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 Theory Problem Problem If a, b, c ∈ (0; 2) such that ab + bc + ca = then b c a + + 2−a 2−b 2−c Proposed by Pham Quoc Sang Problem If a, b, c are positive real numbers such that abc = then a b c + + ab + bc + ca + Proposed by Pham Quoc Sang Problem If a, b, c, k are positive real numbers such that abc = then b2 c2 a2 + + (ab + k) (kab + 1) (bc + k) (kbc + 1) (ca + k) (kca + 1) (k + 1)2 Proposed by Pham Quoc Sang Problem If a, b, c are positive real numbers then a b c √ +√ +√ 2 2 2 3a + 2b + 2c 3b + 2c + 2a 3c + 2a2 + 2b2 √ Proposed by Pham Quoc Sang Problem If a, b, c are positive real numbers and k ≥ then √ ka2 a b c +√ +√ 2 2 2 +b +c kb + c + a kc + a2 + b2 √ k+2 Proposed by Pham Quoc Sang 13 Problem If a, b, c are positive real numbers and k ≥ such that abc = then a b c √ +√ +√ b2 + c + k c + a2 + k a2 + b + k √ k+2 Proposed by Pham Quoc Sang Problem If a, b, c are positive real numbers such that a + b + c = then a2 + b2 + c2 + + + c+5 a+5 b+5 Proposed by Pham Quoc Sang Problem If a, b, c, α, β are positive real numbers and n ∈ N∗ such that a + b + c = then an + n − b n + n − c n + n − + + αb + βc αc + βa αa + βb 3n α+β Proposed by Pham Quoc Sang Problem If a, b, c are positive real numbers such that abc = then a−1 b−1 c−1 + + 2a + 2b + 2c + Proposed by Pham Quoc Sang Problem 10 If a1 , a2 an are positive real numbers such that a1 m + a2 m + + an m m ∈ N∗ , k m, k m Find the minimum value of n n m + i=1 i=1 1, k Proposed by Pham Quoc Sang, RMM 9/2017 Problem 11 If a, b, c, k are positive real numbers then k+ a + b+c k+ b + c+a k+ √ c > 9k + a+b Proposed by Pham Quoc Sang - Nguyen Duc Viet then Problem 12 If a, b, c, k are positive real numbers and k 48 k+ a + b+c k+ b + c+a k+ c a+b √ √ k+1+ k Proposed by Nguyen Duc Viet Problem 13 If a, b, c are positive real numbers then a2 + b b2 + c c + 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 = Find the minimum value of a2n + b2n + c2n 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 = 0, a + b + c = Find of P = a2 b c |ab + bc + ca|3 Proposed by Pham Quoc Sang Problem 16 If a, b, c are positive real numbers such that abc = then a3 + b3 + c3 + (a2 + 2) (b2 + 2) (c2 + 2) a2 + b + c 2 12 Proposed by Pham Quoc Sang, RMM 10/2017 Problem 17 If a, b, c are positive real numbers then (a + b + c) + + abc a + b2 + c Proposed by Pham Quoc Sang Problem 18 If a, b, c are positive real numbers such that a + b + c = then √ a + 2bc + √ b + 2ca + √ √ c + 2ab + a2 + b2 + c2 √ √ 3 3+ Proposed by Pham Quoc Sang Problem 19 If a, b, c, α, β are positive real numbers such that a + b + c = and β 12α then αabc + β ab + bc + ca α+ β Proposed by Pham Quoc Sang Problem 20 If a, b, c are positive real numbers and k ∈ [0; 9] then a3 + b + c ab + bc + ca + k abc a + b2 + c 3+k Proposed by Pham Quoc Sang Note The above inequalities are extended as follows: If a, b, c are positive real numbers and k ≤ then a3 + b + c ab + bc + ca + k abc a + b2 + c 3+k 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 + + + 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 = then 2(a + b)(b + c)(c + a) + 11 ≥ 9(a + b + c) 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 + + b c a (a − b)2 + (b − c)2 + (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 = then (a − b)2 (b − c)2 (c − a)2 + + + (ab + bc + ca) b c a 24 Proposed by Pham Quoc Sang, RMM 11/2017 Problem 25 If a, b, c be positive real number then a2 b2 c2 + + b c a max (a − b)2 , (b − c)2 , (c − a)2 a+b+c a+b+c+ Proposed by Pham Quoc Sang, RMM 11/2017 Problem 26 If a, b, c be positive real number then a2 b c + + b c a a+b+c+ (a − b)2 , (b − c)2 , (c − a)2 a+b+c Proposed by Do Huu Duc Thinh Problem 27 If a, b, c be positive real number such that a a b c + + b c a +3 (a + b + c) b c then 1 + + a b c Proposed by Pham Quoc Sang, RMM 11/2017 Problem 28 If a, b, c be positive real number and k ≥ then a + kb b + kc c + ka + + 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 αa + βb αb + βc αc + βa + + βa + αb βb + αc βc + αa 4αβ then 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 = then 1 + 2+ 2 a b c (a + b) (b + c) (c + a) Proposed by Pham Quoc Sang, TAoM 12/2017 Problem 31 If a, b, c be positive real number such that a + b + c = then ab bc ca + + (2a + bc) (2b + ca) (2b + ca) (2c + ab) (2c + ab) (2a + bc) Proposed by Pham Quoc Sang, RMM 12/2017 Problem 32 If a, b, c are positive real number such that ab + bc + ca = then a3 1 + + 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 ≥ then b2 b c a + + 2 + kbc + c c + kca + a a + kab + b2 (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 = then a2 b2 c2 + + a2 + b2 + c2 + Proposed by Pham Quoc Sang Problem 35 If a, b, c are positive real number such that abc = then a2 a b c + + + bc b + ca c + ab Proposed by Pham Quoc Sang Problem 36 If a, b, c are positive real number such that a2 + b2 + c2 = then a b c a+b+c + + ≥ 2b + 2c + 2a + 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) + + a2 + b b + c2 c + a2 b (a2 + b2 ) c (b2 + c2 ) a (c2 + a2 ) + + a+b b+c c+a Proposed by Pham Quoc Sang Problem 38 Let a, b, c, α, β be positive real number a) If β 2α then b) If α a2 b2 c2 + + αa2 + βbc αb2 + βca αc2 + βab α+β a2 b2 c2 + + αa2 + βbc αb2 + βca αc2 + βab α+β 2β then Proposed by Pham Quoc Sang Problem 39 If a, b, c are positive real number such that a + b + c = then (2a + c)2 (2b + a)2 (2c + b)2 + + a2 + 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 ak + b k + c k bc ca ab + + 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 = then b2 + bc + c2 + ca + a2 + ab + √ √ √ + + 2a2 + 5ab + 2b2 2b2 + 5bc + 2c2 2c2 + 5ca + 2a2 a+b+c 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) = then a2 1 + + 2 + ab + b b + bc + c c + ca + a2 ab + bc + ca 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 + + + b + c + a + 4(a + b + c) Proposed by Pham Quoc Sang Problem 44 If a, b, c are positive real numbers and k is a positive integer then ak+1 + bk+1 + ck+1 ak + b k + c k a2 + b b + c c + a2 + + 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 = then √ bc ca ab +√ +√ a2 + b2 + c2 + 3 Proposed by Pham Quoc Sang Problem 46 If a, b, c are positive real numbers such that ab + bc + ca = then 1 + + (a + 1) (b + 1) (c + 1)2 Proposed by Konstantinos Metaxas, Athens, Greece Problem 47 If a, b, c are positive real numbers then a2 + c b + a2 c + b + + b c a (a + b + c) Inequalities Book Problem 48 If a, b, c are positive real numbers such that abc = then 1 + + 1+a 1+b 1+c + ab + bc + ca Proposed by Pham Quoc Sang Problem 49 If a, b, c are positive real numbers such that a + b + c = then − 2ab − 2bc − 2ca + + a+b b+c c+a Proposed by Pham Quoc Sang, RMM 12/2017 Problem 50 If a, b, c are positive real numbers such that a + b + c + = 4abc then a) ab + bc + ca b) 3abc b + c c + a2 a2 + b + + a b c 6abc Proposed by Pham Quoc Sang Problem 51 If a, b, c are positive real numbers such that abc = then a b c + + + 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 + + b+c c+a a+b ab + bc + ca a+b+c Proposed by Pham Quoc Sang Problem 53 Let a, b, c ≥ and k ≥ k prove that: 27 (a + b + c)3 ab + bc + ca +( ) ≥ 27k + abc a + b2 + c Proposed by Phan Dinh Dan Truong Problem 54 If a, b, c are positive real numbers such that abc = then a+b b+c c+a + + b+c c+a a+b (a + b + c) 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 ka − bc kb − ca kc − ab + + ka + bc kb + ca kc + ab Proposed by Pham Quoc Sang Problem 56 If a, b, c are positive real numbers such that ab + bc + ca = then √ √ √ 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 = then a+ bc + b+ ca + c+ ab 12 Proposed by Pham Quoc Sang Problem 58 If a, b, c are positive real numbers such that ab + bc + ca = then a+ b+c + b+ c+a a+b + c+ 27 Proposed by Pham Quoc Sang Problem 59 If a, b, c are positive real numbers such that a2 + b2 + c2 = then a+ b 2 + b+ c + c+ a 12 Proposed by Pham Quoc Sang Problem 60 If a, b, c are positive real numbers such that a2 + b2 + c2 = then 1 + + 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 = then 1 + + a+1 b+1 c+1 1 + + 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 = then (ab + xy) 1 + ay bx Proposed by Pham Quoc Sang Problem 63 If a, b, c are positive real numbers such that a2 + b2 + c2 = then 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 = then 1 + + ab (a + b) bc (b + c) ca (c + a) Proposed by Pham Quoc Sang Problem 65 If a, b, c are positive real numbers then 1 + + ab(a + b) bc(b + c) ca(c + a)2 729 ab + bc + ca (a + b + c)6 Proposed by Pham Quoc Sang Problem 66 If a, b, c are positive real numbers then 1 + + a(b + c) b(c + a) c(a + b)2 81 4(a + b + c)3 Proposed by Pham Quoc Sang Problem 67 If a, b, c are positive real numbers and k ∈ N, k a(b + c)k + b(c + a)k + then 3k+2 c(a + b)k 2k (a + b + c)k+1 Proposed by Pham Quoc Sang Problem 68 If a, b, c are positive real numbers then a2 1 + + (b + c) b (c + a) c (a + b) 81 2(a + b + c)3 Proposed by Pham Quoc Sang Problem 69 If a, b, c are positive real numbers and k ∈ N, k then 3k+2 1 + + 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 + + a(b + c) b(c + a) c(a + b)2 (a3 + b3 + c ) 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 + + (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 ≥ then a+ b+c b+ c+a c+ a+b 27 Proposed by Pham Quoc Sang Problem 73 If a, b, c are positive real numbers such that abc ≥ then 2a3 + abc 2b3 + abc 2c3 + abc + + (b + c)2 (c + a)2 (a + b)2 Proposed by Pham Quoc Sang Problem 74 If a, b, c are positive real numbers and α, β ∈ N∗ then αa1 1+ βa2 αa2 1+ βa3 αa3 1+ βa4 αan + βa1 α 1+ β n 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 a+b+c+ a+b+c 3 a+b+c + 2abc Proposed by Pham Quoc Sang Problem 76 If a, b, c are positive real numbers such that a + b + c = a2 + b2 + c2 then a) a + b + c + 3 (ab + bc + ca) (a + b + c + 6)3 (a + b + c)2 b) + 729 a2 + b + c ab + bc + ca 2430 Proposed by Pham Quoc Sang Problem 77 If a, b, c are positive real numbers such that a2 + b2 + c2 = then (a + b) (b + c) (c + a) 4abc + Proposed by Pham Quoc Sang Problem 78 If a, b, c are positive real numbers such that a2 + b2 + c2 = then 6+ a+b+c abc 27 a+b+c Proposed by Pham Quoc Sang, RMM 9/2017 Problem 79 If a, b, c are positive real numbers then 1 + + (a + 1) (b + 1) (c + 1)2 1 + + + a (b + c) + b (c + a) + c (a + b) Proposed by Pham Quoc Sang Problem 80 If a, b, c are positive real numbers such that a2 + b2 + c2 = then 1 + + (a + b) (b + c) (c + a)2 2 + + a2 + b + b + c + c + a2 + Proposed by Pham Quoc Sang Problem 81 If a, b, c are positive real numbers then 1 + + > (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 = then a2 + b + c 2 a2 a b c + + + bc b + ca c + ab Proposed by Pham Quoc Sang 11 Problem 83 If a, b, c are positive real numbers such that abc = then a2 1 + + + 2bc b + 2ca c + 2ab (ab + bc + ca) √ √ √ a+ b+ c Proposed by Pham Quoc Sang Problem 84 If a, b, c are positive real numbers then a2 b c a + + + 2bc b + 2ca c + 2ab 1 + + a b c √ √ √ a+ b+ c (a + b + c) Proposed by Pham Quoc Sang Problem 85 If a, b, c are positive real numbers such that a + b + c ≤ then a b c + + 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 = then a2 a b c + + +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 = then a2 b2 c2 + + a+1 b+1 c+1 Proposed by Pham Quoc Sang Problem 88 If a, b, c, k are positive real numbers such that abc = then a+b+c k k+a k+b k+c + + k+b k+c k+a + (1 − k) 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 + + GA1 GB1 GC1 b) S 2S h2a + h2b + h2c hb hc + h2b + h2c h2a 12 Proposed by Pham Quoc Sang, RMM 11/2017 Problem 90 If a, b, c are positive real numbers such that ab + bc + ca = then b2 a b c + + 2 + c + c + a + a + b2 + Proposed by Pham Quoc Sang 1 Problem 91 If a, b, c are positive real numbers such that + + = then a b c √ 1 +√ +√ a3 + b + c b3 + c + a c + a2 + b √ Proposed by Pham Quoc Sang 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 = then √ am 1 +√ m +√ m n p n p +b +c b +c +a c + an + b p Proposed by Pham Quoc Sang 1 Problem 93 If a, b, c are positive real numbers such that + + = then a b c √ a3 +b +√ 1 +√ +c c +a √ 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 a2 + ab + b2 √ + ab a2 + b √ + ab a+b √ a2 − ab + b2 + √ ab ( Luofanxiang - JBMO 2011 - Sqing - Pham Quoc Sang) Problem 96 If a, b, c are positive real numbers such that ab + bc + ca = then a b c + + (b + c) (c + a) (a + b)2 a+b+c Proposed by Do Huu Duc Thinh Problem 97 If a, b are positive real numbers then 1 + 2+ a b (a − b)2 13 ab Proposed by Pham Quoc Sang Problem 98 If a, b are positive real numbers then a2 − ab + b2 1 + 2+ a b (a + b)2 Proposed by Nguyen Viet Hung Problem 99 Let a, b, c be non - negative numbers, such that a = b, b = c, c = a Prove that a+b b+c c+a + + (a − b) (b − c) (c − a)2 a+b+c Hanoi Education TST 2014-2015 Problem 100 Let a, b, c be non - negative numbers, such that a = b, b = c, c = a Prove that (ab + bc + ca) 1 + + (a − b) (a − b) (a − b)2 VMO 2008 Problem 101 If a, b, c are real numbers such that a + b + c = then a2 + b + c 1 + + (a − b) (a − b) (a − b)2 Proposed by Pham Quoc Sang Problem 102 If a, b, c are positive real numbers such that a + b + c = then √ √ √ √ 3abc + a a2 + 2bc + b b2 + 2ca + c c2 + 2ab √ Proposed by Pham Quoc Sang Problem 103 If a, b, c are positive real numbers such that a + b + c = √ Find the max of √ √ √ 27abc + a a2 + 2bc + b b2 + 2ca + c c2 + 2ab √ ( Spain 2017) Problem 104 Let a, b, c > and ab + bc + ac = Prove that 1 + + ≥ (a + b)(b + c)(c + a) a b c Proposed by Le Minh Cuong 14 1 + + = 3, then a b c Problem 105 If a, b, c are positive real numbers such that 1 + + ≤1 a3 + b + b + c + c + a2 + Proposed by Le Minh Cuong Problem 106 If a, b, c are positive real numbers such that 4a3 1 + + = 3, then a b c 1 1 + + ≤ 2 + 3b + 4b + 3c + 4c + 3a + Proposed by Le Minh Cuong, RMM 12/2017 Problem 107 If a, b, c are positive real numbers, then a 4a + 3b + b 4b + 3c + c 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 ≤ 2k + 1, then a ka + b + b kb + c + 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 (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 1 ≥ + + 2 2 a + b + ab b + c + bc c + 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 √ √ ab2 bc2 ca2 2 √ +√ +√ + a +b +c ≥ 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 + 2c3 + 2a3 + Proposed by Le Minh Cuong Problem 113 If a, b, c are positive real numbers such that a2 + b2 + c2 = 3, then √ 3a2 + 4ab + 3b2 + 4bc + 3c2 + 4ca + +√ +√ ≥ 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 ma2 + kab + m a2 ma2 + (k + 2m)ab + mc2 + mb2 + kbc + m a2 mb2 + (k + 2m)bc + ma2 mc2 + kca + m + ≥ a2 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 4(a3 + bc) + b 4(b3 + ca) + c 4(c3 + ab) ≤ 2abc Proposed by Le Minh Cuong Problem 116 If a, b, c are non-negative real numbers, no two of them are zero, then a + b+c b + c+a c a+b +9≥ 27 (a2 + b2 + c2 ) (a + b + c)2 Proposed by Le Minh Cuong Problem 117 < k ≤ then 4k If a, b, c are non-negative real numbers, no two of them are zero, and a + b+c b + c+a c a+b 16 + 27 − 18k ≥ 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 (a + b + c)2 15 1 · ≥ + + + 2 2 2 a +b b +c c +a (a + b)(b + c)(c + a) Proposed by Le Minh Cuong Problem 119 If a, b, c are positive real numbers, then √ √ a2 + ab b2 + bc c2 + ca 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 (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 + + ≤ 2 2 2 6a + b + c 6b + c + a 6c + a + b 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 c + + + + + ≥ 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 + + 2 + a + b + b + 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 = Prove that a2 b2 c2 + + a+b b+c c+a 17 a2 + b + c 2 Proposed by Do Huu Duc Thinh If a, b, c are positive real numbers, then Problem 125 (a + b) (b + c) (c + a) abc a + b b a b + c c b 3 c + a a c Proposed by Do Huu Duc Thinh If a, b, c are positive real numbers such that abc = 1, k Problem 126 a b c + + a+k b+k c+k k+1 2, then 1 + + a+k b+k c+k Proposed by Do Huu Duc Thinh If a, b, c are positive real numbers, then Problem 127 a+b+c 5a2 + 8ab + 5b2 5b2 + 8bc + 5c2 5c2 + 8ca + 5a2 + + 2a2 + ab 2b2 + bc 2c2 + ca 1 + + a+b b+c c+a Proposed by Do Quoc Chinh If x, y, z are positive real numbers, then Problem 128 xy + x2 + y yz + y2 + z2 √ x2 + y + z zx + 15 z + x2 (x + y + z)2 √ 13 2 Proposed by Do Quoc Chinh Let a, b, c be positive real numbers such that a + b + c = Prove that Problem 129 b2 c2 a2 + + + (ab + bc + ca) a+b b+c c+a + (a − b)2 , (b − c)2 , (c − a)2 Proposed by Nguyen Viet Hung Problem 130 Let a, b, c are non - negative numbers such that a + b + c > Prove that a2 b2 c2 + + 8a2 + (b + c)2 8b2 + (c + a)2 8c2 + (a + b)2 Proposed by Nguyen Viet Hung Problem 131 Prove for all positive real numbers a, b, c (1 + a) (b + c) + a + bc (1 + b) (c + a) + b + ca (1 + c) (a + b) 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 + + + a + ab + b + bc + c + ca Proposed by Nguyen Viet Hung 18 Let a, b, c positive real numbers such that abc = Prove that √ √ √ a+ b+ c 1 + + a + b + c a2 + b + c a2 + b + 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 √ + √ + √ ≥√ a 2a + b b 2b + c c 2c + a abc Proposed by Nguyen Duc Viet b c (a + b + c)2 a +√ +√ a2 + b + c b2 − bc + c2 c2 − ca + a2 a2 − ab + b2 Proposed by Do Quoc Chinh, RMM 2017 If a, b, c are positive real numbers then Problem 136 2a2 1 + + + bc 2b + ca 2c + ab a2 + 2 +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 ≤ Prove that a a + bc + Problem 138 Problem 139 + b b + ca + + c 16 c + ab + Proposed by Nguyen Viet Hung Let a, b, c be positive real numbers such that a + b + c ≤ Prove that √ a+b+c 1 √ +√ +√ 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 = Prove that a (a + b) (c + 1) + b c + (b + c) (a + 1) (c + a) (b + 1) Proposed by Nguyen Viet Hung Problem 140 Let a, b, c be positive real numbers such that abc = Prove that a (a + b) (c + 1) + b c + (b + c) (a + 1) (c + a) (b + 1) Proposed by Nguyen Viet Hung Problem 141 Let a, b, c be positive real numbers such that a + b + c = Prove that √ b+c c+a a+b + + 4a + (b − c)2 4b + (c − a)2 4c + (a − b)2 Proposed by Nguyen Viet Hung 19 Problems from Olympic competitions Problem Let a, b, c be positive real numbers such that a+b+c = Find the minimum value of the expression A= − a3 − b − c + + a b c ( JBMO 2015 ) Inequality from AoPS Problem If a, b, c > prove that: b c a a b c 2( + + ) ≥ + 2( + + ) a b c b+c c+a a+b ( sqing ) Solution By C-S and AM-GM, have a b c + + b c a b c a + + a b c b c a √ +√ +√ ca bc ab = a b c + + b c a a b c + + c a b a b c + + b+c c+a a+b Problem If a, b, c > prove that: a b c + + + ≥ a+b b+c c+a a b c + + 2a + b 2b + c 2c + a ( xyy ) Solution it can be written as a b c 9( + + ) 2a + b 2b + c 2c + a + + ≥ 2a + b 2b + c 2c + a + a+b b+c c+a 4 but 2a + b 9a (a − b)2 − − = ≥0 a+b 4(2a + b) 4(a + b)(2a + b) Problem If a, b, c ≥ prove that: √ √ √ √ c a2 + 2ab + 3b2 + a b2 + 2bc + 3c2 + b c2 + 2ac + 3a2 ≥ 6(ab + bc + ca) 20 ( Bara Andrei-Robert) Solution The inequality will become a2 b + ca (a2 + 2ab + 3b2 )(b2 + 2bc + 3c2 ) ≥ cyc cyc a2 bc cyc By CS, (a2 + 2ab + 3b2 )(b2 + 2bc + 3c2 ) = ((a + b)2 + 2b2 )((b + c)2 + 2c2 ) ≥ (a+b)(b+c)+2bc So ca (a2 + 2ab + 3b2 )(b2 + 2bc + 3c2 ) ≥ cyc a2 b2 +5 ca (a+b)(b+c)+2bc = cyc Solution 21 cyc a2 bc cyc ... k+b k+c k+a + (1 − k) 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... an m m ∈ N∗ , k m, k m Find the minimum value of n n m + i=1 i=1 1, k Proposed by Pham Quoc Sang, RMM 9/2017 Problem 11 If a, b, c, k are positive real numbers then k+ a + b+c k+ b + c+a k+... [0; 9] then a3 + b + c ab + bc + ca + k abc a + b2 + c 3+k Proposed by Pham Quoc Sang Note The above inequalities are extended as follows: If a, b, c are positive real numbers and k ≤ then a3