Question

Prove that $(x-y)(y-z)(z-x) \leq \frac{1}{\sqrt{2}}$

score 353 · Answer 1 · 2026-03-23 01:33:23

353

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Prove that $(x-y)(y-z)(z-x) \leq \frac{1}{\sqrt{2}}$

Published on 23 Mar 2026 - 1:33

score 153 · Answer 2 · 2026-03-22 20:17:53

153

Views

Proving $a^4+b^4+c^4+(\sqrt {3}-1)(a^2 b c+a b^2 c+a b c^2 )\ge \sqrt {3} (a^3 b+b^3 c+c^3 a)$ for real $a$, $b$, $c$

Published on 22 Mar 2026 - 20:17

score 670 · Answer 3 · 2026-03-22 05:56:56

670

Views

prove this inequality by $abc=1$

Published on 22 Mar 2026 - 5:56

score 121 · Answer 4 · 2026-03-22 05:42:58

121

Views

Prove $(a+b^2)(b+c^2)(c+a^2)\le13+abc(1-2abc)$ for $a + b + c = 3$

Published on 22 Mar 2026 - 5:42

score 98 · Answer 5 · 2026-03-22 19:34:33

98

Views

Seeking the Maximum of a Product expression using Inequalities

Published on 22 Mar 2026 - 19:34

score 160 · Answer 6 · 2026-03-23 04:46:23

160

Views

If $0⩽x⩽y⩽z⩽w⩽u$ and $x+y+z+w+u=1$, prove $xw+wz+zy+yu+ux⩽\frac15$

Published on 23 Mar 2026 - 4:46

score 196 · Answer 7 · 2026-03-20 04:53:11

196

Views

Inequality from AMM problems section

Published on 20 Mar 2026 - 4:53

score 257 · Answer 8 · 2026-03-19 22:28:43

257

Views

Proving $\sum\limits_{\text{cyc}} \frac{a}{b^2+c^2+d^2} \geq \frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$ for $a, b, c, d >0$

Published on 19 Mar 2026 - 22:28

score 219 · Answer 9 · 2026-03-22 16:20:10

219

Views

Inequality $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+15\sqrt[3]{abc}\geq 6(a+b+c).$

Published on 22 Mar 2026 - 16:20

score 86 · Answer 10 · 2026-03-24 17:20:09

86

Views

Little inequality

Published on 24 Mar 2026 - 17:20

score 453 · Answer 11 · 2026-03-22 17:05:00

453

Views

show this inequality with $\sum_{i=1}^{n}a_{i}=n$

Published on 22 Mar 2026 - 17:05

score 170 · Answer 12 · 2026-03-17 02:32:51

170

Views

to prove this inequality $\sum a^3b+3\ge 2(ab+bc+ca)$

Published on 17 Mar 2026 - 2:32

score 1k · Answer 13 · 2026-03-17 01:03:34

1k

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Given three a-triangle-sidelengths $a,b,c$. Prove that $3\left((a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\right)\geqq b(a+b-c)(a-c)(c-b)$ .

Published on 17 Mar 2026 - 1:03

score 198 · Answer 14 · 2026-03-17 04:06:27

198

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Find $k=constant$ such that $f(a,\,b,\,c\,)=\frac{3a+2b}{\sqrt{5a^2-ab+b^2}}+\frac{3b+2c}{\sqrt{5b^2-bc+c^2}}\leqq f(a,k\,a+\sqrt[3]{abc}-k\,c,c\,)$

Published on 17 Mar 2026 - 4:06

score 221 · Answer 15 · 2026-03-17 04:31:24

221

Views

A nice Nesbitt inequality from a strange inequality

Published on 17 Mar 2026 - 4:31

List Question

Prove that $(x-y)(y-z)(z-x) \leq \frac{1}{\sqrt{2}}$

Proving $a^4+b^4+c^4+(\sqrt {3}-1)(a^2 b c+a b^2 c+a b c^2 )\ge \sqrt {3} (a^3 b+b^3 c+c^3 a)$ for real $a$, $b$, $c$

prove this inequality by $abc=1$

Prove $(a+b^2)(b+c^2)(c+a^2)\le13+abc(1-2abc)$ for $a + b + c = 3$

Seeking the Maximum of a Product expression using Inequalities

If $0⩽x⩽y⩽z⩽w⩽u$ and $x+y+z+w+u=1$, prove $xw+wz+zy+yu+ux⩽\frac15$

Inequality from AMM problems section

Proving $\sum\limits_{\text{cyc}} \frac{a}{b^2+c^2+d^2} \geq \frac{3\sqrt{3}}{2}\frac{1}{\sqrt{a^2+b^2+c^2+d^2}}$ for $a, b, c, d >0$

Inequality $\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}+15\sqrt[3]{abc}\geq 6(a+b+c).$

Little inequality

show this inequality with $\sum_{i=1}^{n}a_{i}=n$

to prove this inequality $\sum a^3b+3\ge 2(ab+bc+ca)$

Given three a-triangle-sidelengths $a,b,c$. Prove that $3\left((a^{2}b(a-b)+b^{2}c(b-c)+c^{2}a(c-a)\right)\geqq b(a+b-c)(a-c)(c-b)$ .

Find $k=constant$ such that $f(a,\,b,\,c\,)=\frac{3a+2b}{\sqrt{5a^2-ab+b^2}}+\frac{3b+2c}{\sqrt{5b^2-bc+c^2}}\leqq f(a,k\,a+\sqrt[3]{abc}-k\,c,c\,)$

A nice Nesbitt inequality from a strange inequality

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