Question

Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$

score 197 · Answer 1 · 2026-03-25 11:13:19

197

Views

Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$

Published on 25 Mar 2026 - 11:13

score 309 · Answer 2 · 2016-10-11 04:50:26

309

Views

AM-GM inequality in a specific question

Published on 11 Oct 2016 - 4:50

score 374 · Answer 3 · 2026-03-25 11:14:59

374

Views

If $abc=1$ so $\sum\limits_{cyc}\frac{a}{\sqrt{a+b^2}}\geq\frac{3}{\sqrt2}$

Published on 25 Mar 2026 - 11:14

score 131 · Answer 4 · 2026-03-26 01:16:43

131

Views

Prove that $(a+b+c)(ab+bc+ca)\geq9abc$ for $a, b, c \geq0.$

Published on 26 Mar 2026 - 1:16

score 156 · Answer 5 · 2016-10-16 19:48:06

156

Views

Tricky inequality no avail to AM-GM

Published on 16 Oct 2016 - 19:48

score 624 · Answer 6 · 2016-10-21 05:06:31

624

Views

How to find the minimum value of this expression using AM-GM Inequality?

Published on 21 Oct 2016 - 5:06

score 211 · Answer 7 · 2016-10-22 02:18:15

211

Views

Arithmetic Mean/Geometrix Mean Inequality of Degree 3

Published on 22 Oct 2016 - 2:18

score 107 · Answer 8 · 2016-10-23 13:38:45

107

Views

Prove that$(\frac{\sin x}{x})^3\ge \cos x$, for all $0\lt x\le \frac{\pi}{2}$

Published on 23 Oct 2016 - 13:38

score 106 · Answer 9 · 2026-03-25 15:41:14

106

Views

Let $x$, $y$, $z$ be positive. Prove an inequality...

Published on 25 Mar 2026 - 15:41

score 1.3k · Answer 10 · 2026-03-25 15:44:10

1.3k

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Prove that, for all non-negative real numbers $x,y,z$ that satisfy $x+y+z=1, x^2y+y^2z+z^2x≤4/27$

Published on 25 Mar 2026 - 15:44

score 173 · Answer 11 · 2016-11-11 17:21:09

173

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Showing $\sum\limits_{cyc}ab\le\frac14\left(\sum\limits_{cyc}a\right)^2$

Published on 11 Nov 2016 - 17:21

score 113 · Answer 12 · 2026-03-25 22:05:05

113

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If $27 + abc = (a +b +c)(ab + ac+ bc)$ so $\frac{a^2}{a+2b} + \frac{b^2}{b+2c} + \frac{c^2}{c+2a }\geq \frac{3}{2}$

Published on 25 Mar 2026 - 22:05

score 142 · Answer 13 · 2026-03-25 19:10:17

142

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Showing $64\le\left(1+\frac1x\right)\left(1+\frac1y\right)\left(1+\frac1z\right)$

Published on 25 Mar 2026 - 19:10

score 70 · Answer 14 · 2026-03-26 01:12:17

70

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Showing $\frac12 3^{2-p}(x+y+z)^{p-1}\le\frac{x^p}{y+z}+\frac{y^p}{x+z}+\frac{z^p}{y+x}$

Published on 26 Mar 2026 - 1:12

score 122 · Answer 15 · 2016-11-27 03:52:39

122

Views

If $\sum\limits_{i=1}^{2017}\frac{1}{x_i+2017}=\frac{1}{2017}$ for $x_i>0$, then prove that $\sqrt[2017]{x_1x_2...x_{2017}}\ge2016\cdot2017$

Published on 27 Nov 2016 - 3:52

List Question

Prove that $(a+b+c-d)(a+c+d-b)(a+b+d-c)(b+c+d-a)\le(a+b)(a+d)(c+b)(c+d)$

AM-GM inequality in a specific question

If $abc=1$ so $\sum\limits_{cyc}\frac{a}{\sqrt{a+b^2}}\geq\frac{3}{\sqrt2}$

Prove that $(a+b+c)(ab+bc+ca)\geq9abc$ for $a, b, c \geq0.$

Tricky inequality no avail to AM-GM

How to find the minimum value of this expression using AM-GM Inequality?

Arithmetic Mean/Geometrix Mean Inequality of Degree 3

Prove that$(\frac{\sin x}{x})^3\ge \cos x$, for all $0\lt x\le \frac{\pi}{2}$

Let $x$, $y$, $z$ be positive. Prove an inequality...

Prove that, for all non-negative real numbers $x,y,z$ that satisfy $x+y+z=1, x^2y+y^2z+z^2x≤4/27$

Showing $\sum\limits_{cyc}ab\le\frac14\left(\sum\limits_{cyc}a\right)^2$

If $27 + abc = (a +b +c)(ab + ac+ bc)$ so $\frac{a^2}{a+2b} + \frac{b^2}{b+2c} + \frac{c^2}{c+2a }\geq \frac{3}{2}$

Showing $64\le\left(1+\frac1x\right)\left(1+\frac1y\right)\left(1+\frac1z\right)$

Showing $\frac12 3^{2-p}(x+y+z)^{p-1}\le\frac{x^p}{y+z}+\frac{y^p}{x+z}+\frac{z^p}{y+x}$

If $\sum\limits_{i=1}^{2017}\frac{1}{x_i+2017}=\frac{1}{2017}$ for $x_i>0$, then prove that $\sqrt[2017]{x_1x_2...x_{2017}}\ge2016\cdot2017$

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