Question

Prove the inequality $a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$ for positive $a, b, c$

score 149 · Answer 1 · 2017-10-04 12:28:42

149

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Prove the inequality $a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$ for positive $a, b, c$

Published on 04 Oct 2017 - 12:28

score 213 · Answer 2 · 2017-10-08 13:49:23

213

Views

Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Published on 08 Oct 2017 - 13:49

score 114 · Answer 3 · 2017-10-10 14:06:35

114

Views

Prove that $ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy} \geq \frac{1}{x} + \frac{1}{y} + \frac{1}{z} $

Published on 10 Oct 2017 - 14:06

score 3.6k · Answer 4 · 2017-10-15 19:42:36

3.6k

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Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

Published on 15 Oct 2017 - 19:42

score 87 · Answer 5 · 2017-10-22 07:21:28

When does $\frac{1}{2n}\left(\sum_{i=1}^{n}x^2_i + \sum_{i=1}^{n}x_ix_{n-i+1}\right) \leq \left(\frac{1}{n}\sum_{i=1}^{n}x_i\right)^2$?

score 86 · Answer 6 · 2017-11-02 14:11:39

86

Views

$\Big(\sum\limits_{i=1}^n \frac{1}{x_{i}^2} \Big) \Big(\sum\limits_{i=1}^n \frac{1}{x_{i}^{1/2}}\Big)^{2}$

Published on 02 Nov 2017 - 14:11

score 153 · Answer 7 · 2026-03-25 13:57:10

153

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Find the minimum of expression: $\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$

Published on 25 Mar 2026 - 13:57

score 428 · Answer 8 · 2018-01-01 04:10:46

428

Views

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

Published on 01 Jan 2018 - 4:10

score 1.9k · Answer 9 · 2026-03-23 06:37:37

1.9k

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Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$

Published on 23 Mar 2026 - 6:37

score 1.8k · Answer 10 · 2020-08-01 23:19:08

1.8k

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If $a^2 + b^2 + c^2 = 1$, what is the the minimum value of $\frac {ab}{c} + \frac {bc}{a} + \frac {ca}{b}$?

Published on 01 Aug 2020 - 23:19

score 71 · Answer 11 · 2026-03-25 15:45:39

71

Views

Is this Factorization?

Published on 25 Mar 2026 - 15:45

score 150 · Answer 12 · 2020-08-23 09:55:00

150

Views

Rearrangement Inequality- $\sum_{c y c} \frac{a^{2}+b c}{b+c} \geq a+b+c$

Published on 23 Aug 2020 - 9:55

score 95 · Answer 13 · 2020-08-25 18:45:22

95

Views

prove $\sum_{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$

Published on 25 Aug 2020 - 18:45

score 220 · Answer 14 · 2020-08-26 05:53:08

220

Views

prove $\sum_{i=1}^{n}\sqrt{a_i}\ge (n-1)\sum_{i=0}^{n}\frac{1}{\sqrt{a_i}}$

Published on 26 Aug 2020 - 5:53

score 68 · Answer 15 · 2020-08-28 06:32:43

68

Views

Doubt:regarding proof $\sum_{cyc}a^{2/3}b\le 3$

Published on 28 Aug 2020 - 6:32

List Question

Prove the inequality $a\frac{a+b}{a+c} + b\frac{b+c}{a+b} + c\frac{c+a}{b+c} \geq a+b+c$ for positive $a, b, c$

Prove $\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x-1}{y-1}+\frac{y-1}{z-1}+\frac{z-1}{x-1}$

Prove that $ \frac{x}{yz} + \frac{y}{xz} + \frac{z}{xy} \geq \frac{1}{x} + \frac{1}{y} + \frac{1}{z} $

Proving that $\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}> \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$

When does $\frac{1}{2n}\left(\sum_{i=1}^{n}x^2_i + \sum_{i=1}^{n}x_ix_{n-i+1}\right) \leq \left(\frac{1}{n}\sum_{i=1}^{n}x_i\right)^2$?

$\Big(\sum\limits_{i=1}^n \frac{1}{x_{i}^2} \Big) \Big(\sum\limits_{i=1}^n \frac{1}{x_{i}^{1/2}}\Big)^{2}$

Find the minimum of expression: $\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$

Find minimum value of $\sum \frac{\sqrt{a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}$

Prove the inequality $a^2bc+b^2cd+c^2da+d^2ab \leq 4$ with $a+b+c+d=4$

If $a^2 + b^2 + c^2 = 1$, what is the the minimum value of $\frac {ab}{c} + \frac {bc}{a} + \frac {ca}{b}$?

Is this Factorization?

Rearrangement Inequality- $\sum_{c y c} \frac{a^{2}+b c}{b+c} \geq a+b+c$

prove $\sum_{cyc}\frac{a^3}{b}\ge ab+bc+ca$ if $a,b,c>0$

prove $\sum_{i=1}^{n}\sqrt{a_i}\ge (n-1)\sum_{i=0}^{n}\frac{1}{\sqrt{a_i}}$

Doubt:regarding proof $\sum_{cyc}a^{2/3}b\le 3$

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