baba

.


3111

++

+++

+++

aac

ccb

bba

393.Cho a,b,c l di ba cnh ca mt tam gic vung .Hy tm gi tr ln nht ca s thc k 3333 )( cbakcba ++++ .

394. Cho nxxx ,...,, 21 l cc s thc dng tho mn iu kin 11

==

n

iix .Chng minh rng

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32

11

1 2

11 +

+

== n

n

xx

n

i i

n

ii .

395. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = 1 .Chng minh rng

22

+

++

++ xyzx

zx

zxyz

yz

yzxy

xy .

396. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 3.Chng minh rng 222

222

111cba

cba++++ .


++

++

+

++

bac

acb

cba

ac

cb

ba

23

2

.

398. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1.Chng minh rng )(3 222

222

cbaac

cb

ba

++++ .

399. Cho x,y,z l cc s thc dng tho mn iu kin xy + yz + zx = 1 .Chng minh rng 36)())()((

427 2 ++++++++ xzzyyxxzzyyx .

400. Cho a,b,c l cc s thc dng .Chng minh rng 3

22

22

22

333

++

+

++

+

++

bcac

abcb

caba

.

401.Cho naaaZk ,...,,, 21+ l cc s thc dng tho mn iu kin 1...21 =+++ naaa . Chng minh

rng nk

n

iki

ki n

a

a)1(

1

1

--

=

.

402. Cho naaa ,...,, 21 l cc s thc khng m tho mn iu kin 1...21 =+++ naaa . Chng minh rng

41

... 13221 +++ - nn aaaaaa .

403. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1.Chng minh rng )1)(1)(1(5 cba

ac

cb

ba

++++++ .


acc

cbb

baa

accbba+

++

++

+++++222

3 333 333 33 444444444 .

405.Cho a,b,c > 1 tho mn iu kin .11

11

11

1222

=-

+-

+- cba

Chng minh rng

11

11

11

1

++

++

+ cba .


223

1222222

++

++

+ 1. Chng minh rng

12111

-+

-+

- ac

c

b

b

a.


cbcbabacaac

cb

ba

++++333

.

422. Cho x,y,z l cc s thc dng tho mn iu kin x3 + y3 + z3 = 1.Chng minh rng

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34

2111 2

2

2

2

2

2

-

+-

+- z

z

y

y

x

x .

423.Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 1.Tm gi tr ln nht ca biu thc S = ab + 2bc + 3ca .

424. Cho x,y,z l ba s thc dng tho mn iu kin x3 + y3 + z3 = 3.Tm gi tr ln nht ca biu thc P = 3(xy + yz + zx) xyz .

425.Cho 1,, cba . Chng minh rng

91

11

11

12)()()(

222

++

++

+++++++

cbabacacbcba .

426. Cho a,b,c l ba s thc dng tho mn iu kin a2 + b2 + c2 = 3. Tm gi tr nh nht ca biu thc

M = 2

222

)( cabcabcabcab

++++

.

427. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Hy xc nh gi tr ln nht ca s thc k tho mn bt ng thc ))(1(3

111222

cbakkcba

++++++ .

428.Cho x,y,z [ ]2;1 .Chng minh rng

+

++

++

++++

yxz

xzy

zyx

zyxzyx 6

111)( .

429. Cho a,b,c l cc s thc dng tho mn iu kin abc 1. Chng minh rng

2

3

++

++

+ bca

c

abc

b

acb

a .

430.Cho 1,,21

zyx . Tm gi tr nh nht v gi tr ln nht ca biu thc

P = yxz

xzy

zyx

++

+++

+++

111 .

431. Cho x,y,z l ba s thc khng m tho mn iu kin x + y + z = 4. Tm gi tr nh nht v gi tr ln nht ca biu thc P = 141312 +++++ zyx .

432. Cho a,b,c l cc s thc khng m tho mn iu kin a + b + c = 1. Tm gi tr nh nht v gi tr ln nht ca biu thc P = 333 )()()( bacacbcba -+-+- .


9)(

1

)(

1

)(

14

222

++

++

++

+

++

++ c

baa

cbb

ca

bacacbcbaabc .

434. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = abc .Chng minh rng

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35

4)1()1()1(4

33 cbaabc

abcab

cabca

bc ++

++

++

+ .

435.Cho

22

;0, yx . Tm gi tr ln nht ca biu thc

P = 22 11 x

yy

x+

++

.

436. Cho x,y,z l cc s thc dng tho mn iu kin 2011222222 =+++++ xzzyyx

Tm gi tr nh nht ca biu thc

A = yx

zxz

yzy

x+

++

++

222

.


1111

222

-+

+-+

+-+ ca

cbc

bab

a .

438.Cho x,y,z ( )1;0 tho mn iu kin xyz = (1 x)(1 y)(1 z) . Chng minh rng

43222 ++ zyx .

439.Cho a,b,c 1- tho mn iu kin a + b + c = 143 - . Tm gi tr nh nht ca biu thc P = 333 cba ++ . 440. Cho a,b,c l cc s thc dng .Tm gi tr nh nht ca biu thc

P = abc

cbacbacabcab 3

222

)( +++

++++

.

441.Cho n N .K hiu (2n + 1) !! l tch cc s nguyn dng l t 1 n 2n + 1. Chng minh rng nn nn p!)!12()12( 1 ++ + .

442. Cho a,b,c l cc s thc dng tho mn iu kin abccabcab 3++ . Chng minh rng

1222

444

+

++

++ ac

accb

cbba

ba .

443.Cho a,b,c,d l cc s thc phn bit tho mn iu kin .,4 bdac

ad

dc

cb

ba

==+++

Tm gi tr ln nht ca biu thc P =

2)( cdababcd

db

ac

db

ca

+-+++ .

444. Cho a,b,c l cc s thc dng tho mn iu kin 1abc . Chng minh rng .

11

11

11

ac

cb

ba

cba++

+++

+++

++

445. Cho x,y,z l cc s thc dng .Chng minh rng ).(58)(2 222 zyxzyxxyz ++++++

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36

ng thc xy ra khi no ? 446.Cho a,b,c l ba s thc i mt khc nhau .Chng minh rng

( )49

)(

1

)(

1

)(

1222

222

-

+-

+-

+++++accbba

cabcabcba

ng thc xy ra khi no ? 447. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1.Chng minh rng

).(2111 cbabca

abc

cab

+++++++


abcccab

bbca

aabc

++

++

++

111.

449. Cho x,y,z l cc s thc dng .Chng minh rng

1)(1)(1)(1)(1)(1)(

222

++++

++++

+++

++++

++++

+++ zyxzz

zyxyy

zyxxx

zyxzxy

zyxyzx

zyxxyz

450. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng .333 cbaaccbba ++++ 451. Cho x,y,z l cc s thc dng tho mn iu kin x + y + z = 1.Chng minh rng .

10

9

1

1

1

1

1

1

++

++

+ zxyzxy

452.Cho [ ]1;1, -yx . Chng minh rng

.2

12112

22

+--+-

yxyx

453. Cho a,b,c l cc s thc dng tho mn iu kin ab + bc + ca = 1. Chng minh rng

23

111 222

++

++

+ c

c

b

b

a

a.

454.Cho a,b,x,y l cc s thc dng tho mn a < b. Chng minh rng .)()( abbbaa yxyx ++

455.Cho

21

;0,, cba . Chng minh rng

3

13

11

11

11

-

++

-

-

-

cbacba.

456. Cho a,b,c l cc s thc dng .Chng minh rng 1

)()()( 222222

+++

+++

++ bac

c

acb

b

cba

a .

457. Cho a,b,c l cc s thc dng .Chng minh rng cbacba cbacacbcbaba +++++++ )()2()2()2( 222222 .

458.Chng minh rng nu xy + yz + zx = 5 th 1033 222 ++ zyx .

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37

459. Cho a,b,c l cc s thc dng tho mn iu kin b2 + c2 a2. Tm gi tr nh nht ca biu thc

P =

+++

22222

2

11)(

1cb

acba

460. Cho x,y,z l cc s thc dng tho mn iu kin xy + yz + zx = 1 . Tm gi tr nh nht ca biu thc

P =xyx

z

xzz

y

yzy

x

++

++

+ 333 .

461.Cho a,b,c [ ]1;0 . Chng minh rng

25

111+

++

++

+abc

abc

cab

bca

.

462.Cho a,b,c l cc s thc khng m . Chng minh rng

))()((43

3

333

accbbaabccba

---+++

.

463.Cho x,y,z l ba s thc tho mn 2x + 3y + z = 40. Tm gi tr nh nht ca biu thc P = 3616312 222 +++++ zyx .

464. Cho x, y, z l cc s thc dng tho mn x + y + z = 1

1 1 1 27

:1 1 1 8

CMRxy yz xz+ +

- - -

465. Cho cc s thc x,y tha mn x2 + xy + y2 3 . Chng minh rng 3343334 22 ----- yxyx 466.Cho x,y,z l cc s nguyn dng thay i c tng bng 2002. Tm gi tr nh nht v gi tr ln nht ca biu thc P = x!y!z!

467. Gi s , , a b c l cc s thc dng tha mn 2012 2012 2012

2010 2010 2010 2011a b cb c a

+ + < . Chng minh rng lun tn

ti s t nhin n sao cho 3 3 3 2 2 2

1 1 1

20112010

n n n n n n

n n n n n n

a b c a b cb c a b c a

+ + + + + +

+ + ++ + + + +

468. Cho , , a b c l cc s thc dng. Chng minh rng vi mi s t nhin m ta c bt ng thc

3 3 3 2 2 2

1 1 1

m m m m m m

m m m m m m

a b c a b cb c a b c a

+ + + + + +

+ + ++ + + + .

469. Chng minh rng nu ,x y l cc s thc dng th ( ) ( )2 2

1 1 111 1 xyx y

+ ++ +

470. Cho hm s ( ) ( ): 0; 0;f + + tha mn iu kin ( ) ( )13 2 22

f x f f x x +

vi mi 0x > .

Chng minh rng ( )f x x vi mi 0x > . 471. Cho a, b, c l cc s thc khng m tha mn iu kin a + b + c = 1. Tm gi tr ln nht ca biu thc P = abcaccbba -++ .

472.Cho a,b l cc s khng m tho mn 222 + ba . Tm gi tr ln nht ca biu thc

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38

M = )2(3)2(3 ababbaba +++ .

473.Cho cc s a,b,c,d [ ]1;0 v x,y,z,t

21

;0 tho mn iu kin a + b + c + d = x + y + z + t = 1.

Chng minh rng ax + by + cz + dt 54abcd . 474. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 3.Chng minh rng ))()((5333 accbbacba ++++++ .

ng thc xy ra khi no ?

475.Cho cc s thc x,y,z tho mn

=++

2

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