550 Bdt Chon Loc
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Transcript of 550 Bdt Chon Loc
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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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
++
+++
+++
++
27. Cho a,b,c l cc s thc dng. Chng minh rng
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
32. Cho a,b,c l cc s thc dng. Chng minh rng
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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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
77. Cho a,b,c l cc s thc dng. Chng minh rng
)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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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
89. Cho a,b,c l cc s thc dng. Chng minh rng
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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(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
104. Cho a,b,c l cc s thc dng. Chng minh rng
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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
++
++
++
+
115. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng
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
117. 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(
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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.
128. Cho a,b,c l cc s thc dng. Chng minh rng 1
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 .
131. Cho a,b,c l cc s thc dng. Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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
.
149. Cho a,b,c l cc s thc dng. Chng minh rng 1
222
++
++
+ bac
acb
cba
.
150. Cho a,b,c l cc s thc dng. Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
12
155. Cho x,y,z l ba s thc thuc on
3;31
v zxyx , .Chng minh rng
57
+
++
++ xz
zzy
yyx
x .
156. Cho a,b,c l cc s thc dng. Chng minh rng
+++
+
+
+
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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
.
175. Cho a,b,c l cc s thc dng. Chng minh rng 1
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
.
181. Cho a,b,c l cc s thc dng. Chng minh rng
++++
++
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
14
184. Cho a,b,c,d l cc s thc dng. Chng minh rng
abcd
dcbadcba
++++++
3333
1111 .
185.Cho [ ]1;0,, cba . Chng minh rng
2111
++
++
+ abc
cab
bca
.
186. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng
3
1++ accbba .
187. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng
cc
bb
aa
ca
ac
ab
-+
+-+
+-+
++
11
11
11
2 .
188. Cho a,b,c l cc s thc dng. Chng minh rng
cba
bacba
ac
cb
ba
++-
+++++2222 )(4
.
189. Cho a,b,c l cc s thc dng tho mn iu kin abc = 8. Chng minh rng
34
)1)(1()1)(1()1)(1( 33
2
33
2
33
2
++
+++
+++ ac
c
cb
b
ba
a.
190. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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
.
198. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng
32
)1()1()1( -+-+- bcaabccab .
199. Cho a,b,c l cc s thc dng. Chng minh rng
)(23
cbaba
c
ac
b
cb
a++
++
++
+ .
200. Cho a,b,c l cc s thc dng tho mn iu kin a4 + b4 + c4 = 3. Chng minh rng
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 .
205. Cho a,b,c,d l cc s thc dng. Chng minh rng
42
2
2
2
2
2
2
2
abcd
dcbaad
dc
cb
ba +++
+++ .
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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
16
206.Cho [ ]2;0x . Chng minh rng
433 334 ++- xxxx .
207. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1. Chng minh rng
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 .
211. Cho a,b,c l cc s thc dng. Chng minh rng
++
++
+
++++
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.
213. Cho a,b,c l cc s thc dng. Chng minh rng
cbacba
acac
cbcb
baba
++++
++
+++
+++ 222222222
.3 .
214. Cho a,b,c l cc s thc dng. Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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.
219. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1. Chng minh rng
4
331+
++
++
+ abcabc
cabb
bcaa
.
220. Cho a,b,c l cc s thc dng. Chng minh rng
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 .
222. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 6. Chng minh rng
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= + + + .
226. Cho a,b,c l cc s thc dng. Chng minh rng
++
++
+
++
++
+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 .
229. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 6. Chng minh rng
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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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 .
273. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1.Chng minh rng
.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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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
296. Cho a,b,c l cc s thc dng .Chng minh rng
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.
298. Cho a,b,c l cc s thc dng .Chng minh rng
..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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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 .
304. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1.Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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 .
334. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1.Chng minh rng
2
211
1
211
1
211
1
+++
+++
++c
ab
ca
b
.
335. Cho a,b,c l cc s thc dng .Chng minh rng
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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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 -
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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
28
347. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 1.Chng minh rng
611
11
11
11
11
11
--+--+--accbba
.
348. Cho a,b,c l cc s thc dng tho mn iu kin a + b + c = 2.Chng minh rng
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 .
350. Cho a,b,c l cc s thc dng .Chng minh rng
.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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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 +++++ .
361. Cho a,b,c l cc s thc dng .Chng minh rng
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 .
367. Cho a,b,c l cc s thc dng .Chng minh rng
))(())(())((
222
bcacabc
cbabcab
cababca
bac
acb
cba
+++
+++
++
+++
+
++
++
.
368. Cho a,b,c l cc s thc khng m .Chng minh rng
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++++ .
370. Cho a,b,c l cc s thc dng .Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
30
2 (4 ab bc cd da) )12( + (4 a b c d) .
372. Cho a,b,c l cc s thc dng .Chng minh rng
cabcab
cba
abc
c
cab
b
bca
a
++++
+
++
++ 222 222
.
373. Cho a,b,c l cc s thc khng m .Chng minh rng
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
-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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 .
386. Cho a,b,c l cc s thc dng .Chng minh rng
)(2
111111222 cbacabcabaccbbacba ++
+++
++
++
+++ .
387. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1.Chng minh rng
cabcabaccbba
cabcabbacacbcba
+++++
+++
++
++
+ ))()(()(4
)(1
)(1
)(1
333 .
388. Cho a,b,c l cc s thc dng .Chng minh rng
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
.
392. Cho a,b,c l cc s thc dng tho mn iu kin abc = 1.Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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++++ .
397. Cho a,b,c l cc s thc dng .Chng minh rng
++
++
+
++
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
++++++ .
404. Cho a,b,c l cc s thc dng .Chng minh rng
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 .
406. Cho a,b,c l cc s thc dng .Chng minh rng
223
1222222
++
++
+ 1. Chng minh rng
12111
-+
-+
- ac
c
b
b
a.
421. Cho a,b,c l cc s thc dng .Chng minh rng
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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 -+-+- .
433. Cho a,b,c l cc s thc dng .Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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
.
437. Cho a,b,c l cc s thc dng tho mn iu kin a2 + b2 + c2 = 1. Chng minh rng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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
+++++++
448. Cho a,b,c l cc s thc dng .Chng minh rng
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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550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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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-
550 Bi Ton Chn Lc Bt ng Thc Trn Mnh Cng
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