Question

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

score 143 · Answer 1 · 2026-05-07 09:53:17

143

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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 07 May 2026 - 9:53

score 182 · Answer 2 · 2026-03-25 12:50:04

182

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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 25 Mar 2026 - 12:50

score 108 · Answer 3 · 2026-05-07 13:05:36

108

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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 07 May 2026 - 13:05

score 280 · Answer 4 · 2026-03-25 14:22:52

280

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

Published on 25 Mar 2026 - 14:22

score 98 · Answer 5 · 2026-03-25 12:49:53

98

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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 25 Mar 2026 - 12:49

score 782 · Answer 6 · 2026-03-25 12:49:45

782

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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 25 Mar 2026 - 12:49

score 173 · Answer 7 · 2026-03-25 11:16:26

173

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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 25 Mar 2026 - 11:16

score 101 · Answer 8 · 2026-05-07 12:29:42

101

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

Published on 07 May 2026 - 12:29

score 82 · Answer 9 · 2026-05-10 20:26:46

82

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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 10 May 2026 - 20:26

score 175 · Answer 10 · 2026-03-25 12:49:57

175

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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 25 Mar 2026 - 12:49

score 197 · Answer 11 · 2026-03-25 11:13:19

197

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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 25 Mar 2026 - 11:13

score 374 · Answer 12 · 2026-03-25 11:14:59

374

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

Published on 25 Mar 2026 - 11:14

score 226 · Answer 13 · 2026-03-25 11:16:33

226

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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 25 Mar 2026 - 11:16

score 1.1k · Answer 14 · 2026-03-25 12:49:50

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 25 Mar 2026 - 12:49

score 275 · Answer 15 · 2026-03-25 14:22:11

275

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

Published on 25 Mar 2026 - 14:22

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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