AM GM BCS THE ART OF INEQUALITY 2018

Pham Quoc Sang - Christos EythymioyLe Minh Cuong The art of Mathematics - TAoM THE ART OF INEQUALITY AM-GM, BCS, Holder..... If a, b, c be positive real number then Problem 27... If a, b

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

a2(ab + k) (kab + 1) +

b2(bc + k) (kbc + 1)+

c2(ca + k) (kca + 1) > 3

(k + 1)2Proposed by Pham Quoc SangProblem 4 If a, b, c are positive real numbers then

4 such that abc = 1 thena

Proposed by Pham Quoc SangProblem 9 If a, b, c are positive real numbers such that abc = 1 then

a − 12a + 1 +

b − 12b + 1+

c − 12c + 1 6 0

Proposed by Pham Quoc SangProblem 10 If a1, a2 an are positive real numbers such that a1m+ a2m+ + anm 6 1,

m ∈ N∗, k > m, k .m Find the minimum value of

nXi=1

aim+

nXi=1

1

aikProposed by Pham Quoc Sang, RMM 9/2017Problem 11 If a, b, c, k are positive real numbers then

48 thenr

a2n+ b2n+ c2n

1(a − b)2n +

1(b − c)2n +

1(c − a)2n

Proposed by Pham Quoc Sang, RMM 10/2017Problem 17 If a, b, c are positive real numbers then

1abc +

5 (a + b + c)

3 +

3

a2+ b2+ c2 > 7

Proposed by Pham Quoc Sang

Problem 18 If a, b, c are positive real numbers such that a + b + c = 3 then

Problem 19 If a, b, c, α, β are positive real numbers such that a + b + c = 3 and β> 12αthen

αabc + β

ab + bc + ca > α + β

3Proposed by Pham Quoc Sang

Problem 20 If a, b, c are positive real numbers and k ∈ [0; 9] then

a3+ b3+ c3abc + k.

ab + bc + ca

a2+ b2+ c2 > 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 ≤ 9 then

a3+ b3+ c3abc + k.

ab + bc + ca

a2+ b2+ c2 > 3 + kThis 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

a2a2+ bc+

b2b2+ ca +

c2c2+ ab > 1Proposed by Pham Quoc Sang, RMM 11/2017Problem 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)

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

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

Problem 26 If a, b, c be positive real number then

Problem 27 If a, b, c be positive real number such that a6 b 6 c then

+ 3 > (a + b + c) 1

a +

1

b +

1c



Proposed by Pham Quoc Sang, RMM 11/2017

Problem 28 If a, b, c be positive real number and k ≥ 2 then

Problem 29 If a, b, c, α, β be positive real number and α2+ β2 > 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 = 3 then

Problem 31 If a, b, c be positive real number such that a + b + c = 3 then

ab(2a + bc) (2b + ca)+

bc(2b + ca) (2c + ab)+

ca(2c + ab) (2a + bc) 6 1

3Proposed by Pham Quoc Sang, RMM 12/2017Problem 32 If a, b, c are positive real number such that ab + bc + ca = 3 then

a2b + 1 +

b2c + 1+

c2a + 1 ≥ a + b + c

3Proposed by Pham Quoc SangProblem 37 If a, b, c are positive real number then

Problem 38 Let a, b, c, α, β be positive real number

Problem 39 If a, b, c are positive real number such that a + b + c = 3 then

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+ bk+ ck > 2

3

ab

Proposed by Pham Quoc Sang

Problem 41 If a, b, c are positive real number such that a2+ b2+ c2 = 3 then

2 + ca + 1

√2c2+ 5ca + 2a2 > a + b + cProposed 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

Problem 43 If a, b, c are positive real numbers then

4Proposed by Pham Quoc Sang

Problem 44 If a, b, c are positive real numbers and k is a positive integer then

Proposed by Pham Quoc Sang

Problem 45 If a, b, c are positive real numbers such that a + b + c = 3 then

Problem 46 If a, b, c are positive real numbers such that ab + bc + ca = 3 then

1(a + 1)2 +

1(b + 1)2 +

1(c + 1)2 > 3

4Proposed by Konstantinos Metaxas, Athens, Greece

Problem 47 If a, b, c are positive real numbers then

Proposed by Pham Quoc Sang

Problem 49 If a, b, c are positive real numbers such that a + b + c = 1 then

Problem 50 If a, b, c are positive real numbers such that a + b + c + 1 = 4abc thena) ab + bc + ca> 3abc

Proposed by Pham Quoc Sang

Problem 51 If a, b, c are positive real numbers such that abc = 1 then

Problem 53 Let a, b, c ≥ 0 and k ≥ 4

27 prove that:

k.(a + b + c)

3abc + (

ab + bc + ca

a2+ b2+ c2)2 ≥ 27k + 1

Proposed by Phan Dinh Dan Truong

Problem 54 If a, b, c are positive real numbers such that abc = 1 then

Proposed by Phan Ngoc Chau

Problem 55 If a, b, c, k are positive real numbers such that a + b + c = k then

2+



b + 1ca

2+



c + 1ab

2

> 12Proposed by Pham Quoc Sang

Problem 58 If a, b, c are positive real numbers such that ab + bc + ca = 3 then



a + 1

b + c

2+



b + 1

c + a

2+



a + 1b

2+



b + 1c

2+



c + 1a

2

> 12Proposed by Pham Quoc SangProblem 60 If a, b, c are positive real numbers such that a2+ b2+ c2 = 3 then

12ab + bc + ca +

12bc + ca + ab+

12ca + ab + bc 6 3

4abcProposed by Pham Quoc SangProblem 61 If a, b, c are positive real numbers such that a + b + c = 3 then



> 4Proposed by Pham Quoc SangProblem 63 If a, b, c are positive real numbers such that a2+ b2+ c2 = 3 then

1ab(a + b)2 +

1bc(b + c)2 +

1ca(c + a)2 > 729

4 .

ab + bc + ca(a + b + c)6Proposed by Pham Quoc SangProblem 66 If a, b, c are positive real numbers then

1a(b + c)2 +

1b(c + a)2 +

1c(a + b)2 > 81

4(a + b + c)3

Proposed by Pham Quoc SangProblem 67 If a, b, c are positive real numbers and k ∈ N, k > 2 then

1a(b + c)k +

1b(c + a)k +

1c(a + b)k > 3

k+2

2k(a + b + c)k+1Proposed by Pham Quoc SangProblem 68 If a, b, c are positive real numbers then

1a(b + c)2 +

1b(c + a)2 +

1c(a + b)2 > 9

4 (a3+ b3+ c3)Proposed by Pham Quoc SangProblem 71 If a, b, c are positive real numbers then

2a +√

bc(b + c)2 +

2b +√

ca(c + a)2 +

2c +√

ab(a + b)2 > 81

4 .

ab + bc + ca(a + b + c)3Proposed by Pham Quoc SangProblem 72 (Prove or deny) If a, b, c are positive real numbers such that abc ≥ 1 then

2a3+ abc(b + c)2 +

2b3+ abc(c + a)2 +

2c3+ abc(a + b)2 > 9

4Proposed by Pham Quoc SangProblem 74 If a, b, c are positive real numbers and α, β ∈ N∗ then

n

Proposed by Pham Quoc Sang

Problem 75 If a, b, c are positive real numbers such that a + b + c = a3+ b3+ c3 then

1

(a + 1)2 +

1(b + 1)2 +

1(c + 1)2 > 1

1

(a + b)2 +

1(b + c)2 +

1(c + a)2 > 2

1(a + b)2 +

1(b + c)2 +

1(c + a)2 >

3

2 (ab + bc + ca)Proposed by Pham Quoc SangProblem 82 If a, b, c are positive real numbers such that abc = 1 then

a2+ b2+ c2 > 2

a

Problem 83 If a, b, c are positive real numbers such that abc = 1 then



√

a +√

b +√c

A1 = AG ∩ (O) , B1 = BG ∩ (O) , C1 = CG ∩ (O) Prove that

Proposed by Pham Quoc Sang, RMM 11/2017Problem 90 If a, b, c are positive real numbers such that ab + bc + ca = 3 then

a(b + c)2 +

b(c + a)2 +

c(a + b)2 > a + b + c

4Proposed by Do Huu Duc ThinhProblem 97 If a, b are positive real numbers then

1

a2 + 1

b2 + 1(a − b)2 > 4

ab

Proposed by Pham Quoc SangProblem 98 If a, b are positive real numbers then

a2− ab + b2
 1

a2 + 1

b2 + 1(a + b)2



> 94Proposed 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 Provethat

a + b(a − b)2 +

b + c(b − c)2 +

c + a(c − a)2 > 9

a + b + cHanoi Education TST 2014-2015Problem 100 Let a, b, c be non - negative numbers, such that a 6= b, b 6= c, c 6= a Provethat

(ab + bc + ca)

1(a − b)2 +

1(a − b)2 +

1(a − b)2



> 4

VMO 2008Problem 101 If a, b, c are real numbers such that a + b + c = 0 then

a2+ b2+ c2

1(a − b)2 +

1(a − b)2 +

1(a − b)2



> 94Proposed by Pham Quoc SangProblem 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√3Proposed by Pham Quoc SangProblem 103 If a, b, c are positive real numbers such that a + b + c = √1

3.Find the max of

Problem 105 If a, b, c are positive real numbers such that 1

Proposed by Le Minh Cuong

Problem 106 If a, b, c are positive real numbers such that 1

4a3 + 3b2+ 2 +

14b3+ 3c2 + 2 +

14c3 + 3a2+ 2 ≤ 1

3Proposed by Le Minh Cuong, RMM 12/2017

Problem 107 If a, b, c are positive real numbers, then

a4a + 3b

2+

b4b + 3c

2+

c4c + 3a

2

≥ 349Proposed by Le Minh Cuong, RMM 12/2017

Problem 108 If k and a, b, c are positive real numbers such that 2k2 ≤ 2k + 1, then

a

ka + b

2+

b

kb + c

2+

c

kc + a

2

≥ 3(k + 1)2Proposed 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(a + b + ab)2 + 1 + c + a

(b + c + bc)2 + 1 + a + b

(c + a + ca)2 ≥ 1Proposed by Le Minh Cuong, RMM 12/2017

Problem 110 If a, b, c are positive real numbers such that a + b + c = 3, then

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

4 a

2+ b2+ c2
≥ 7

√34

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

√2b3+ 7 +

b

√2c3+ 7 +

c

√2a3+ 7 ≤ 1

Proposed by Le Minh Cuong

Problem 113 If a, b, c are positive real numbers such that a2+ b2+ c2 = 3, then

2+ 4ca + 9

√3c2+ 10ca + 3b2 ≥ 4(a + b + c)

Proposed by Le Minh Cuong

Problem 114 If a, b, c, m, k are positive real numbers such that m ≥ 2k, then

ma2+ kab + mP a2

pma2+ (k + 2m)ab + mc2 + mb

2+ kbc + mP a2pmb2+ (k + 2m)bc + ma2 + mc

2+ kca + mP a2pmc2+ (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

Problem 116 If a, b, c are non-negative real numbers, no two of them are zero, then

4

ra

b + c +

rb

c + a +

rc

a + b

!+ 9 ≥ 27 (a

2+ b2+ c2)(a + b + c)2Proposed by Le Minh Cuong

Problem 117 If a, b, c are non-negative real numbers, no two of them are zero, and

0 < k ≤ 3

2 then

4k3

ra

b + c +

rb

c + a +

rc

a + b

!+ 27 − 18k2 ≥ 27 (a

2+ 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

2Proposed by Le Minh Cuong

Problem 119 If a, b, c are positive real numbers, then

ab + bc + ca ≥ 3√3Proposed by Le Minh Cuong

Problem 120 If a, b, c are positive real numbers, then

2a2(a + b)2 + 2b

2(b + c)2 + 2c

2(c + a)2 ≥ b

Problem 121 If a, b, c are positive real numbers such that a + b + c = 3, then

a6a + b2+ c2 + b

6b + c2+ a2 + c

6c + a2+ b2 ≤ 3

8Proposed by Le Minh Cuong

Problem 122 If a, b, c are positive real numbers such that abc = 1, then

Proposed by Do Huu Duc ThinhProblem 125 If a, b, c are positive real numbers, then

!

3

rb

c +

3

r cb

5c2+ 8ca + 5a22c2+ ca



> 4

1

z2 + x2 + 15√

2.x

2+ y2+ z2(x + y + z)2 > 13

√22Proposed by Do Quoc ChinhProblem 129 Let a, b, c be positive real numbers such that a + b + c = 1 Prove that

1

8 6 a

28a2+ (b + c)2 +

b28b2+ (c + a)2 +

c28c2+ (a + b)2 6 1

4Proposed by Nguyen Viet HungProblem 131 Prove for all positive real numbers a, b, c

r

(1 + a) (b + c)

a + bc +

r(1 + b) (c + a)

b + ca +

r(1 + c) (a + b)

c + ab > 3√2Proposed by Nguyen Viet HungProblem 132 Prove that for any positive real numbers a, b, c the inequality holds

Problem 133 Let a, b, c positive real numbers such that abc = 1 Prove that

1

b√2b + c +

1

c√2c + a ≥

√3

√abcProposed by Nguyen Duc VietProblem 135 If a, b, c are positive real numbers, then

3



12a2+ bc +

12b2+ ca +

12c2+ ab



> 7

a2+ b2+ c2 + 2

ab + bc + caProposed by Do Quoc ChinhProblem 137 Let a, b, c be positive real numbers such that a + b + c ≤ 1 Prove that

a(a + b)2(c + 1)+

b(b + c)2(a + 1) +

c(c + a)2(b + 1) 6 3

8Proposed by Nguyen Viet HungProblem 140 Let a, b, c be positive real numbers such that abc = 1 Prove that

a(a + b)2(c + 1)+

b(b + c)2(a + 1) +

c(c + a)2(b + 1) 6 3

8Proposed by Nguyen Viet HungProblem 141 Let a, b, c be positive real numbers such that a + b + c = 1 Prove that

b + cq

4a + (b − c)2

+q c + a4b + (c − a)2

+ q a + b4c + (a − b)2

>√3

Proposed by Nguyen Viet Hung

3 Problems from Olympic competitions.

Problem 1 Let a, b, c be positive real numbers such that a+b+c = 3 Find the minimumvalue of the expression

4 Inequality from AoPS

Problem 2 If a, b, c > 0 prove that:

By C-S and AM-GM, have



>

a

2

> 4

a

b2b + c +

c2c + a



( xyy )Solution

b2b + c +

c2c + a)

94but

2a + b

a + b −3

4 − 9a4(2a + b) =

(a − b)24(a + b)(2a + b) ≥ 0

Problem 4 If a, b, c ≥ 0 prove that:

( Bara Andrei-Robert)Solution.

The inequality will become