Question

Simple symmetric inequality involving four variables

score 73 · Answer 1 · 2026-03-25 15:46:16

73

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Simple symmetric inequality involving four variables

Published on 25 Mar 2026 - 15:46

score 275 · Answer 2 · 2026-03-25 15:44:11

275

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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 25 Mar 2026 - 15:44

score 87 · Answer 3 · 2026-03-25 15:43:41

87

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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 25 Mar 2026 - 15:43

score 117 · Answer 4 · 2026-03-25 15:44:07

117

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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 25 Mar 2026 - 15:44

score 137 · Answer 5 · 2026-03-25 15:42:31

137

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

Published on 25 Mar 2026 - 15:42

score 667 · Answer 6 · 2026-03-25 15:41:13

667

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

Published on 25 Mar 2026 - 15:41

score 106 · Answer 7 · 2026-03-25 15:41:14

106

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

Published on 25 Mar 2026 - 15:41

score 316 · Answer 8 · 2026-03-25 15:42:21

316

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

Published on 25 Mar 2026 - 15:42

score 1.3k · Answer 9 · 2026-03-25 15:44:10

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 25 Mar 2026 - 15:44

score 153 · Answer 10 · 2026-03-25 15:43:48

153

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

Published on 25 Mar 2026 - 15:43

score 740 · Answer 11 · 2026-05-10 18:45:05

740

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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 10 May 2026 - 18:45

score 123 · Answer 12 · 2026-03-25 15:41:13

123

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

Published on 25 Mar 2026 - 15:41

score 223 · Answer 13 · 2026-05-10 18:43:45

223

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

Published on 10 May 2026 - 18:43

score 2k · Answer 14 · 2026-03-25 15:42:31

2k

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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 25 Mar 2026 - 15:42

score 416 · Answer 15 · 2026-03-25 15:47:20

416

Views

Two inequalities involving the rearrangement inequality

Published on 25 Mar 2026 - 15:47

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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