Question

Hard inequality for positive numbers

score 443 · Answer 1 · 2019-12-23 13:17:43

443

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Hard inequality for positive numbers

Published on 23 Dec 2019 - 13:17

score 255 · Answer 2 · 2026-03-25 04:43:01

255

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A nontrvial inequality $\sum\limits_{\rm cyc}a^2 c(4a-3b-c)^2\ge 20abc\sum\limits_{\rm cyc}a(a-b)$

Published on 25 Mar 2026 - 4:43

score 286 · Answer 3 · 2020-02-14 01:50:48

286

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How to prove $\frac a{\sqrt{a^2+3b^2+3c^2}}+\frac b{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}$ when $a,b,c>0$

Published on 14 Feb 2020 - 1:50

score 132 · Answer 4 · 2020-02-29 18:22:06

132

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For $a,b,c \ge 0$ real numbers, prove $\frac{2}{(1+a)^2}+\frac{2}{(1+b)^2}+\frac{2}{(1+c)^2} \geq \frac{9}{3+ab+bc+ca}$

Published on 29 Feb 2020 - 18:22

score 122 · Answer 5 · 2020-03-08 06:32:56

122

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Given positives $a, b, c$ such that $a + b + c = 3$, prove that $\sum_{cyc}\frac{1}{a^2 + 4b^2 + c^2} \le \frac{1}{2}$.

Published on 08 Mar 2020 - 6:32

score 196 · Answer 6 · 2020-03-19 08:30:03

196

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$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$

Published on 19 Mar 2020 - 8:30

score 220 · Answer 7 · 2020-03-21 22:47:20

220

Views

Inequality for Olympiad students

Published on 21 Mar 2020 - 22:47

score 160 · Answer 8 · 2020-04-01 22:27:46

160

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Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$

Published on 01 Apr 2020 - 22:27

score 97 · Answer 9 · 2020-04-16 07:57:29

97

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Inequality with 4 variables

Published on 16 Apr 2020 - 7:57

score 305 · Answer 10 · 2026-03-23 23:47:13

305

Views

MOP 2011 inequality

Published on 23 Mar 2026 - 23:47

score 139 · Answer 11 · 2026-03-25 09:40:00

139

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Prove $\Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$

Published on 25 Mar 2026 - 9:40

score 117 · Answer 12 · 2020-05-04 15:09:45

117

Views

show this inequality $\sum_{cyc}\frac{1}{5-2xy}\le 1$

Published on 04 May 2020 - 15:09

score 111 · Answer 13 · 2020-05-06 09:49:09

111

Views

Prove $\frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a}$

Published on 06 May 2020 - 9:49

score 161 · Answer 14 · 2020-05-09 10:58:07

161

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prove that $3(a+b+c) \geq 8(a b c)^{1 / 3}+\left(\frac{a^{3}+b^{3}+c^{3}}{3}\right)^{1 / 3}$

Published on 09 May 2020 - 10:58

score 147 · Answer 15 · 2026-03-25 09:43:13

147

Views

Prove $\frac{a^2}{(b+c)^2}+\frac{b^2}{(c+a)^2}+\frac{c^2}{(a+b)^2}+\frac{1}{4}\ge \frac{a^2+b^2+c^2}{ab+bc+ca}$

Published on 25 Mar 2026 - 9:43

List Question

Hard inequality for positive numbers

A nontrvial inequality $\sum\limits_{\rm cyc}a^2 c(4a-3b-c)^2\ge 20abc\sum\limits_{\rm cyc}a(a-b)$

How to prove $\frac a{\sqrt{a^2+3b^2+3c^2}}+\frac b{\sqrt{3a^2+b^2+3c^2}}+\frac{c}{\sqrt{3a^2+3b^2+c^2}}\le\frac3{\sqrt7}$ when $a,b,c>0$

For $a,b,c \ge 0$ real numbers, prove $\frac{2}{(1+a)^2}+\frac{2}{(1+b)^2}+\frac{2}{(1+c)^2} \geq \frac{9}{3+ab+bc+ca}$

Given positives $a, b, c$ such that $a + b + c = 3$, prove that $\sum_{cyc}\frac{1}{a^2 + 4b^2 + c^2} \le \frac{1}{2}$.

$\frac{a}{b}+ \frac{b}{c} + \frac{c}{a} \geq \frac{9(a^2+b^2+c^2)}{(a+b+c)^2}$

Inequality for Olympiad students

Prove that : $m_{a}m_{b}m_{c}\leq\frac{Rs^{2}}{2}$

Inequality with 4 variables

MOP 2011 inequality

Prove $\Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$

show this inequality $\sum_{cyc}\frac{1}{5-2xy}\le 1$

Prove $\frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a}$

prove that $3(a+b+c) \geq 8(a b c)^{1 / 3}+\left(\frac{a^{3}+b^{3}+c^{3}}{3}\right)^{1 / 3}$

Prove $\frac{a^2}{(b+c)^2}+\frac{b^2}{(c+a)^2}+\frac{c^2}{(a+b)^2}+\frac{1}{4}\ge \frac{a^2+b^2+c^2}{ab+bc+ca}$

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