Question

Prove $\frac{1}{2a+2bc+1} + \frac{1}{2b+2ca+1} + \frac{1}{2c+2ab+1} \ge 1$

score 1k · Answer 1 · 2013-04-02 22:59:09

1k

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Prove $\frac{1}{2a+2bc+1} + \frac{1}{2b+2ca+1} + \frac{1}{2c+2ab+1} \ge 1$

Published on 02 Apr 2013 - 22:59

score 593 · Answer 2 · 2013-07-10 15:29:29

593

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How prove that $xyz+\sqrt{x^2y^2+y^2z^2+x^2z^2}\ge \frac{4}{3}\sqrt{xyz(x+y+z)}$

Published on 10 Jul 2013 - 15:29

score 379 · Answer 3 · 2013-08-18 08:13:54

379

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How prove that $10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\le\dfrac{9}{4}$

Published on 18 Aug 2013 - 8:13

score 1k · Answer 4 · 2013-09-30 17:19:49

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How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

Published on 30 Sep 2013 - 17:19

score 1.8k · Answer 5 · 2026-03-25 18:57:51

1.8k

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How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

Published on 25 Mar 2026 - 18:57

score 157 · Answer 6 · 2017-04-14 13:23:06

157

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Prove this $(ab)^2+(bc)^2+(ca)^2\le ab+bc+ac$

Published on 14 Apr 2017 - 13:23

score 127 · Answer 7 · 2017-04-18 04:16:20

127

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Prove this inequality $(\sqrt{a}+\sqrt{b}+\sqrt{c})^2(a+b+c)^3\ge 27(ab+bc+ac)^2$

Published on 18 Apr 2017 - 4:16

score 320 · Answer 8 · 2017-06-06 17:01:25

320

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Prove that $\sum_{cyc}\sqrt{a^2+\frac{1}{\left(b+1\right)^2}}\ge \frac{\sqrt{181}}{5}$

Published on 06 Jun 2017 - 17:01

score 439 · Answer 9 · 2017-06-11 10:02:13

439

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For $a,b,c>0$ . Prove $ (a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 9+8\frac{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}{(a+b+c)^{2}}$

Published on 11 Jun 2017 - 10:02

score 706 · Answer 10 · 2017-06-12 09:34:51

706

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If $a^3+b^3+c^3=3$ so $2(a^2+b^2+c^2)-(ab+bc+ca) \geq 3$

Published on 12 Jun 2017 - 9:34

score 77 · Answer 11 · 2017-06-20 07:02:34

77

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prove that : if $a, b \in \mathbb{R}^+$ : then : $a^2b^2(a^2+b^2-2)\geq (a^2+b^2)(ab-1)$

Published on 20 Jun 2017 - 7:02

score 214 · Answer 12 · 2017-07-07 11:35:35

214

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Find the minimum value of $P=\sum _{cyc}\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}$

Published on 07 Jul 2017 - 11:35

score 147 · Answer 13 · 2017-07-11 11:48:20

147

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Prove inequality $\sum _{cyc}\frac{2ab}{\sqrt{c+3}}\le 3$

Published on 11 Jul 2017 - 11:48

score 184 · Answer 14 · 2017-07-14 06:20:37

184

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show that $a^2+b^2+c^2+2abc+3\ge (1+a)(1+b)(1+c)$

Published on 14 Jul 2017 - 6:20

score 222 · Answer 15 · 2017-07-16 23:25:16

222

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$\sqrt{2(x^2+y^2)}+\sqrt{2(y^2+z^2)}+\sqrt{2(z^2+x^2)}\ge \sqrt[3]{9(x+y)^3+9(y+z)^3+9(z+x)^3}$?

Published on 16 Jul 2017 - 23:25

List Question

Prove $\frac{1}{2a+2bc+1} + \frac{1}{2b+2ca+1} + \frac{1}{2c+2ab+1} \ge 1$

How prove that $xyz+\sqrt{x^2y^2+y^2z^2+x^2z^2}\ge \frac{4}{3}\sqrt{xyz(x+y+z)}$

How prove that $10(a^3+b^3+c^3)-9(a^5+b^5+c^5)\le\dfrac{9}{4}$

How prove this inequality $\frac{2}{(a+b)(4-ab)}+\frac{2}{(b+c)(4-bc)}+\frac{2}{(a+c)(4-ac)}\ge 1$

How prove this inequality $\sum\limits_{cyc}\frac{x+y}{\sqrt{x^2+xy+y^2+yz}}\ge 2+\sqrt{\frac{xy+yz+xz}{x^2+y^2+z^2}}$

Prove this $(ab)^2+(bc)^2+(ca)^2\le ab+bc+ac$

Prove this inequality $(\sqrt{a}+\sqrt{b}+\sqrt{c})^2(a+b+c)^3\ge 27(ab+bc+ac)^2$

Prove that $\sum_{cyc}\sqrt{a^2+\frac{1}{\left(b+1\right)^2}}\ge \frac{\sqrt{181}}{5}$

For $a,b,c>0$ . Prove $ (a+b+c)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 9+8\frac{(a-b)^{2}+(b-c)^{2}+(c-a)^{2}}{(a+b+c)^{2}}$

If $a^3+b^3+c^3=3$ so $2(a^2+b^2+c^2)-(ab+bc+ca) \geq 3$

prove that : if $a, b \in \mathbb{R}^+$ : then : $a^2b^2(a^2+b^2-2)\geq (a^2+b^2)(ab-1)$

Find the minimum value of $P=\sum _{cyc}\frac{\left(x+1\right)^2\left(y+1\right)^2}{z^2+1}$

Prove inequality $\sum _{cyc}\frac{2ab}{\sqrt{c+3}}\le 3$

show that $a^2+b^2+c^2+2abc+3\ge (1+a)(1+b)(1+c)$

$\sqrt{2(x^2+y^2)}+\sqrt{2(y^2+z^2)}+\sqrt{2(z^2+x^2)}\ge \sqrt[3]{9(x+y)^3+9(y+z)^3+9(z+x)^3}$?

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