Question

Prove $\sum \sqrt{{\frac {2{a}^{2}b}{a+c}}} \leqq a+b+c$ for $a,b,c>0$

score 141 · Answer 1 · 2026-02-23 04:22:29

141

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Prove $\sum \sqrt{{\frac {2{a}^{2}b}{a+c}}} \leqq a+b+c$ for $a,b,c>0$

Published on 23 Feb 2026 - 4:22

score 181 · Answer 2 · 2020-05-17 03:04:08

181

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Prove $\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} +\frac{81abc}{4(a+b+c)^2} \geqq \frac{7}{4} (a+b+c)$

Published on 17 May 2020 - 3:04

score 106 · Answer 3 · 2026-02-23 04:24:34

106

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Prove $2\left(x^2+y^2+z^2+1)(x^3y+y^3z+z^3x+xyz\right) \le \left(x^2+y^2+z^2+3xyz\right)^2.$

Published on 23 Feb 2026 - 4:24

score 279 · Answer 4 · 2020-06-04 02:53:05

279

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Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$

Published on 04 Jun 2020 - 2:53

score 97 · Answer 5 · 2020-06-04 03:44:13

97

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Prove$:$ $\sum\limits_{cyc} (\frac{a}{b+c}-\frac{1}{2}) \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{1}{(a+b)^2}\Big]-\frac{9}{4}$

Published on 04 Jun 2020 - 3:44

score 781 · Answer 6 · 2020-06-07 14:17:50

781

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Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$

Published on 07 Jun 2020 - 14:17

score 172 · Answer 7 · 2020-06-14 14:10:53

172

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Find maximum $k \in \mathbb{R}^{+}$ such that $ \frac{a^3}{(b-c)^2} + \frac{b^3}{(c-a)^2} + \frac{c^3}{(a-b)^2} \geq k (a+b+c) $

Published on 14 Jun 2020 - 14:10

score 99 · Answer 8 · 2026-02-23 04:22:02

99

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Proving a non-homogeneous inequality with $x,y,z>0$

Published on 23 Feb 2026 - 4:22

score 80 · Answer 9 · 2020-07-11 21:09:54

80

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on possible generalisations of $1\le\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\le 2$

Published on 11 Jul 2020 - 21:09

score 174 · Answer 10 · 2020-07-26 00:45:53

174

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Proving $\frac {a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a} \geqslant \frac 32 \cdot \sqrt[6]{\frac{ab+bc+ca}{a^2+b^2+c^2}}$

Published on 26 Jul 2020 - 0:45

score 196 · Answer 11 · 2016-10-08 11:07:57

196

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Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$

Published on 08 Oct 2016 - 11:07

score 373 · Answer 12 · 2016-10-11 17:56:27

373

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If $abc=1$ so $\sum\limits_{cyc}\frac{a}{\sqrt{a+b^2}}\geq\frac{3}{\sqrt2}$

Published on 11 Oct 2016 - 17:56

score 225 · Answer 13 · 2016-11-22 15:03:30

225

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Prove that: $\sum\limits_{cyc}\frac{1}{a}\sum\limits_{cyc}\frac{1}{1+a^2}\geq\frac{16}{1+abcd}$

Published on 22 Nov 2016 - 15:03

score 1.1k · Answer 14 · 2016-12-12 15:32:56

1.1k

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Let $a,b,c$ be the length of sides of a triangle then prove that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$

Published on 12 Dec 2016 - 15:32

score 274 · Answer 15 · 2016-12-19 21:58:55

274

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If $a+b+c+d=1$ so $\sum\limits_{cyc}\sqrt{a+b+c^2}\geq3$

Published on 19 Dec 2016 - 21:58

List Question

Prove $\sum \sqrt{{\frac {2{a}^{2}b}{a+c}}} \leqq a+b+c$ for $a,b,c>0$

Prove $\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} +\frac{81abc}{4(a+b+c)^2} \geqq \frac{7}{4} (a+b+c)$

Prove $2\left(x^2+y^2+z^2+1)(x^3y+y^3z+z^3x+xyz\right) \le \left(x^2+y^2+z^2+3xyz\right)^2.$

Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$

Prove$:$ $\sum\limits_{cyc} (\frac{a}{b+c}-\frac{1}{2}) \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{1}{(a+b)^2}\Big]-\frac{9}{4}$

Prove that $\sum_{\mathrm{cyc}} \frac{214x^4}{133x^3 + 81y^3} \ge x + y + z$ for $x, y, z > 0$

Find maximum $k \in \mathbb{R}^{+}$ such that $ \frac{a^3}{(b-c)^2} + \frac{b^3}{(c-a)^2} + \frac{c^3}{(a-b)^2} \geq k (a+b+c) $

Proving a non-homogeneous inequality with $x,y,z>0$

on possible generalisations of $1\le\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\le 2$

Proving $\frac {a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a} \geqslant \frac 32 \cdot \sqrt[6]{\frac{ab+bc+ca}{a^2+b^2+c^2}}$

Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$

If $abc=1$ so $\sum\limits_{cyc}\frac{a}{\sqrt{a+b^2}}\geq\frac{3}{\sqrt2}$

Prove that: $\sum\limits_{cyc}\frac{1}{a}\sum\limits_{cyc}\frac{1}{1+a^2}\geq\frac{16}{1+abcd}$

Let $a,b,c$ be the length of sides of a triangle then prove that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge0$

If $a+b+c+d=1$ so $\sum\limits_{cyc}\sqrt{a+b+c^2}\geq3$

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