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国外数学竞赛中的不等式题目(高中)


Problems of Vasc and Arqady
Amir Hossain Parvardi ? i January 6, 2011

Edited by: Sayan Mukherjee 1. Suppose that a, b, c are positive real numbers, prove that √ a b c 3 2 1<

; √ +√ +√ ≤ 2 a2 + b2 b2 + c2 c2 + a2 2. If a, b, c are nonnegative real numbers, no two of which are zero, then ? ? 2 b3 c3 a3 + + + (a + b + c)2 ≥ 4(a2 + b2 + c2 ); b+c c+a a+b a2 b2 c2 3(a2 + b2 + c2 ) + + ≤ . a+b b+c c+a 2(a + b + c)

3. For all reals a, b and c prove that:
2

(a ? b)(a ? c)
cyc cyc

a (a ? b)(a ? c) ≥
cyc

2

a(a ? b)(a ? c)

4. Let a, b and c are non-negatives such that a + b + c + ab + ac + bc = 6. Prove that: 4(a + b + c) + abc ≥ 13 5. Let a, b and c are non-negatives. Prove that: (a2 +b2 ?2c2 ) c2 + ab+(a2 +c2 ?2b2 ) b2 + ac+(b2 +c2 ?2a2 ) a2 + bc ≤ 0 6. If a, b, c are nonnegative real numbers, then ?
cyc

a 3a2 + 5(ab + bc + ca) ≥



2(a + b + c)2 ;

?
cyc

a 2a(a + b + c) + 3bc ≥ (a + b + c)2 ; a 5a2 + 9bc + 11a(b + c) ≥ 2(a + b + c)2 .
cyc

?
? Email:

ahpwsog@gmail.com , blog: http://www.math-olympiad.blogsky.com/

1

7. If a, b, c are nonnegative real numbers, then ?
cyc

a a
cyc

2(a2 + b2 + c2 ) + 3bc ≥ (a + b + c)2 ; 4a2 + 5bc ≥ (a + b + c)2 .

?

8. If a, b, c are nonnegative real numbers, then √ ? a ab + 2bc + ca ≥ 2(ab + bc + ca);
cyc

?
cyc

a

a2 + 4b2 + 4c2 ≥ (a + b + c)2 .

9. If a, b, c are nonnegative real numbers, then a(b + c)(a2 + bc) ≥ 2(ab + bc + ca)
cyc

10. If a, b, c are positive real numbers such that ab + bc + ca = 3, then ?
cyc

a(a + b)(a + c) ≥ 6 a(4a + 5b)(4a + 5c) ≥ 27
cyc

?

11. If a, b, c are nonnegative real numbers, then ?
cyc

a (a + b)(a + c) ≥ 2(ab + bc + ca) a (a + 2b)(a + 2c) ≥ 3(ab + bc + ca)
cyc

? ?
cyc

a (a + 3b)(a + 3c) ≥ 4(ab + bc + ca)

12. If a, b, c are nonnegative real numbers, then ?
cyc

a a
cyc

(2a + b)(2a + c) ≥ (a + b + c)2 (a + b)(a + c) ≥ 2 (a + b + c)2 3

? ?
cyc

a

(4a + 5b)(4a + 5c) ≥ 3(a + b + c)2

2

13. If a, b, c are positive real numbers, then ? ? ? a3 + b3 + c3 + abc + 8 ≥ 4(a + b + c); a3 + b3 + c3 + 3abc + 12 ≥ 6(a + b + c); 4(a3 + b3 + c3 ) + 15abc + 54 ≥ 27(a + b + c).

14. If a, b, c are positive real numbers, then ? ? ? b c a +√ +√ ≤1 2 2 + 3bc + 2c2 2 + 3ca + 2a2 + 3ab + 2b 4b 4c a b c √ +√ +√ ≤1 2 + 2ab + 3b2 2 + 2bc + 3c2 2 + 2ca + 3a2 4a 4b 4c a b c √ +√ +√ ≤1 2 + ab + 4b2 2 + bc + 4c2 2 + ca + 4a2 4a 4b 4c √ 4a2

The last is a known inequality. 15. If a, b, c are positive real numbers, then 1+ a b c + + ≥2 b c a 1+ b c a + + a b c

16. Let x, y, z be real numbers such that x + y + z = 0. Find the maximum value of yz zx xy E= 2+ 2 + 2 x y z 17. If a.b.c are distinct real numbers, then ab bc ca 1 + + + ≥0 (a ? b)2 (b ? c)2 (c ? a)2 4 18. If a and b are nonnegative real numbers such that a + b = 2, then ? ? ? aa bb + ab ≥ 2; aa bb + 3ab ≤ 4; ab ba + 2 ≥ 3ab.

19. Let a, b, c, d and k be positive real numbers such that (a + b + c + d) 1 1 1 1 + + + a b c d =k

Find the range of k such that any three of a, b, c, d are triangle side-lengths. 20. If a, b, c, d, e are positive real numbers such that a + b + c + d + e = 5, then 1 1 1 1 1 14 + 2 + 2 + 2 + 2 +9≥ a2 b c d e 5 3 1 1 1 1 1 + + + + a b c d e

21. Let a, b and c are non-negatives such that ab + ac + bc = 3. Prove that: a b c 1 + + + abc ≤ 2 a+b b+c c+a 2 22. Let a, b, c and d are positive numbers such that a4 + b4 + c4 + d4 = 4. Prove that: b3 c3 d3 a3 + + + ≥4 bc cd da ab 23. Let a ≥ b ≥ c ≥ 0 Prove that: ? ?
cyc

(a ? b)5 + (b ? c)5 + (c ? a)5 ≤ 0 (5a2 + 11ab + 5b2 )(a ? b)5 ≤ 0

24. Let a, b and c are positive numbers. Prove that: a2 b c 3 a + 2 + 2 ≤ √ 3 + bc b + ac c + ab 2 abc

25. Let a, b and c are positive numbers. Prove that: a+b + c b+c + a c+a ≥ b 11(a + b + c) √ ? 15 3 abc

26. Let a, b and c are non-negative numbers. Prove that: 9a2 b2 c2 + a2 b2 + a2 c2 + b2 c2 ? 4(ab + ac + bc) + 2(a + b + c) ≥ 0 27. Let a, b, c, d be nonnegative real numbers such that a ≥ b ≥ c ≥ d and 3(a2 + b2 + c2 + d2 ) = (a + b + c + d)2 . Prove that ? ? ? a ≤ 3b; a ≤ 4c; √ b ≤ (2 + 3)c.

28. If a, b, c are nonnegative real numbers, no two of which are zero, then ? ? bc ca ab a2 + b2 + c2 + 2 + 2 ≤ ; 2a2 + bc 2b + ca 2c + ab ab + bc + ca 2bc 2ca 2ab a2 + b2 + c2 + 2 + 2 + ≥ 3. 2 + 2bc a b + 2ca c + 2ab ab + bc + ca

4

29. Let a1 , a2 , ..., an be real numbers such that a1 , a2 , ..., an ≥ n ? 1 ? Prove that 1 + (n ? 1)2 , a1 + a2 + ... + an = n.

1 1 n 1 + + ··· + 2 ≥ . a2 + 1 a2 + 1 an + 1 2 1 2

30. Let a, b, c be nonnegative real numbers such that a + b + c = 3. For given real p = ?2, ?nd q such that the inequality holds a2 1 1 3 1 + 2 + 2 ≤ , + pa + q b + pb + q c + pc + q 1+p+q

With two equality cases. Some particular cases: ? 1 1 1 1 + + ≤ , a2 + 2a + 15 b2 + 2b + 15 c2 + 2c + 15 6
3 With equality for a = 0 and b = c = 2 ;

?

1 1 1 1 + + ≤ , 8a2 + 8a + 65 8b2 + 8b + 65 8c2 + 8c + 65 27 With equality for a =
5 2 1 and b = c = 4 ;

? 8a2 ?

1 1 1 1 + 2 + 2 ≤ , ? 8a + 9 8b ? 8b + 9 8c ? 8c + 9 3
3 2 3 and b = c = 4 ;

With equality for a =

1 1 1 1 + + ≤ , 8a2 ? 24a + 25 8b2 ? 24b + 25 8c2 ? 24c + 25 3 With equality for a =
1 2 5 and b = c = 4 ;

?

1 1 1 1 + + ≤ , 2a2 ? 8a + 15 2b2 ? 8b + 15 2c2 ? 8c + 15 3 With equality for a = 3 and b = c = 0.

31. If a, b, c are the side-lengths of a triangle, then a3 (b + c) + bc(b2 + c2 ) ≥ a(b3 + c3 ). 32. Find the minimum value of k > 0 such that a2 a b c 9 + 2 + 2 ≥ , + kbc b + kca c + kab (1 + k)(a + b + c) 1 1 1 , , , is known. a b c

for any positive a, b, c. See the nice case k = 8. PS. Actually, this inequality, with a, b, c replaced by 5

33. If a, b, c, d are nonnegative real numbers such that a + b + c + d = 4, then a3 + b3 + c3 + d3 ≤ 16. 34. If a ≥ b ≥ c ≥ 0, then ? ? 35. If a ≥ b ≥ 0, then a?b ab?a ≤ 1 + √ . a 36. If a, b ∈ (0, 1], then ab?a + ba?b ≤ 2. 37. If a, b, c are positive real numbers such that a + b + c = 3, then 1 24 + ≥ 9. a2 b + b2 c + c2 a abc 38. Let x, y, z be positive real numbers belonging to the interval [a, b]. Find the best M (which does not depend on x, y, z) such that x + y + z ≤ 3M √ 3 xyz. √ 64(a ? b)2 3 a + b + c ? 3 abc ≥ ; 7(11a + 24b) √ 25(b ? c)2 3 . a + b + c ? 3 abc ≥ 7(3b + 11c) a2 + b2 + c2 + d2 = 7,

39. Let a and b be nonnegative real numbers. a?b (a) If 2a2 + b2 = 2a + b, then 1 ? ab ≥ ; 3 3 3 4 4 4 4 (b) If a + b = 2, then 3(a + b ) + 2a b ≤ 8. 40. Let a, b and c are non-negative numbers. Prove that: √ √ √ √ a + b + c + ab + ac + bc + 3 abc 7 (a + b + c)(a + b)(a + c)(b + c)abc ≥ 7 24 41. Let a, b, c and d are non-negative numbers such that abc+abd+acd+bcd = 4. Prove that: 1 1 1 1 3 7 + + + ? ≤ a+b+c a+b+d a+c+d b+c+d a+b+c+d 12

6

42. Let a, b, c and d are positive numbers such that ab+ac+ad+bc+bd+cd = 6. Prove that: 1 1 1 1 + + + ≤1 a+b+c+1 a+b+d+1 a+c+d+1 b+c+d+1 43. Let x ≥ 0. Prove without calculus: (ex ? 1) ln(1 + x) ≥ x2 . 44. Let a, b and c are positive numbers. Prove that: √ c 24 3 abc a b + + + ≥ 11 b c a a+b+c 45. For all reals a, b and c such that
cyc

(a2 + 5ab) ≥ 0 prove that:

(a + b + c)6 ≥ 36(a + b)(a + c)(b + c)abc The equality holds also when a, b and c are roots of the equation: 2x3 ? 6x2 ? 6x + 9 = 0 46. Let a, b and c are non-negative numbers such that ab + ac + bc = 0. Prove that: a2 (b + c)2 (c + a)2 3 (a + b)2 + 2 + 2 ≥ 2 2 + 3ab + 4b b + 3bc + 4c c + 3ca + 4a2 2

47. a, b and c are real numbers such that a + b + c = 3. Prove that: 1 1 1 1 + + ≤ 2 + 14 2 + 14 2 + 14 (a + b) (b + c) (c + a) 6 48. Let a, b and c are real numbers such that a + b + c = 1. Prove that: b c 9 a + + ≤ a2 + 1 b2 + 1 c2 + 1 10 49. Let a, b and c are positive numbers such that 4abc = a + b + c + 1. Prove that: b2 + c 2 c2 + a2 b2 + a 2 + + ≥ 2(a2 + b2 + c2 ) a b c 50. Let a, b and c are positive numbers. Prove that: (a + b + c) 1 1 1 + + a b b ≥ 1 + 2 3 6(a2 + b2 + c2 ) 1 1 1 + 2+ 2 2 a b c + 10

7

51. Let a, b and c are positive numbers. Prove that: a2 b2 c2 37(a2 + b2 + c2 ) ? 19(ab + ac + bc) + + ≥ b c a 6(a + b + c) 52. Let a, b and c are positive numbers such that abc = 1. Prove that a3 + b3 + c3 + 4 1 1 1 + 3+ 3 a3 b c + 48 ≥ 7(a + b + c) 1 1 1 + + a b c

53. Let a, b and c are non-negative numbers such that ab + ac + bc = 3. Prove that: 1 1 1 3 ? + + ≥ 2 2 2 1+a 1+b 1+c 2 1 1 3 1 + + ≥ ? 3 3 3 2 + 3a 2 + 3b 2 + 3c 5 1 1 1 3 ? + + ≥ 3 + 5a4 3 + 5b4 3 + 5c4 8 54. Let a, b and c are non-negative numbers such that ab + ac + bc = 0. Prove that a+b+c a b c a3 + b3 + c3 ≤ 2 + 2 + 2 ≤ 2 2 ab + ac + bc b + bc + c2 a + ac + c2 a + ab + b2 a b + a2 c2 + b2 c2 55. Let a, b and c are non-negative numbers such that ab + ac + bc = 3. Prove that 2 2 2 a+b+c 5 a b + b c + c a ? ≥ 3 3 3 b + b3 c + c3 a a+b+c 11 a ? ≥ 3 3 56. Let a, b and c are non-negative numbers. Prove that (a2 + b2 + c2 )2 ≥ 4(a ? b)(b ? c)(c ? a)(a + b + c) 57. Let a, b and c are non-negative numbers. Prove that: (a + b + c)8 ≥ 128(a5 b3 + a5 c3 + b5 a3 + b5 c3 + c5 a3 + c5 b3 ) 58. Let a, b and c are positive numbers. Prove that a2 ? bc b2 ? ac c2 ? ab + + ≥0 3a + b + c 3b + a + c 3c + a + b It seems that
cyc

a3 ? bcd ≥ 0 is true too for positive a, b, c and d. 7a + b + c + d

8

59. Let a, b and c are non-negative numbers such that ab + ac + bc = 3. Prove that: ? a2 + b2 + c2 + 3abc ≥ 6 ? a4 + b4 + c4 + 15abc ≥ 18

60. Let a, b and c are positive numbers such that abc = 1. Prove that a2 b + b2 c + c2 a ≥ 3(a2 + b2 + c2 )

61. Let a, b and c are non-negative numbers such that ab + ac + bc = 0. Prove that: a3 1 1 1 81 + 3 + 3 ≥ 3 3 3 + 3abc + b a + 3abc + c b + 3abc + c 5(a + b + c)3

62. Let ma , mb and mc are medians of triangle with sides lengths a, b and c. Prove that ma + mb + mc ≥ 3 2 2(ab + ac + bc) ? a2 ? b2 ? c2

63. Let a, b and c are positive numbers. Prove that: a+b+c a2 b2 c2 √ ≥ + 2 + 2 3 2 + 5bc 4a 4b + 5ca 4c + 5ab 9 abc 64. Let {a, b, c, d} ? [1, 2]. Prove that 16(a2 + b2 )(b2 + c2 )(c2 + d2 )(d2 + a2 ) ≤ 25(ac + bd)4 65. Let a, b and c are positive numbers. Prove that a2 ? ab + b2 ≤
cyc

10(a2 + b2 + c2 ) ? ab ? ac ? bc 3(a + b + c)

66. Let a, b and c are non-negative numbers. Prove that: 2(a2 + b2 ) ≥
cyc
3

9
cyc

(a + b)3

67. Let a, b and c are positive numbers. Prove that: c a b + + ≥ b c a 15(a2 + b2 + c2 ) ?6 ab + bc + ca

68. Let a, b, c, d and e are non-negative numbers. Prove that (a + b)(b + c)(c + d)(d + e)(e + a) 32 9
128



a+b+c+d+e 5

125

(abcde)103

69. Let a, b and c are positive numbers. Prove that a b+a c+b a+b c+c a+c b≥6
2 2 2 2 2 2

a2 + b2 + c2 ab + ac + bc

4 5

abc

70. Let a, b and c are non-negative numbers such that a3 + b3 + c3 = 3. prove that (a + b2 c2 )(b + a2 c2 )(c + a2 b2 ) ≥ 8a2 b2 c2 71. Given real di?erent numbers a, b and c. Prove that: (a2 + b2 + c2 ? ab ? bc ? ca)3 (a ? b)(b ? c)(c ? a) 1 1 1 + + 3 3 (a ? b) (b ? c) (c ? a)3 ≤? 405 16

72. Let x = 1, y = 1 and x = 1 such that xyz = 1. Prove that: x2 (x ? 1)
2

+

y2 (y ? 1)
2

+

z2 (z ? 1)
2

≥1

When does the equality occur? 73. Let a, b, and c are non-negative numbers such that a + b + c = 3. Prove that: a5 + b5 + c5 + 6 ≥ 3(a3 + b3 + c3 ) 74. a > 1, b > 1 and c > 1. Find the minimal value of the expression: a3 b3 c3 + + a+b?2 b+c?2 c+a?2 75. For all non-negative a, b and c prove that: (ab ? c2 )(a + b ? c)3 + (ac ? b2 )(a + c ? b)3 + (bc ? a2 )(b + c ? a)3 ≥ 0 76. Let a, b, c and d are positive numbers such that a4 + b4 + c4 + d4 = 4. Prove that a2 b2 c2 d2 + + + ≥4 b c d a Remark. This inequality is not true for the condition a5 +b5 +c5 +d5 = 4. 1 1 1 77. Let a, b and c are positive numbers such that √ + √ + √ = 3. Prove a c b that 1 1 1 3 + + ≤ a+b a+c b+c 2 78. Let a, b and c are positive numbers such that abc = 1. Prove that: (a + b + c)3 ≥ 63 1 1 1 + + 5a3 + 2 5b3 + 3 5c3 + 2 10

79. Let a, b and c are positive numbers such that max{ab, bc, ca} ≤ and a + b + c = 3. Prove that a2 + b2 + c2 ≥ a2 b2 + b2 c2 + c2 a2

ab + ac + bc 2

80. Let a, b and c are positive numbers such that a + b + c = 3. Prove that: a2 b2 c2 3 + + ≥ 3a + b2 3b + c2 3c + a2 4 81. Let a, b and c are non-negative numbers and k ≥ 2. Prove that ? ?
cyc

2a2 + 5ab + 2b2 + a2 + kab + b2 ≤

2a2 + 5ac + 2c2 +

2b2 + 5bc + 2c2 ≤ 3(a + b + c);

4(a2 + b2 + c2 ) + (3k + 2)(ab + ac + bc).

82. Let x, y and z are non-negative numbers such that x2 + y 2 + z 2 = 3. Prove that: √ y z x + + ≤ 3 x2 + y + z x + y2 + z x + y + z2 83. Let a, b and c are non-negative numbers such that a + b + c = 3. Prove that a+b a+c b+c 3 + + ≥ ab + 9 ac + 9 bc + 9 5 84. If x, y, z be positive reals, then √ y z x +√ +√ ≥ x+y y+z z+x
4

27(yz + zx + xy) 4

85. For positive numbers a, b, c, d, e, f and g prove that: a+b+c+d c+d+e+f e+f +a+b + > a+b+c+d+f +g c+d+e+f +b+g e+f +a+b+d+g 86. Let a, b and c are non-negative numbers. Prove that: a 4a2 + 5b2 + b 4b2 + 5c2 + c 4c2 + 5a2 ≥ (a + b + c)2

87. Let a, b and c are positive numbers. Prove that: √ √ a b c (4 2 ? 3)(ab + ac + bc) + + + ≥4 2 2 + b2 + c2 b c a a 88. Let a, b and c are non-negative numbers such, that a4 + b4 + c4 = 3. Prove that: a5 b + b5 c + c5 a ≤ 3 11

89. Let a and b are positive numbers, n ∈ N. Prove that: (n + 1)(an+1 + bn+1 ) ≥ (a + b)(an + an?1 b + · · · + bn ) 90. Find the maximal α, for which the following inequality holds for all nonnegative a, b and c such that a + b + c = 4. a3 b + b3 c + c3 a + αabc ≤ 27 91. Let a, b and c are non-negative numbers. Prove that
10 + c10 a10 + c10 10 b + 2 2 √ √ 92. Let a and b are positive numbers and 2 ? 3 ≤ k ≤ 2 + 3. Prove that

3

9

a9 + b9 + c9 ≥ 3

10

a10 + b10 + 2

10



√ a+ b



1 1 +√ a + kb b + ka

≤√

4 1+k

93. Let a, b and c are nonnegative numbers, no two of which are zeros. Prove that: b c 3(a + b + c) a + 2 + 2 ≥ 2 . b2 + c2 a + c2 a + b2 a + b2 + c2 + ab + ac + bc 94. Let x, y and z are positive numbers such that xy + xz + yz = 1. Prove that y3 z3 (x + y + z)3 x3 + + ≥ . 2 xz 2 yx 2 yz 1 ? 4y 1 ? 4z 1 ? 4x 5 95. Let a, b and c are positive numbers such that a6 + b6 + c6 = 3. Prove that: (ab + ac + bc) a2 b2 c2 + 2+ 2 b2 c a ≥ 9.

96. Let a, b and c are positive numbers. Prove that
3

a + 2b + 25c

3

b + 2c + 25a

3

c ≥ 1. 2a + 25b

97. Let a, b and c are sides lengths of triangle. Prove that (2a + b)(2b + c)(2c + a) (a + b)(a + c)(b + c) ≥ . 8 27 98. Let a, b and c are non-negative numbers. Prove that
3

(2a + b)(2b + c)(2c + a) ≥ 27 12

ab + ac + bc . 3

99. Let a, b and c are positive numbers. Prove that a3 + + (c + a)3 b3 + + (a + b)3 c3 ≥ 1. + (b + c)3

b3

c3

a3

100. Let x, y and z are non-negative numbers such that xy +xz +yz = 9. Prove that (1 + x2 )(1 + y 2 )(1 + z 2 ) ≥ 64.

13


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