Question

How prove this inequality with $a+b+c+2=abc$

score 238 · Answer 1 · 2026-03-25 17:19:24

238

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How prove this inequality with $a+b+c+2=abc$

Published on 25 Mar 2026 - 17:19

score 130 · Answer 2 · 2026-03-25 17:26:56

130

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For $a, b, c$ is the length of three sides of a triangle. Prove that $\left|\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}\right|<\frac{1}{8}$

Published on 25 Mar 2026 - 17:26

score 81 · Answer 3 · 2026-03-25 17:21:49

81

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Minimize $S=\dfrac{x}{y^2}+\dfrac{y}{z^2}+\dfrac{z}{x^2}$ when $x+y+z=xyz$.

Published on 25 Mar 2026 - 17:21

score 148 · Answer 4 · 2026-03-25 17:19:44

148

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Let $a,b,c$ be positive numbers. Prove that $a^2b+b^2c+c^2a\leq a^3+b^3+c^3$

Published on 25 Mar 2026 - 17:19

score 150 · Answer 5 · 2026-03-25 17:21:47

150

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Show that $\forall (a, b, c) \ \in \mathbf{R}: \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \ge \frac{b}{a} + \frac{c}{b} + \frac{a}{c}$

Published on 25 Mar 2026 - 17:21

score 43 · Answer 6 · 2026-03-25 17:17:05

43

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Simple Combinatorial Optimization problem, analytical solution.

Published on 25 Mar 2026 - 17:17

score 94 · Answer 7 · 2026-03-25 17:21:45

94

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$a,b,c\in\mathbb R^+$, prove $\frac{a^3}{bc} +\frac{b^3}{ca} + \frac{c^3}{ab} \geq \frac{a^2 + b^2}{2c} + \frac{b^2 + c^2}{2a} + \frac{c^2 + a^2}{2b}$

Published on 25 Mar 2026 - 17:21

score 84 · Answer 8 · 2026-03-25 17:22:53

84

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Prove that $(\Sigma a^2)^5\ge27~(\Sigma a^3b^2)^2$

Published on 25 Mar 2026 - 17:22

score 42 · Answer 9 · 2026-03-25 17:18:41

42

Views

Isoperimetric inequality using rearrangements

Published on 25 Mar 2026 - 17:18

score 64 · Answer 10 · 2026-03-25 18:53:38

64

Views

Inequality from Riasat Inequalitus

Published on 25 Mar 2026 - 18:53

score 96 · Answer 11 · 2026-03-25 17:21:45

96

Views

Maximize $P=xy^2+yz^2+zx^2+xyz$ if $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$

Published on 25 Mar 2026 - 17:21

score 604 · Answer 12 · 2026-03-25 17:26:15

604

Views

How to prove the equations have only one real solution?

Published on 25 Mar 2026 - 17:26

score 73 · Answer 13 · 2026-03-25 17:26:54

73

Views

Showing $\sum_{i=1}^n\tan\alpha_i\geq (n-1)\cdot \sum_{i=1}^n\cot\alpha_i$, for real $\alpha_i\in(0,\pi/2)$ with $\sum_{i=1}^n\cos^2\alpha_i=1$

Published on 25 Mar 2026 - 17:26

score 49 · Answer 14 · 2026-03-25 17:21:32

49

Views

$\frac{1}{9-ab}+\frac{1}{9-bc}+\frac{1}{9-ca} \leq \frac{3}{8}.$

Published on 25 Mar 2026 - 17:21

score 129 · Answer 15 · 2026-03-25 17:26:51

129

Views

$x, y, z > 0, xyz = 1 => xy^2 + yz^2 + zx^2 \geq x+y+z$ with the help of the rearrangement inequality

Published on 25 Mar 2026 - 17:26

List Question

How prove this inequality with $a+b+c+2=abc$

For $a, b, c$ is the length of three sides of a triangle. Prove that $\left|\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}\right|<\frac{1}{8}$

Minimize $S=\dfrac{x}{y^2}+\dfrac{y}{z^2}+\dfrac{z}{x^2}$ when $x+y+z=xyz$.

Let $a,b,c$ be positive numbers. Prove that $a^2b+b^2c+c^2a\leq a^3+b^3+c^3$

Show that $\forall (a, b, c) \ \in \mathbf{R}: \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} \ge \frac{b}{a} + \frac{c}{b} + \frac{a}{c}$

Simple Combinatorial Optimization problem, analytical solution.

$a,b,c\in\mathbb R^+$, prove $\frac{a^3}{bc} +\frac{b^3}{ca} + \frac{c^3}{ab} \geq \frac{a^2 + b^2}{2c} + \frac{b^2 + c^2}{2a} + \frac{c^2 + a^2}{2b}$

Prove that $(\Sigma a^2)^5\ge27~(\Sigma a^3b^2)^2$

Isoperimetric inequality using rearrangements

Inequality from Riasat Inequalitus

Maximize $P=xy^2+yz^2+zx^2+xyz$ if $(x^2+y^2)(y^2+z^2)(z^2+x^2)=2$

How to prove the equations have only one real solution?

Showing $\sum_{i=1}^n\tan\alpha_i\geq (n-1)\cdot \sum_{i=1}^n\cot\alpha_i$, for real $\alpha_i\in(0,\pi/2)$ with $\sum_{i=1}^n\cos^2\alpha_i=1$

$\frac{1}{9-ab}+\frac{1}{9-bc}+\frac{1}{9-ca} \leq \frac{3}{8}.$

$x, y, z > 0, xyz = 1 => xy^2 + yz^2 + zx^2 \geq x+y+z$ with the help of the rearrangement inequality

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