Question

Simple symmetric inequality involving four variables

score 72 · Answer 1 · 2016-08-19 15:03:39

72

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Simple symmetric inequality involving four variables

Published on 19 Aug 2016 - 15:03

score 274 · Answer 2 · 2016-08-30 17:49:59

274

Views

Prove the inequality $\left(\frac1a+\frac1b+\frac1c\right)\left(\frac1{1+a}+\frac1{1+b}+\frac1{1+c}\right)\ge\frac9{1+abc}$

Published on 30 Aug 2016 - 17:49

score 86 · Answer 3 · 2016-09-05 15:25:29

86

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Prove that $ a^{k} + b^{k} + c^{k} \ge a^{k-1}b + b^{k-1}c + c^{k-1}a$

Published on 05 Sep 2016 - 15:25

score 116 · Answer 4 · 2016-09-24 06:44:15

116

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Inequality $\frac{3}{\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}}\geq1+ \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$ for positive $a$, $b$, $c$

Published on 24 Sep 2016 - 6:44

score 136 · Answer 5 · 2016-10-08 06:34:41

136

Views

Prove that $\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2$

Published on 08 Oct 2016 - 6:34

score 666 · Answer 6 · 2016-10-08 13:21:02

666

Views

How to approach Inequalities that can be solved using Rearrangement Inequality?

Published on 08 Oct 2016 - 13:21

score 105 · Answer 7 · 2016-10-31 10:16:16

105

Views

Let $x$, $y$, $z$ be positive. Prove an inequality...

Published on 31 Oct 2016 - 10:16

score 315 · Answer 8 · 2016-11-02 13:05:00

315

Views

How to prove that $\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$

Published on 02 Nov 2016 - 13:05

score 1.3k · Answer 9 · 2016-11-05 04:22:31

1.3k

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Prove that, for all non-negative real numbers $x,y,z$ that satisfy $x+y+z=1, x^2y+y^2z+z^2x≤4/27$

Published on 05 Nov 2016 - 4:22

score 152 · Answer 10 · 2016-11-27 05:36:13

152

Views

Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $

Published on 27 Nov 2016 - 5:36

score 739 · Answer 11 · 2026-03-25 09:29:06

739

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Proving $\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$

Published on 25 Mar 2026 - 9:29

score 122 · Answer 12 · 2017-01-27 12:36:45

122

Views

Given $ x_1 + x_2 + .... + x_{1994} = 1994$ find all $x_i$

Published on 27 Jan 2017 - 12:36

score 222 · Answer 13 · 2026-03-25 09:26:56

222

Views

Prove that $\sum\limits_{cyc} a^7 \geq \sum\limits_{cyc}a^4b^3$

Published on 25 Mar 2026 - 9:26

score 2k · Answer 14 · 2017-02-02 18:49:35

2k

Views

Prove the inequality $\frac{a^8+b^8+c^8}{a^3b^3c^3}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ for $a,b,c>0$

Published on 02 Feb 2017 - 18:49

score 415 · Answer 15 · 2017-02-02 19:38:42

415

Views

Two inequalities involving the rearrangement inequality

Published on 02 Feb 2017 - 19:38

List Question

Simple symmetric inequality involving four variables

Prove the inequality $\left(\frac1a+\frac1b+\frac1c\right)\left(\frac1{1+a}+\frac1{1+b}+\frac1{1+c}\right)\ge\frac9{1+abc}$

Prove that $ a^{k} + b^{k} + c^{k} \ge a^{k-1}b + b^{k-1}c + c^{k-1}a$

Inequality $\frac{3}{\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}}\geq1+ \frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$ for positive $a$, $b$, $c$

Prove that $\frac{x^2y}{z}+\frac{y^2z}{x}+\frac{z^2x}{y} \geq x^2+y^2+z^2$

How to approach Inequalities that can be solved using Rearrangement Inequality?

Let $x$, $y$, $z$ be positive. Prove an inequality...

How to prove that $\frac x{\sqrt y}+\frac y{\sqrt x}\ge\sqrt x+\sqrt y$

Prove that, for all non-negative real numbers $x,y,z$ that satisfy $x+y+z=1, x^2y+y^2z+z^2x≤4/27$

Algebraic proof that if $a>0$ then $1+a^9 \leq \frac{1}{a}+a^{10} $

Proving $\left(\frac{a}{a+b+c}\right)^2+\left(\frac{b}{b+c+d}\right)^2+\left(\frac{c}{c+d+a}\right)^2+\left(\frac{d}{d+a+b}\right)^2\ge\frac{4}{9}$

Given $ x_1 + x_2 + .... + x_{1994} = 1994$ find all $x_i$

Prove that $\sum\limits_{cyc} a^7 \geq \sum\limits_{cyc}a^4b^3$

Prove the inequality $\frac{a^8+b^8+c^8}{a^3b^3c^3}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ for $a,b,c>0$

Two inequalities involving the rearrangement inequality

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