Question

How prove this inequality $\sum\limits_{cyc}\frac{a^3}{b+c+d}\ge \dfrac{1}{3}$,if $\sum\limits_{cyc}a\sqrt{bc}\ge 1$

score 253 · Answer 1 · 2026-03-25 11:24:34

253

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How prove this inequality $\sum\limits_{cyc}\frac{a^3}{b+c+d}\ge \dfrac{1}{3}$,if $\sum\limits_{cyc}a\sqrt{bc}\ge 1$

Published on 25 Mar 2026 - 11:24

score 295 · Answer 2 · 2014-07-13 11:05:09

295

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Inequality using AM-GM

Published on 13 Jul 2014 - 11:05

score 332 · Answer 3 · 2026-03-25 03:07:00

332

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Proof of an inequality

Published on 25 Mar 2026 - 3:07

score 361 · Answer 4 · 2014-08-01 13:19:57

361

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Prove $a+b+c \geq ab+bc+ca$, given an additional constraint

Published on 01 Aug 2014 - 13:19

score 736 · Answer 5 · 2014-09-01 08:13:10

736

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Prove $\frac{ab}{\sqrt{2c+a+b}}+\frac{bc}{\sqrt{2a+b+c}}+\frac{ca}{\sqrt{2b+c+a}}\le\sqrt\frac{2}{3}$ if $a+b+c=2$

Published on 01 Sep 2014 - 8:13

score 107 · Answer 6 · 2014-09-18 09:14:52

107

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I'm new to proving inequalities. How does one prove this?

Published on 18 Sep 2014 - 9:14

score 237 · Answer 7 · 2026-03-25 07:46:29

237

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How prove this Nice inequality $\sum\limits_\text{cyc}\frac{x^2}{y}\ge 3+\frac{x^4+y^4+z^4-x^2-y^2-z^2}{x^3+y^3+z^3-xyz}$

Published on 25 Mar 2026 - 7:46

score 458 · Answer 8 · 2026-03-25 11:21:48

458

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How to prove the inequality $ \frac{a}{\sqrt{1+a}}+\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} \ge \frac{3\sqrt{2}}{2}$

Published on 25 Mar 2026 - 11:21

score 431 · Answer 9 · 2026-03-25 11:21:28

431

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Minimum of $\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}$

Published on 25 Mar 2026 - 11:21

score 1.7k · Answer 10 · 2014-11-11 17:18:50

1.7k

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Inequality with five variables

Published on 11 Nov 2014 - 17:18

score 83 · Answer 11 · 2026-03-25 03:03:48

83

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Find Min $P=\frac{a^4}{a+2b}+\frac{b^4}{b+2c}+\frac{c^4}{c+2a}$

Published on 25 Mar 2026 - 3:03

score 900 · Answer 12 · 2026-03-25 12:15:14

900

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Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$

Published on 25 Mar 2026 - 12:15

score 489 · Answer 13 · 2014-12-29 12:43:12

489

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Inequality between AM-GM

Published on 29 Dec 2014 - 12:43

score 83 · Answer 14 · 2026-03-25 18:08:25

83

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Given three non-negative numbers $a,b,c$. Prove that $\frac{a+b+c}{k}\geqq\sum\limits_{cyc}\frac{a-b}{b+ k}$ for $k= constant$ so that $k> 0$ .

Published on 25 Mar 2026 - 18:08

score 215 · Answer 15 · 2019-10-07 03:39:58

215

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I made an inequality to solve myself, but someone pointed out my solution is wrong.

Published on 07 Oct 2019 - 3:39

List Question

How prove this inequality $\sum\limits_{cyc}\frac{a^3}{b+c+d}\ge \dfrac{1}{3}$,if $\sum\limits_{cyc}a\sqrt{bc}\ge 1$

Inequality using AM-GM

Proof of an inequality

Prove $a+b+c \geq ab+bc+ca$, given an additional constraint

Prove $\frac{ab}{\sqrt{2c+a+b}}+\frac{bc}{\sqrt{2a+b+c}}+\frac{ca}{\sqrt{2b+c+a}}\le\sqrt\frac{2}{3}$ if $a+b+c=2$

I'm new to proving inequalities. How does one prove this?

How prove this Nice inequality $\sum\limits_\text{cyc}\frac{x^2}{y}\ge 3+\frac{x^4+y^4+z^4-x^2-y^2-z^2}{x^3+y^3+z^3-xyz}$

How to prove the inequality $ \frac{a}{\sqrt{1+a}}+\frac{b}{\sqrt{1+b}}+\frac{c}{\sqrt{1+c}} \ge \frac{3\sqrt{2}}{2}$

Minimum of $\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}$

Inequality with five variables

Find Min $P=\frac{a^4}{a+2b}+\frac{b^4}{b+2c}+\frac{c^4}{c+2a}$

Proving the inequality $4\ge a^2b+b^2c+c^2a+abc$

Inequality between AM-GM

Given three non-negative numbers $a,b,c$. Prove that $\frac{a+b+c}{k}\geqq\sum\limits_{cyc}\frac{a-b}{b+ k}$ for $k= constant$ so that $k> 0$ .

I made an inequality to solve myself, but someone pointed out my solution is wrong.

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