Question

Inequality sum of square root $3a^3+bc$ when $a^2+b^2+c^2=3$.

score 83 · Answer 1 · 2023-12-03 03:03:07

83

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Inequality sum of square root $3a^3+bc$ when $a^2+b^2+c^2=3$.

Published on 03 Dec 2023 - 3:03

score 132 · Answer 2 · 2026-03-25 16:20:36

132

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Find a nice proof for an elegant problem.

Published on 25 Mar 2026 - 16:20

score 83 · Answer 3 · 2023-12-06 00:22:08

83

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Proving $\sqrt{31a+b+c} +\sqrt{31b+a+c} +\sqrt{31c+b+a}\le 3\sqrt{3}\cdot\sqrt{a+b+c+8}$? when $a+b+c+abc=4.$

Published on 06 Dec 2023 - 0:22

score 118 · Answer 4 · 2023-12-06 08:48:28

118

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Proving $\sum_{cyc}\frac{a}{\sqrt{8a+bc}}\le \frac{\sqrt{a+b+c+abc}}{2}$ if $ab+bc+ca=3.$

Published on 06 Dec 2023 - 8:48

score 125 · Answer 5 · 2023-12-07 12:20:36

125

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$a+b+c−3≥k(a−b)(b−c)(c−a)$

Published on 07 Dec 2023 - 12:20

score 149 · Answer 6 · 2023-12-08 03:01:35

149

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Proving $5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.$

Published on 08 Dec 2023 - 3:01

score 116 · Answer 7 · 2015-01-24 07:48:50

116

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Symmetric and homogeneous three variable inequality with radicals.

Published on 24 Jan 2015 - 7:48

score 2.3k · Answer 8 · 2015-05-14 23:53:34

2.3k

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Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$

Published on 14 May 2015 - 23:53

score 277 · Answer 9 · 2015-12-25 01:44:32

277

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Inequality exercise (olympiad)

Published on 25 Dec 2015 - 1:44

score 2.6k · Answer 10 · 2026-03-21 21:52:03

2.6k

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Prove that $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$

Published on 21 Mar 2026 - 21:52

score 1.7k · Answer 11 · 2016-02-19 19:09:55

1.7k

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Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$

Published on 19 Feb 2016 - 19:09

score 171 · Answer 12 · 2026-03-23 00:38:04

171

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How to prove $|a^{2} + b^{2} + c^{2} - 2\left( ab+bc+ac\right)| \le \frac{1}{27}$

Published on 23 Mar 2026 - 0:38

score 370 · Answer 13 · 2016-05-30 20:09:42

370

Views

Prove $\frac{x+y+z}{3}+\frac{3}{\frac1x+\frac1y+\frac1z}\geq5\sqrt[3]{\frac{xyz}{16}}$

Published on 30 May 2016 - 20:09

score 180 · Answer 14 · 2016-06-04 16:04:47

180

Views

Prove $(a+b)^3(b+c)^3(c+d)^3(d+a)^3\ge 16a^2b^2c^2d^2$

Published on 04 Jun 2016 - 16:04

score 164 · Answer 15 · 2016-06-28 14:36:07

164

Views

How prove this inequality: $\frac1{1-a}+\frac1{1-b}+\frac1{1-c}\ge \frac1{ab+bc+ac}+\frac1{2(a^2+b^2+c^2)}$ for $a+b+c=1$?

Published on 28 Jun 2016 - 14:36

List Question

Inequality sum of square root $3a^3+bc$ when $a^2+b^2+c^2=3$.

Find a nice proof for an elegant problem.

Proving $\sqrt{31a+b+c} +\sqrt{31b+a+c} +\sqrt{31c+b+a}\le 3\sqrt{3}\cdot\sqrt{a+b+c+8}$? when $a+b+c+abc=4.$

Proving $\sum_{cyc}\frac{a}{\sqrt{8a+bc}}\le \frac{\sqrt{a+b+c+abc}}{2}$ if $ab+bc+ca=3.$

$a+b+c−3≥k(a−b)(b−c)(c−a)$

Proving $5(a^2+b^2+c^2)+ab+bc+ca\ge 2\sum_{cyc}a\sqrt{5a^2+b^2+c^2+ab+ac}.$

Symmetric and homogeneous three variable inequality with radicals.

Given $a,b,c,d>0$ and $a^2+b^2+c^2+d^2=1$, prove $a+b+c+d\ge a^3+b^3+c^3+d^3+ab+ac+ad+bc+bd+cd$

Inequality exercise (olympiad)

Prove that $a^2+b^2+c^2+3\sqrt[3]{a^2b^2c^2} \geq 2(ab+bc+ca).$

Prove that $(x+y+z)^2(yz+zx+xy)^2 \leq 3(y^2+yz+z^2)(z^2+zx+x^2)(x^2+xy+y^2)$

How to prove $|a^{2} + b^{2} + c^{2} - 2\left( ab+bc+ac\right)| \le \frac{1}{27}$

Prove $\frac{x+y+z}{3}+\frac{3}{\frac1x+\frac1y+\frac1z}\geq5\sqrt[3]{\frac{xyz}{16}}$

Prove $(a+b)^3(b+c)^3(c+d)^3(d+a)^3\ge 16a^2b^2c^2d^2$

How prove this inequality: $\frac1{1-a}+\frac1{1-b}+\frac1{1-c}\ge \frac1{ab+bc+ac}+\frac1{2(a^2+b^2+c^2)}$ for $a+b+c=1$?

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