Tài liệu Am_gm_bcs_the_art_of_inequality_2018

Pham Quoc Sang - Christos Eythymioy Le Minh Cuong The art of Mathematics - TAoM THE ART OF INEQUALITY AM-GM, BCS, Holder... Ho Chi Minh City - Athens - 2018 1 Theory. 2 Problem. Problem 1. If a, b, c ∈ (0; 2) such that ab + bc + ca = 3 then b c a + + >3 2−a 2−b 2−c Proposed by Pham Quoc Sang Problem 2. If a, b, c are positive real numbers such that abc = 1 then a b c 3 + + > ab + 1 bc + 1 ca + 1 2 Proposed by Pham Quoc Sang Problem 3. If a, b, c, k are positive real numbers such that abc = 1 then b2 c2 3 a2 + + > (ab + k) (kab + 1) (bc + k) (kbc + 1) (ca + k) (kca + 1) (k + 1)2 Proposed by Pham Quoc Sang Problem 4. If a, b, c are positive real numbers then a b c 3 √ +√ +√ 6√ 2 2 2 2 2 2 2 2 2 3a + 2b + 2c 3b + 2c + 2a 3c + 2a + 2b 7 Proposed by Pham Quoc Sang Problem 5. If a, b, c are positive real numbers and k ≥ 1 then √ ka2 a b c 3 +√ +√ 6√ 2 2 2 2 2 2 2 2 k+2 +b +c kb + c + a kc + a + b Proposed by Pham Quoc Sang 13 Problem 6. If a, b, c are positive real numbers and k ≥ such that abc = 1 then 4 a b c 3 √ +√ +√ >√ k+2 b2 + c 2 + k c 2 + a2 + k a2 + b 2 + k Proposed by Pham Quoc Sang Problem 7. If a, b, c are positive real numbers such that a + b + c = 3 then a2 + 1 b2 + 1 c2 + 1 + + >1 c+5 a+5 b+5 Proposed by Pham Quoc Sang Problem 8. If a, b, c, α, β are positive real numbers and n ∈ N∗ such that a + b + c = 3 then an + n − 1 b n + n − 1 c n + n − 1 3n + + > αb + βc αc + βa αa + βb α+β 1 Proposed by Pham Quoc Sang Problem 9. If a, b, c are positive real numbers such that abc = 1 then a−1 b−1 c−1 + + 60 2a + 1 2b + 1 2c + 1 Proposed by Pham Quoc Sang Problem 10. If a1 , a2 ...an are positive real numbers such that a1 m + a2 m + ... + an m 6 1, . m ∈ N∗ , k > m, k ..m. Find the minimum value of n X ai m i=1 n X 1 + ak i=1 i Proposed by Pham Quoc Sang, RMM 9/2017 Problem 11. If a, b, c, k are positive real numbers then r r r √ a b c k+ + k+ + k+ > 9k 2 + 4 b+c c+a a+b Proposed by Pham Quoc Sang - Nguyen Duc Viet 1 then Problem 12. If a, b, c, k are positive real numbers and k 6 48 r r r √ √ a b c k+ + k+ + k+ >2 k+1+ k b+c c+a a+b Proposed by Nguyen Duc Viet Problem 13. If a, b, c are positive real numbers then a2 + b 2
b2 + c 2

8 c2 + a2 > (ab + bc + ca)3 27 Proposed by Pham Quoc Sang, RMM 11/2017 Problem 14. Let n ∈ N∗ and a, b, c are all different real numbers such that a + b + c = 0. Find the minimum value of a 2n +b 2n +c 2n
 1 1 1 2n + 2n + (a − b) (b − c) (c − a)2n  Proposed by Pham Quoc Sang Problem 15. Let a, b, c be real numbers such that abc 6= 0, a + b + c = 0 . Find min of P = 1 a2 b 2 c 2 .|ab + bc + ca|3 Proposed by Pham Quoc Sang Problem 16. If a, b, c are positive real numbers such that abc = 1 then a3 + b3 + c3 + (a2 + 2) (b2 + 2) (c2 + 2) > 12 a2 + b 2 + c 2 2 Proposed by Pham Quoc Sang, RMM 10/2017 Problem 17. If a, b, c are positive real numbers then 1 5 (a + b + c) 3 + + 2 >7 abc 3 a + b2 + c 2 Proposed by Pham Quoc Sang Problem 18. If a, b, c are positive real numbers such that a + b + c = 3 then √ a + 2bc + √ b + 2ca + √ √ √ √ 3 3 c + 2ab + a2 + b2 + c2 6 3 3 + 3 Proposed by Pham Quoc Sang Problem 19. If a, b, c, α, β are positive real numbers such that a + b + c = 3 and β > 12α then αabc + β β >α+ ab + bc + ca 3 Proposed by Pham Quoc Sang Problem 20. If a, b, c are positive real numbers and k ∈ [0; 9] then a3 + b 3 + c 3 ab + bc + ca + k. 2 >3+k abc a + b2 + c 2 Proposed by Pham Quoc Sang Note. The above inequalities are extended as follows: If a, b, c are positive real numbers and k ≤ 9 then a3 + b 3 + c 3 ab + bc + ca + k. 2 >3+k abc a + b2 + c 2 This interesting extension was proposed by Nguyen Trung Hieu. Problem 21. Let a, b, c be sides of triangle such that a + b + c = a2 + b2 + c2 . Prove that 2a2 a b c + 2 + 2 >1 + bc 2b + ca 2c + ab Proposed by Pham Quoc Sang, RMM 11/2017 Problem 22. If a, b, c be positive real number such that abc = 1 then 2(a + b)(b + c)(c + a) + 11 ≥ 9(a + b + c) 3 Proposed by Pham Quoc Sang, RMM 11/2017 Problem 23. If a, b, c be positive real number such that a + b + c = ab + bc + ca then (a − b)2 (b − c)2 (c − a)2 (a − b)2 + (b − c)2 + (c − a)2 + + > b c a 2 Proposed by Pham Quoc Sang, RMM 10/2017 Problem 24. If a, b, c be positive real number such that a + b + c = 6 then (a − b)2 (b − c)2 (c − a)2 + + + 2 (ab + bc + ca) > 24 b c a Proposed by Pham Quoc Sang, RMM 11/2017 Problem 25. If a, b, c be positive real number then  4 a2 b2 c2 + + >a+b+c+ .max (a − b)2 , (b − c)2 , (c − a)2 b c a a+b+c Proposed by Pham Quoc Sang, RMM 11/2017 Problem 26. If a, b, c be positive real number then  6 a2 b 2 c 2 + + >a+b+c+ .min (a − b)2 , (b − c)2 , (c − a)2 b c a a+b+c Proposed by Do Huu Duc Thinh Problem 27. If a, b, c be positive real number such that a 6 b 6 c then     a b c 1 1 1 2 + + + 3 > (a + b + c) + + b c a a b c Proposed by Pham Quoc Sang, RMM 11/2017 Problem 28. If a, b, c be positive real number and k ≥ 2 then a + kb b + kc c + ka + + >3 ka + b kb + c kc + a Proposed by Pham Quoc Sang - Le Minh Cuong, TAoM 11/2017 Problem 29. If a, b, c, α, β be positive real number and α2 + β 2 > 4αβ then αa + βb αb + βc αc + βa + + >3 βa + αb βb + αc βc + αa 4 Proposed by Pham Quoc Sang - Le Minh Cuong, TAoM 11/2017 Problem 30. If a, b, c be positive real number such that a + b + c = 3 then 1 1 1 3 + 2 + 2 > (a + b) (b + c) (c + a) 2 a b c 8 Proposed by Pham Quoc Sang, TAoM 12/2017 Problem 31. If a, b, c be positive real number such that a + b + c = 3 then ab bc ca 1 + + 6 (2a + bc) (2b + ca) (2b + ca) (2c + ab) (2c + ab) (2a + bc) 3 Proposed by Pham Quoc Sang, RMM 12/2017 Problem 32. If a, b, c are positive real number such that ab + bc + ca = 3 then a3 1 1 1 + 3 + 3 61 2 2 + b + c b + c + a c + a2 + b Proposed by Pham Quoc Sang, TAoM 12/2017 Problem 33. If a, b, c are positive real number and k ≥ 2 then b2 b c 9 a + 2 + 2 > 2 2 2 + kbc + c c + kca + a a + kab + b (k + 2) (a + b + c) Proposed by Pham Quoc Sang, RMM 12/2017 Problem 34. If a, b, c are positive real number such that ab + bc + ca = 3 then a2 b2 c2 + + >1 a2 + 2 b2 + 2 c2 + 2 Proposed by Pham Quoc Sang Problem 35. If a, b, c are positive real number such that abc = 1 then a2 a b c 3 + 2 + 2 6 + bc b + ca c + ab 2 Proposed by Pham Quoc Sang Problem 36. If a, b, c are positive real number such that a2 + b2 + c2 = 3 then a b c a+b+c + + ≥ 2b + 1 2c + 1 2a + 1 3 Proposed by Pham Quoc Sang Problem 37. If a, b, c are positive real number then a3 (a + b) b3 (b + c) c3 (c + a) b (a2 + b2 ) c (b2 + c2 ) a (c2 + a2 ) + + > + + a2 + b 2 b2 + c 2 c 2 + a2 a+b b+c c+a Proposed by Pham Quoc Sang Problem 38. Let a, b, c, α, β be positive real number. 5 a) If β > 2α then a2 b2 c2 3 + + > αa2 + βbc αb2 + βca αc2 + βab α+β b) If α > 2β then a2 b2 c2 3 + + 6 2 2 2 αa + βbc αb + βca αc + βab α+β Proposed by Pham Quoc Sang Problem 39. If a, b, c are positive real number such that a + b + c = 3 then (2a + c)2 (2b + a)2 (2c + b)2 + 2 + 2 69 a2 + 2 b +2 c +2 Proposed by Pham Quoc Sang Problem 40. If a, b, c are positive real number and k is a positive integer then   ak+1 + bk+1 + ck+1 2 bc ca ab > + + ak + b k + c k 3 a+b b+c c+a Proposed by Pham Quoc Sang Problem 41. If a, b, c are positive real number such that a2 + b2 + c2 = 3 then b2 + bc + 1 c2 + ca + 1 a2 + ab + 1 √ √ √ + + >a+b+c 2a2 + 5ab + 2b2 2b2 + 5bc + 2c2 2c2 + 5ca + 2a2 Proposed by Pham Quoc Sang - Le Minh Cuong, TAoM 12/2017 Problem 42. If a, b, c are positive real number such that (a + b)(b + c)(c + a) = 8 then a2 1 1 1 ab + bc + ca + 2 + 2 > 2 2 2 + ab + b b + bc + c c + ca + a a+b+c Proposed by Pham Quoc Sang, TAoM 12/2017 Problem 43. If a, b, c are positive real numbers then a b c 9 9 + + + > b + 1 c + 1 a + 1 4(a + b + c) 4 Proposed by Pham Quoc Sang 6 Problem 44. If a, b, c are positive real numbers and k is a positive integer then   ak+1 + bk+1 + ck+1 1 a2 + b 2 b 2 + c 2 c 2 + a2 + + > ak + b k + c k 3 a+b b+c c+a Proposed by Pham Quoc Sang Problem 45. If a, b, c are positive real numbers such that a + b + c = 3 then √ bc ca ab 3 +√ +√ 6 2 a2 + 3 b2 + 3 c2 + 3 Proposed by Pham Quoc Sang Problem 46. If a, b, c are positive real numbers such that ab + bc + ca = 3 then 1 1 1 3 2 + 2 + 2 > 4 (a + 1) (b + 1) (c + 1) Proposed by Konstantinos Metaxas, Athens, Greece Problem 47. If a, b, c are positive real numbers then a2 + c 2 b 2 + a2 c 2 + b 2 + + > 2 (a + b + c) b c a Inequalities Book Problem 48. If a, b, c are positive real numbers such that abc = 1 then 1 1 9 1 + + > 1+a 1+b 1+c 3 + ab + bc + ca Proposed by Pham Quoc Sang Problem 49. If a, b, c are positive real numbers such that a + b + c = 1 then 1 − 2ab 1 − 2bc 1 − 2ca 7 + + > a+b b+c c+a 2 Proposed by Pham Quoc Sang, RMM 12/2017 Problem 50. If a, b, c are positive real numbers such that a + b + c + 1 = 4abc then a) ab + bc + ca > 3abc b) b 2 + c 2 c 2 + a2 a2 + b 2 + + > 6abc a b c 7 Proposed by Pham Quoc Sang Problem 51. If a, b, c are positive real numbers such that abc = 1 then a b c 6 + + + >6 b c a a+b+c−1 Proposed by Pham Quoc Sang Problem 52. If a, b, c are positive real numbers then √ √ √ 2a2 + a bc 2b2 + b ca 2c2 + c ab 9 ab + bc + ca + + > . b+c c+a a+b 2 a+b+c Proposed by Pham Quoc Sang Problem 53. Let a, b, c ≥ 0 and k ≥ k. 4 prove that: 27 (a + b + c)3 ab + bc + ca 2 +( 2 ) ≥ 27k + 1 abc a + b2 + c 2 Proposed by Phan Dinh Dan Truong Problem 54. If a, b, c are positive real numbers such that abc = 1 then 3 (a + b + c) a+b b+c c+a + + > b+c c+a a+b ab + bc + ca Proposed by Phan Ngoc Chau Problem 55. If a, b, c, k are positive real numbers such that a + b + c = k then 3 ka − bc kb − ca kc − ab + + 6 ka + bc kb + ca kc + ab 2 Proposed by Pham Quoc Sang Problem 56. If a, b, c are positive real numbers such that ab + bc + ca = 3 then √ √ √ √ 3 3 3 3 a + bc + b + ca + c + ab > 3 2abc Proposed by Pham Quoc Sang Problem 57. If a, b, c are positive real numbers such that ab + bc + ca = 3 then  2  2  2 1 1 1 a+ + b+ + c+ > 12 bc ca ab Proposed by Pham Quoc Sang 8 Problem 58. If a, b, c are positive real numbers such that ab + bc + ca = 3 then  2  2  2 1 1 1 27 a+ + b+ + c+ > b+c c+a a+b 4 Proposed by Pham Quoc Sang Problem 59. If a, b, c are positive real numbers such that a2 + b2 + c2 = 3 then  2  2  2 1 1 1 a+ + b+ + c+ > 12 b c a Proposed by Pham Quoc Sang Problem 60. If a, b, c are positive real numbers such that a2 + b2 + c2 = 3 then 1 1 1 3 + + 6 2ab + bc + ca 2bc + ca + ab 2ca + ab + bc 4abc Proposed by Pham Quoc Sang Problem 61. If a, b, c are positive real numbers such that a + b + c = 3 then   1 1 1 1 1 1 1 + + 6 . + + a+1 b+1 c+1 4 a b c Proposed by Pham Quoc Sang Problem 62. If a, b, x, y are positive real numbers such that a + b = x + y = 1 then   1 1 (ab + xy) + >4 ay bx Proposed by Pham Quoc Sang Problem 63. If a, b, c are positive real numbers such that a2 + b2 + c2 = 3 then 1 1 1 + + >1 a + ab + abc b + bc + bca c + ca + cab Proposed by Pham Quoc Sang Problem 64. If a, b, c are positive real numbers such that a6 + b6 + c6 = 3 then 1 1 1 3 + + > ab (a + b) bc (b + c) ca (c + a) 2 Proposed by Pham Quoc Sang Problem 65. If a, b, c are positive real numbers then 1 1 1 729 ab + bc + ca . 2 + 2 + 2 > 4 (a + b + c)6 ab(a + b) bc(b + c) ca(c + a) Proposed by Pham Quoc Sang Problem 66. If a, b, c are positive real numbers then 1 1 1 81 2 + 2 + 2 > a(b + c) b(c + a) c(a + b) 4(a + b + c)3 9 Proposed by Pham Quoc Sang Problem 67. If a, b, c are positive real numbers and k ∈ N, k > 2 then 1 a(b + c)k + 1 b(c + a)k + 1 c(a + b)k > 3k+2 2k (a + b + c)k+1 Proposed by Pham Quoc Sang Problem 68. If a, b, c are positive real numbers then a2 1 1 81 1 + 2 + 2 > (b + c) b (c + a) c (a + b) 2(a + b + c)3 Proposed by Pham Quoc Sang Problem 69. If a, b, c are positive real numbers and k ∈ N, k > 2 then 1 1 1 3k+2 + + > ak (b + c) bk (c + a) ck (a + b) 2(a + b + c)k+1 Proposed by Pham Quoc Sang Problem 70. If a, b, c are positive real numbers then 1 1 1 9 2 + 2 + 2 > 3 4 (a + b3 + c3 ) a(b + c) b(c + a) c(a + b) Proposed by Pham Quoc Sang Problem 71. If a, b, c are positive real numbers then √ √ √ 81 ab + bc + ca 2a + bc 2b + ca 2c + ab . 2 + 2 + 2 > 4 (a + b + c)3 (b + c) (c + a) (a + b) Proposed by Pham Quoc Sang Problem 72. (Prove or deny) If a, b, c are positive real numbers such that abc ≥ 1 then     1 1 1 27 a+ b+ c+ > b+c c+a a+b 8 Proposed by Pham Quoc Sang Problem 73. If a, b, c are positive real numbers such that abc ≥ 1 then 2a3 + abc 2b3 + abc 2c3 + abc 9 2 + 2 + 2 > 4 (b + c) (c + a) (a + b) Proposed by Pham Quoc Sang Problem 74. If a, b, c are positive real numbers and α, β ∈ N∗ then        n αa1 αa2 αa3 αan α 1+ 1+ 1+ ... 1 + > 1+ βa2 βa3 βa4 βa1 β Proposed by Pham Quoc Sang 10 Problem 75. If a, b, c are positive real numbers such that a + b + c = a3 + b3 + c3 then r 6 3 a + b + c a+b+c+ >3 + 2abc a+b+c 3 Proposed by Pham Quoc Sang Problem 76. If a, b, c are positive real numbers such that a + b + c = a2 + b2 + c2 then p a) a + b + c + 6 > 3 3 9 (ab + bc + ca) (a + b + c + 6)3 (a + b + c)2 b) > 2430 + 729. a2 + b 2 + c 2 ab + bc + ca Proposed by Pham Quoc Sang Problem 77. If a, b, c are positive real numbers such that a2 + b2 + c2 = 3 then (a + b) (b + c) (c + a) 6 4abc + 4 Proposed by Pham Quoc Sang Problem 78. If a, b, c are positive real numbers such that a2 + b2 + c2 = 3 then 6+ 27 a+b+c > abc a+b+c Proposed by Pham Quoc Sang, RMM 9/2017 Problem 79. If a, b, c are positive real numbers then 1 1 1 1 1 1 + + 2 + 2 + 2 > 2 + a (b + c) 2 + b (c + a) 2 + c (a + b) (a + 1) (b + 1) (c + 1) Proposed by Pham Quoc Sang Problem 80. If a, b, c are positive real numbers such that a2 + b2 + c2 = 3 then 1 1 1 2 2 2 + + 2 + 2 + 2 > 2 a + b 2 + 6 b 2 + c 2 + 6 c 2 + a2 + 6 (a + b) (b + c) (c + a) Proposed by Pham Quoc Sang Problem 81. If a, b, c are positive real numbers then 1 1 1 3 2 + 2 + 2 > 2 (ab + bc + ca) (a + b) (b + c) (c + a) Proposed by Pham Quoc Sang Problem 82. If a, b, c are positive real numbers such that abc = 1 then   a b c 2 2 2 a +b +c >2 + + a2 + bc b2 + ca c2 + ab Proposed by Pham Quoc Sang 11 Problem 83. If a, b, c are positive real numbers such that abc = 1 then a2 1 1 1 3 (ab + bc + ca) + 2 + 2 6 √ √ √ 2 + 2bc b + 2ca c + 2ab a+ b+ c Proposed by Pham Quoc Sang Problem 84. If a, b, c are positive real numbers then  1 1 1 (a + b + c) + + b c a a b c + + 6 √ √ √ 2 a2 + 2bc b2 + 2ca c2 + 2ab a+ b+ c  Proposed by Pham Quoc Sang Problem 85. If a, b, c are positive real numbers such that a + b + c ≤ 3 then a b c + + 61 a+2 b+2 c+2 Proposed by Pham Quoc Sang Problem 86. If a, b, c are positive real numbers such that a + b + c = 3 then a2 a b c + 2 + 2 61 +2 b +2 c +2 Proposed by Pham Quoc Sang Problem 87. If a, b, c are positive real numbers such that ab + bc + ca = 3 then a2 b2 c2 3 + + > a+1 b+1 c+1 2 Proposed by Pham Quoc Sang Problem 88. If a, b, c, k are positive real numbers such that abc = 1 then   k+a k+b k+c a+b+c>k + + + 3 (1 − k) k+b k+c k+a Proposed by Pham Quoc Sang Problem 89. Let the triangle ABC intersect the center circle O, G is the center of the triangle. A1 = AG ∩ (O) , B1 = BG ∩ (O) , C1 = CG ∩ (O). Prove that a) 1 1 1 3 p 2 + + 6 ha + h2b + h2c GA1 GB1 GC1 2S ha hb hc b) S > p 2 ha + h2b + h2c 12 Proposed by Pham Quoc Sang, RMM 11/2017 Problem 90. If a, b, c are positive real numbers such that ab + bc + ca = 3 then b2 a b c 3 + 2 + 2 > 2 2 2 +c +6 c +a +6 a +b +6 8 Proposed by Pham Quoc Sang 1 1 1 Problem 91. If a, b, c are positive real numbers such that + + = 3 then a b c √ √ 1 1 1 +√ +√ 6 3 a3 + b 2 + c b3 + c 2 + a c 3 + a2 + b Proposed by Pham Quoc Sang 1 1 1 Problem 92. If a, b, c are positive real numbers and m, n, p ∈ N such that + + = 3, a b c m + n + p = 6 then √ am 1 1 1 +√ m +√ m n p n p +b +c b +c +a c + an + b p Proposed by Pham Quoc Sang 1 1 1 Problem 93. If a, b, c are positive real numbers such that + + = 3 then a b c √ 1 a3 +b +√ 1 1 3 +√ 6√ 3 +c c +a 2 b3 ( Baltic Way 2014) Problem 94. If a, b are non-negative real numbers then √ a2 − ab + b2 + √ ab > a + b Proposed by Pham Quoc Sang Problem 95. If a, b are non-negative real numbers then r r √ √ a2 + ab + b2 √ a2 + b 2 √ + ab 6 + ab 6 a + b 6 a2 − ab + b2 + ab 3 2 ( Luofanxiang - JBMO 2011 - Sqing - Pham Quoc Sang) Problem 96. If a, b, c are positive real numbers such that ab + bc + ca = 3 then a b c a+b+c 2 + 2 + 2 > 4 (b + c) (c + a) (a + b) Proposed by Do Huu Duc Thinh Problem 97. If a, b are positive real numbers then 1 1 1 4 + 2+ 2 > 2 a b ab (a − b) 13 Proposed by Pham Quoc Sang Problem 98. If a, b are positive real numbers then  
1 9 1 1 2 2 a − ab + b + 2+ 2 > 2 a b 4 (a + b) Proposed by Nguyen Viet Hung Problem 99. Let a, b, c be non - negative numbers, such that a 6= b, b 6= c, c 6= a. Prove that a+b b+c c+a 9 2 + 2 + 2 > a+b+c (a − b) (b − c) (c − a) Hanoi Education TST 2014-2015 Problem 100. Let a, b, c be non - negative numbers, such that a 6= b, b 6= c, c 6= a. Prove that   1 1 1 (ab + bc + ca) + + >4 (a − b)2 (a − b)2 (a − b)2 VMO 2008 Problem 101. If a, b, c are real numbers such that a + b + c = 0 then  
1 1 1 9 2 2 2 a +b +c 2 + 2 + 2 > 4 (a − b) (a − b) (a − b) Proposed by Pham Quoc Sang Problem 102. If a, b, c are positive real numbers such that a + b + c = 3 then √ √ √ √ √ 3 3abc + a a2 + 2bc + b b2 + 2ca + c c2 + 2ab 6 6 3 Proposed by Pham Quoc Sang 1 Problem 103. If a, b, c are positive real numbers such that a + b + c = √ . 3 Find the max of √ √ √ √ 27abc + a a2 + 2bc + b b2 + 2ca + c c2 + 2ab 6 6 3 ( Spain 2017) Problem 104. Let a, b, c > 0 and ab + bc + ac = 3. Prove that 1 1 3 1 + 2 + 2 ≥ (a + b)(b + c)(c + a) 2 a b c 8 Proposed by Le Minh Cuong 14 Problem 105. If a, b, c are positive real numbers such that 1 1 1 + + = 3, then a b c 1 1 1 + + ≤1 a3 + b 2 + 1 b 3 + c 2 + 1 c 3 + a2 + 1 Proposed by Le Minh Cuong Problem 106. If a, b, c are positive real numbers such that 4a3 1 1 1 + + = 3, then a b c 1 1 1 1 + 3 + 3 ≤ 2 2 2 + 3b + 2 4b + 3c + 2 4c + 3a + 2 3 Proposed by Le Minh Cuong, RMM 12/2017 Problem 107. If a, b, c are positive real numbers, then 2  2  2  b c 3 a + + ≥ 4a + 3b 4b + 3c 4c + 3a 49 Proposed by Le Minh Cuong, RMM 12/2017 Problem 108. If k and a, b, c are positive real numbers such that 2k 2 ≤ 2k + 1, then  2  2  2 a b c 3 + + ≥ ka + b kb + c kc + a (k + 1)2 Proposed by Le Minh Cuong, RMM 12/2017 Problem 109. If a, b, c are positive real numbers such that a + b + c = 3, then 1+b+c 1+c+a 1+a+b + + ≥1 2 2 (a + b + ab) (b + c + bc) (c + a + ca)2 Proposed by Le Minh Cuong, RMM 12/2017 Problem 110. If a, b, c are positive real numbers such that a + b + c = 3, then 2a + b2 + c2 2b + c2 + a2 2c + b2 + a2 + + a2 (b2 + bc + c2 )2 b2 (c2 + ca + a2 )2 c2 (a2 + ab + b2 )2   4 1 1 1 ≥ + + 3 a2 + b2 + ab b2 + c2 + bc c2 + a2 + ac Proposed by Le Minh Cuong, RMM 12/2017 Problem 111. If a, b, c are positive real numbers such that a + b + c = 3, then √ √
7 3 ab2 bc2 ca2 3 2 2 2 √ +√ +√ + a +b +c ≥ 4 4 b2 + bc + c2 c2 + ca + a2 a2 + ab + b2 15 Proposed by Le Minh Cuong, RMM 12/2017 Problem 112. If a, b, c are positive numbers such that a2 + b2 + c2 = 3, then √ a b c +√ +√ ≤1 2b3 + 7 2c3 + 7 2a3 + 7 Proposed by Le Minh Cuong Problem 113. If a, b, c are positive real numbers such that a2 + b2 + c2 = 3, then √ 3a2 + 4ab + 9 3b2 + 4bc + 9 3c2 + 4ca + 9 +√ +√ ≥ 4(a + b + c) 3a2 + 10ab + 3c2 3b2 + 10bc + 3a2 3c2 + 10ca + 3b2 Proposed by Le Minh Cuong Problem 114. If a, b, c, m, k are positive real numbers such that m ≥ 2k, then P P P ma2 + kab + m a2 mb2 + kbc + m a2 mc2 + kca + m a2 p +p +p ma2 + (k + 2m)ab + mc2 mb2 + (k + 2m)bc + ma2 mc2 + (k + 2m)ca + mb2 √ ≥ 4m + k.(a + b + c) Proposed by Le Minh Cuong Problem 115. If a, b, c are positive numbers such that a + b + c = 3, then a b c 3 p +p +p ≤ 3 3 3 3 3 3 2abc 4(a + bc) 4(b + ca) 4(c + ab) Proposed by Le Minh Cuong If a, b, c are non-negative real numbers, no two of them are zero, then ! r r r a b c 27 (a2 + b2 + c2 ) 4 + + +9≥ b+c c+a a+b (a + b + c)2 Problem 116. Proposed by Le Minh Cuong If a, b, c are non-negative real numbers, no two of them are zero, and Problem 117. 3 0 < k ≤ then 2 4k 3 r a + b+c r b + c+a r c a+b ! 16 + 27 − 18k 2 ≥ 27 (a2 + b2 + c2 ) (a + b + c)2 Proposed by Le Minh Cuong Problem 118. If a, b, c are nonnegative real numbers such that ab + bc + ca = 1, then 1 (a + b + c)2 15 1 1 5 · ≥ + + + 2 2 2 2 2 2 a +b b +c c +a 2 (a + b)(b + c)(c + a) 2 Proposed by Le Minh Cuong Problem 119. If a, b, c are positive real numbers, then √ √ a2 + ab b2 + bc c2 + ca 3 3 √ +√ +√ + ≥3 3 a2 + ab + b2 b2 + bc + c2 c2 + ca + a2 ab + bc + ca Proposed by Le Minh Cuong Problem 120. If a, b, c are positive real numbers, then 2a2 2b2 2c2 b c a + + ≥ + + 2 2 2 (a + b) (b + c) (c + a) a+b b+c c+a Proposed by Le Minh Cuong Problem 121. If a, b, c are positive real numbers such that a + b + c = 3, then a b c 3 + + ≤ 2 2 2 2 2 2 6a + b + c 6b + c + a 6c + a + b 8 Proposed by Le Minh Cuong Problem 122. If a, b, c are positive real numbers such that abc = 1, then a b c a3 b 3 c 3 + + + + + ≥ 2(a2 + b2 + c2 ) b c a b c a Proposed by Le Minh Cuong Problem 123. If a, b, c are positive real numbers such that abc = 1, then a4 b c a + 4 + 4 61 2 2 + a + 1 b + b + 1 c + c2 + 1 Proposed by Do Huu Duc Thinh Problem 124. Let a, b, c be non - negative numbers, no two of them are zero such that a + b + c = 3. Prove that a2 b2 c2 a2 + b 2 + c 2 + + 6 a+b b+c c+a 2 17 Proposed by Do Huu Duc Thinh If a, b, c are positive real numbers, then r ! r r r ! r r  (a + b) (b + c) (c + a) a b b c c a 3 3 3 3 3 > + + + 3 abc b a c b a c Problem 125. Proposed by Do Huu Duc Thinh Problem 126. If a, b, c are positive real numbers such that abc = 1, k > 2, then a b c 3 1 1 1 + + > > + + a+k b+k c+k k+1 a+k b+k c+k Proposed by Do Huu Duc Thinh Problem 127. If a, b, c are positive real numbers, then  2    5a + 8ab + 5b2 5b2 + 8bc + 5c2 5c2 + 8ca + 5a2 1 1 1 1 + + >4 + + a+b+c 2a2 + ab 2b2 + bc 2c2 + ca a+b b+c c+a Proposed by Do Quoc Chinh If x, y, z are positive real numbers, then √ r r √ x2 + y 2 + z 2 13 2 xy yz zx > + + + 15 2. x2 + y 2 y2 + z2 z 2 + x2 2 (x + y + z)2 Problem 128. r Proposed by Do Quoc Chinh Problem 129. Let a, b, c be positive real numbers such that a + b + c = 1. Prove that  b2 c2 3 3 a2 + + + (ab + bc + ca) > 1 + .min (a − b)2 , (b − c)2 , (c − a)2 a+b b+c c+a 2 8 Proposed by Nguyen Viet Hung Problem 130. Let a, b, c are non - negative numbers such that a + b + c > 0. Prove that 1 a2 b2 c2 1 6 + + 2 2 2 6 2 2 2 8 4 8a + (b + c) 8b + (c + a) 8c + (a + b) Proposed by Nguyen Viet Hung Problem 131. Prove for all positive real numbers a, b, c. r r r √ (1 + a) (b + c) (1 + b) (c + a) (1 + c) (a + b) + + >3 2 a + bc b + ca c + ab Proposed by Nguyen Viet Hung Problem 132. Prove that for any positive real numbers a, b, c the inequality holds a b c + + 61 1 + a + ab 1 + b + bc 1 + c + ca Proposed by Nguyen Viet Hung 18 Let a, b, c positive real numbers such that abc = 1. Prove that √ √ √ !2 1 a+ b+ c 1 1 6 + + a + b 2 + c 2 a2 + b + c 2 a2 + b 2 + c a+b+c Problem 133. Proposed by Nguyen Duc Viet Problem 134. Let a, b, c positive real numbers. Prove that Problem 135. If a, b, c are positive real numbers, then √ √ 1 1 1 3 √ + √ + √ ≥√ a 2a + b b 2b + c c 2c + a abc Proposed by Nguyen Duc Viet b c (a + b + c)2 a +√ +√ > 2 a + b2 + c 2 b2 − bc + c2 c2 − ca + a2 a2 − ab + b2 Proposed by Do Quoc Chinh, RMM 2017 Problem 136. If a, b, c are positive real numbers then   1 1 7 2 1 + 2 + 2 > 2 + 3 2 2 2 2a + bc 2b + ca 2c + ab a +b +c ab + bc + ca Proposed by Do Quoc Chinh Problem 137. Let a, b, c be positive real numbers such that a + b + c ≤ 1. Prove that a 4 a + bc + 3 Problem 138. Problem 139. + b 4 b + ca + 3 + c 9 6 4 16 c + ab + 3 Proposed by Nguyen Viet Hung Let a, b, c be positive real numbers such that a + b + c ≤ 1. Prove that √ a+b+c 1 1 1 √ +√ +√ 6 3. ab + bc + ca a2 + 2bc b2 + 2ca c2 + 2ab Proposed by Nguyen Viet Hung Let a, b, c be positive real numbers such that abc = 1. Prove that a 2 (a + b) (c + 1) + b c 3 + 6 2 8 (b + c) (a + 1) (c + a) (b + 1) 2 Proposed by Nguyen Viet Hung Problem 140. Let a, b, c be positive real numbers such that abc = 1. Prove that a 2 (a + b) (c + 1) + b c 3 + 6 2 8 (b + c) (a + 1) (c + a) (b + 1) 2 Proposed by Nguyen Viet Hung Let a, b, c be positive real numbers such that a + b + c = 1. Prove that √ b+c c+a a+b q +q +q > 3 4a + (b − c)2 4b + (c − a)2 4c + (a − b)2 Problem 141. Proposed by Nguyen Viet Hung 19