List questions from a-m-g-m-inequality

Published on 08 Dec 2016 - 19:30

101

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An Inequality $\left(\sum\limits_{k=1}^n a_k^{1/2}\right)^2\le\left(\sum\limits_{k=1}^n a_k^{1/3}\right)^3$

Published on 14 Dec 2016 - 16:27

357

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Prove $\sum\limits_\text{cyc}\frac{a}{a+(n-1)b}\geq 1$

Published on 14 Dec 2016 - 17:42

135

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Prove that $\prod\limits_{i=1}^nx_i^{x_i}\ge\prod\limits_{i=1}^n{x_i^a}$

Published on 25 Mar 2026 - 22:06

321

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Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \geq \sqrt{(bc)^a(ac)^b(ab)^c}$

Published on 25 Mar 2026 - 22:06

287

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prove $\frac{a}{b}+\frac{b}{a}+\frac{c}{d}+\frac{d}{c}\le\frac{1}{64abcd}$

Published on 21 Dec 2016 - 14:23

272

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AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$

Published on 26 Dec 2016 - 0:02

560

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AM - GM inequality

Published on 26 Dec 2016 - 13:30

77

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Find the biggest number $M$ such that the inequality $a^2+b^{1389} \ge Mab$ holds for every $a,b \in [0,1]$.

Published on 03 Jan 2017 - 20:02

204

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Inequality with a+b+c=1

Published on 26 Mar 2026 - 3:01

192

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Verify $\cos{x}<\left(\frac{\sin{x}}{x}\right)^3$ for $0<x<\pi/2$.

Published on 08 Jan 2017 - 12:24

95

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$a\sqrt{b} + b\sqrt{c} + c\sqrt{d} + d\sqrt{a} \le 4$

Published on 08 Jan 2017 - 13:28

226

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Prove that $x+y+z=3$ $\implies$ $x^xy^yz^z\ge1$

Published on 25 Mar 2026 - 23:38

82

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Maximum of $f(x) =(1+\cos x)\cdot \sin(\frac{x}{2})$ on $x \in (0, \pi)$

Published on 10 Jan 2017 - 4:52

329

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Inequality with square roots : $\frac{\sqrt{b+c}}{a}+\frac{\sqrt{c+a}}{b}+\frac{\sqrt{a+b}}{c}\geq \frac{4(a+b+c)}{\sqrt{(a+b)(b+c)(c+a)}}$

Published on 25 Mar 2026 - 23:42

List Question

How to solve $x^6 - 6x + 5 > 0$?

An Inequality $\left(\sum\limits_{k=1}^n a_k^{1/2}\right)^2\le\left(\sum\limits_{k=1}^n a_k^{1/3}\right)^3$

Prove $\sum\limits_\text{cyc}\frac{a}{a+(n-1)b}\geq 1$

Prove that $\prod\limits_{i=1}^nx_i^{x_i}\ge\prod\limits_{i=1}^n{x_i^a}$

Prove that $(\frac{bc+ac+ab}{a+b+c})^{a+b+c} \geq \sqrt{(bc)^a(ac)^b(ab)^c}$

prove $\frac{a}{b}+\frac{b}{a}+\frac{c}{d}+\frac{d}{c}\le\frac{1}{64abcd}$

AM-GM inequality: $\frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{d} + \frac{d^2}{a} \geq a + b + c + d$

AM - GM inequality

Find the biggest number $M$ such that the inequality $a^2+b^{1389} \ge Mab$ holds for every $a,b \in [0,1]$.

Inequality with a+b+c=1

Verify $\cos{x}<\left(\frac{\sin{x}}{x}\right)^3$ for $0<x<\pi/2$.

$a\sqrt{b} + b\sqrt{c} + c\sqrt{d} + d\sqrt{a} \le 4$

Prove that $x+y+z=3$ $\implies$ $x^xy^yz^z\ge1$

Maximum of $f(x) =(1+\cos x)\cdot \sin(\frac{x}{2})$ on $x \in (0, \pi)$

Inequality with square roots : $\frac{\sqrt{b+c}}{a}+\frac{\sqrt{c+a}}{b}+\frac{\sqrt{a+b}}{c}\geq \frac{4(a+b+c)}{\sqrt{(a+b)(b+c)(c+a)}}$

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