Question

Find the maximum of the value $F=x^3y+y^3z+z^3x$

score 425 · Answer 1 · 2026-03-26 12:08:26

425

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Find the maximum of the value $F=x^3y+y^3z+z^3x$

Published on 26 Mar 2026 - 12:08

score 404 · Answer 2 · 2021-08-31 13:21:34

404

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Let $a,b,c$ be non-negative real numbers .Prove that : $ a^2+b^2+c^2 +\sqrt{2} abc + 2\sqrt{2} +3 \geq (2+\sqrt {2} )(a+b+c) $

Published on 31 Aug 2021 - 13:21

score 527 · Answer 3 · 2021-09-12 11:45:43

527

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Prove that: $\sum\limits_{cyc}\frac{1}{\sqrt{2a^2+5ab+2b^2}} \geq\sqrt{\frac{3}{ab+ac+bc}}$

Published on 12 Sep 2021 - 11:45

score 452 · Answer 4 · 2021-09-27 04:16:51

452

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Prove that $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \ge ab + bc + ca$

Published on 27 Sep 2021 - 4:16

score 161 · Answer 5 · 2021-10-02 02:02:41

161

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How do I prove that $\sum_{cyc}\left(\dfrac{1}{x^2-xy+y^2}\right)+15\ge6(\sqrt{xy}+\sqrt{yz}+\sqrt{zx})$ given $x,y,z > 0$ and $x+y+z=3$?

Published on 02 Oct 2021 - 2:02

score 124 · Answer 6 · 2021-10-13 12:59:47

124

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Inequality without BW and Uvw, just AM GM

Published on 13 Oct 2021 - 12:59

score 240 · Answer 7 · 2021-10-14 05:42:25

240

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Prove that: $2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$

Published on 14 Oct 2021 - 5:42

score 189 · Answer 8 · 2021-10-15 12:28:47

189

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Prove that $\frac{a^4+2a+b(2a^2+b)}{a^2(b^2+2a)}+\frac{b^4+2b+c(2b^2+c)}{b^2(c^2+2b)}+\frac{c^4+2c+a(2c^2+a)}{c^2(a^2+2c)}\ge 6$

Published on 15 Oct 2021 - 12:28

score 63 · Answer 9 · 2026-03-25 23:42:38

63

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Smallest positive $k$ such that $\sum_{cyc}\left(\frac{(k-1)(z+x)}{y}-\frac{3}{2}\right)(x-y)^2\geq 0$

Published on 25 Mar 2026 - 23:42

score 72 · Answer 10 · 2022-08-14 19:39:15

72

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Prove $\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}\geq x^3+y^3+z^3$ where $x,y,z>0$ and $x+y+z=3$.

Published on 14 Aug 2022 - 19:39

score 168 · Answer 11 · 2026-03-25 23:42:38

168

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$3$-var inequality: $\frac{bc}{\sqrt{a}+3}+\frac{ca}{\sqrt{b}+3}+\frac{ab}{\sqrt{c}+3} \leq \frac{3}{4}$ for $a+b+c=3$.

Published on 25 Mar 2026 - 23:42

score 225 · Answer 12 · 2023-01-18 14:34:13

225

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Prove that $(\sum x^2)^3\ge9\sum x^4yz$

Published on 18 Jan 2023 - 14:34

List Question

Find the maximum of the value $F=x^3y+y^3z+z^3x$

Let $a,b,c$ be non-negative real numbers .Prove that : $ a^2+b^2+c^2 +\sqrt{2} abc + 2\sqrt{2} +3 \geq (2+\sqrt {2} )(a+b+c) $

Prove that: $\sum\limits_{cyc}\frac{1}{\sqrt{2a^2+5ab+2b^2}} \geq\sqrt{\frac{3}{ab+ac+bc}}$

Prove that $\sqrt[3]{a} + \sqrt[3]{b} + \sqrt[3]{c} \ge ab + bc + ca$

How do I prove that $\sum_{cyc}\left(\dfrac{1}{x^2-xy+y^2}\right)+15\ge6(\sqrt{xy}+\sqrt{yz}+\sqrt{zx})$ given $x,y,z > 0$ and $x+y+z=3$?

Inequality without BW and Uvw, just AM GM

Prove that: $2(a+b+c)+\sqrt{ab}+\sqrt{bc}+\sqrt{ca}\ge\sqrt{5ab+4ac}+\sqrt{5bc+4ba}+\sqrt{5ca+4cb}$

Prove that $\frac{a^4+2a+b(2a^2+b)}{a^2(b^2+2a)}+\frac{b^4+2b+c(2b^2+c)}{b^2(c^2+2b)}+\frac{c^4+2c+a(2c^2+a)}{c^2(a^2+2c)}\ge 6$

Smallest positive $k$ such that $\sum_{cyc}\left(\frac{(k-1)(z+x)}{y}-\frac{3}{2}\right)(x-y)^2\geq 0$

Prove $\frac{x}{y^2}+\frac{y}{z^2}+\frac{z}{x^2}\geq x^3+y^3+z^3$ where $x,y,z>0$ and $x+y+z=3$.

$3$-var inequality: $\frac{bc}{\sqrt{a}+3}+\frac{ca}{\sqrt{b}+3}+\frac{ab}{\sqrt{c}+3} \leq \frac{3}{4}$ for $a+b+c=3$.

Prove that $(\sum x^2)^3\ge9\sum x^4yz$

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