Question

Stuck on this cyclic 3-variables-inequality with constraint

score 74 · Answer 1 · 2026-03-25 03:07:24

74

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Stuck on this cyclic 3-variables-inequality with constraint

Published on 25 Mar 2026 - 3:07

score 45 · Answer 2 · 2019-09-29 17:07:38

45

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Find minimize of function $P=5(a^4+b^4+a^2b^2)-3(a^2+b^2)+4$

Published on 29 Sep 2019 - 17:07

score 90 · Answer 3 · 2019-09-30 15:29:23

90

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Given $0 < x \le 1 \le y$. Calculate the minimum value of $\frac{x}{y + 1} + \frac{y}{x + 1} + \frac{x^2 + 3y^2 + 2}{6xy(xy + 1)}$.

Published on 30 Sep 2019 - 15:29

score 1.2k · Answer 4 · 2014-04-05 08:48:49

1.2k

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Hard Olympiad Inequality

Published on 05 Apr 2014 - 8:48

score 174 · Answer 5 · 2026-03-25 03:22:13

174

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Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

Published on 25 Mar 2026 - 3:22

score 1k · Answer 6 · 2014-04-14 05:29:59

1k

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How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$

Published on 14 Apr 2014 - 5:29

score 108 · Answer 7 · 2026-03-25 12:21:11

108

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Show $ \frac {a^3}{a^2+b^2+c^2-bc}+\frac {b^3}{a^2+b^2+c^2-ca} +\frac {c^3}{a^2+b^2+c^2-ab} \le \frac {(a+b+c)^2 (a^4+b^4+c^4)}{6abc(ab+bc+ca)} $

Published on 25 Mar 2026 - 12:21

score 154 · Answer 8 · 2026-03-27 13:03:23

154

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Prove or disprove inequality $\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\le\frac{a^4+b^4+c^4}{2abc}$.

Published on 27 Mar 2026 - 13:03

score 626 · Answer 9 · 2026-03-25 12:49:48

626

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How to prove $\sqrt{\frac{ab}{2a^2+bc+ca}}+\sqrt{\frac{bc}{2b^2+ca+ab}}+\sqrt{\frac{ca}{2c^2+ab+bc}}\ge\frac{81}{2}\cdot\frac{abc}{(a+b+c)^3}$

Published on 25 Mar 2026 - 12:49

score 253 · Answer 10 · 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 1k · Answer 11 · 2026-03-28 01:46:18

1k

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Prove that this is a metric

Published on 28 Mar 2026 - 1:46

score 171 · Answer 12 · 2014-06-28 23:34:49

171

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If $x,y,z>0$, then $\frac{x}{4x+4y+z} + \frac{y}{4y+4z+x}+\frac{z}{4z+4x+y}\le \frac13$

Published on 28 Jun 2014 - 23:34

score 3.2k · Answer 13 · 2026-03-25 11:04:55

3.2k

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Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$

Published on 25 Mar 2026 - 11:04

score 453 · Answer 14 · 2026-03-26 16:06:08

453

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Prove $\sum\limits_{\mathrm{cyc}}\sqrt{a^2+bc}\leq{3\over2}(a+b+c)$ with $a,b,c$ are nonnegative

Published on 26 Mar 2026 - 16:06

score 136 · Answer 15 · 2026-03-25 07:42:04

136

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Proving Cauchy inequality involving four expression

Published on 25 Mar 2026 - 7:42

List Question

Stuck on this cyclic 3-variables-inequality with constraint

Find minimize of function $P=5(a^4+b^4+a^2b^2)-3(a^2+b^2)+4$

Given $0 < x \le 1 \le y$. Calculate the minimum value of $\frac{x}{y + 1} + \frac{y}{x + 1} + \frac{x^2 + 3y^2 + 2}{6xy(xy + 1)}$.

Hard Olympiad Inequality

Prove that positive $x,y,z$ satisfy $\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}\ge \sqrt{2x}+\sqrt{2y}+\sqrt{2z}$

How to show $n\sum_{i=1}^n {x_i^2} \ge (\sum_{i=1}^n{x_i})^2$

Show $ \frac {a^3}{a^2+b^2+c^2-bc}+\frac {b^3}{a^2+b^2+c^2-ca} +\frac {c^3}{a^2+b^2+c^2-ab} \le \frac {(a+b+c)^2 (a^4+b^4+c^4)}{6abc(ab+bc+ca)} $

Prove or disprove inequality $\frac{a^2}{b+c}+\frac{b^2}{a+c}+\frac{c^2}{a+b}\le\frac{a^4+b^4+c^4}{2abc}$.

How to prove $\sqrt{\frac{ab}{2a^2+bc+ca}}+\sqrt{\frac{bc}{2b^2+ca+ab}}+\sqrt{\frac{ca}{2c^2+ab+bc}}\ge\frac{81}{2}\cdot\frac{abc}{(a+b+c)^3}$

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

Prove that this is a metric

If $x,y,z>0$, then $\frac{x}{4x+4y+z} + \frac{y}{4y+4z+x}+\frac{z}{4z+4x+y}\le \frac13$

Proof of the inequality $\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b} \geq \frac{3}{2}$

Prove $\sum\limits_{\mathrm{cyc}}\sqrt{a^2+bc}\leq{3\over2}(a+b+c)$ with $a,b,c$ are nonnegative

Proving Cauchy inequality involving four expression

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