List questions from muirhead-inequality

Published on 06 Feb 2019 - 11:02

127

Views

Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

Published on 10 Feb 2019 - 12:20

129

Views

Inequality with $a+b+c=1$, a,b,c positive numbers

Published on 22 Mar 2026 - 12:20

69

Views

A three variable inequality doubt , can I consider the three variables into just one variable , and show the inequality.

Published on 04 May 2019 - 15:57

162

Views

Find $f(x)$ so that $\sum\limits_{cyc}a^{2}-f(x)\left(\prod_{sym}a-\prod_{sym}(1-a)\right)\geqq3(\frac{x}{2})^{2}$ .

Published on 14 May 2019 - 11:46

327

Views

show this inequality with $xy+yz+zx=3$

Published on 22 Mar 2026 - 15:14

117

Views

Nice refinement of an inequality by Michael Rozenberg

Published on 22 Mar 2026 - 19:33

116

Views

Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

Published on 08 Jun 2019 - 2:15

149

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Prove $x+y+z \ge xy+yz+zx$

Published on 22 Mar 2026 - 13:24

272

Views

Given three positive numbers $a,b,c$. Prove that $\sum\limits_{cyc}\sqrt{\frac{a+b}{b+1}}\geqq3\sqrt[3]{\frac{4\,abc}{3\,abc+1}}$ .

Published on 15 Jul 2019 - 3:04

59

Views

an interesting inequality with Muirhead

Published on 22 Mar 2026 - 12:07

210

Views

How can we not use Muirhead's Inequality for proving the following inequality?

Published on 18 Aug 2019 - 4:11

191

Views

How can I solve prove that $8(1-a)(1-b)(1-c)\le abc$ with the conditions below?

Published on 21 Aug 2019 - 5:04

85

Views

An inequality with four variables

Published on 22 Mar 2026 - 1:21

266

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A hard inequality indian olympiad problem

Published on 22 Mar 2026 - 8:42

List Question

Show that $(a^2+3)(b^2+3)(c^2+3)(d^2+3)\geq 256$.

Show that $\frac{1}{a^2+b^2+1}+\frac{1}{b^2+c^2+1}+\frac{1}{c^2+a^2+1}\leq 1$.

Inequality with $a+b+c=1$, a,b,c positive numbers

A three variable inequality doubt , can I consider the three variables into just one variable , and show the inequality.

Find $f(x)$ so that $\sum\limits_{cyc}a^{2}-f(x)\left(\prod_{sym}a-\prod_{sym}(1-a)\right)\geqq3(\frac{x}{2})^{2}$ .

show this inequality with $xy+yz+zx=3$

Nice refinement of an inequality by Michael Rozenberg

Calculate the maximum value of $\frac{x^2}{x^4 + yz} + \frac{y^2}{y^4 + zx} + \frac{z^2}{z^4 + xy}$ where $x, y, z > 0$ and $x^2 + y^2 + z^2 = 3xyz$.

Prove $x+y+z \ge xy+yz+zx$

Given three positive numbers $a,b,c$. Prove that $\sum\limits_{cyc}\sqrt{\frac{a+b}{b+1}}\geqq3\sqrt[3]{\frac{4\,abc}{3\,abc+1}}$ .

an interesting inequality with Muirhead

How can we not use Muirhead's Inequality for proving the following inequality?

How can I solve prove that $8(1-a)(1-b)(1-c)\le abc$ with the conditions below?

An inequality with four variables

A hard inequality indian olympiad problem

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