Question

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

score 237 · Answer 1 · 2026-03-23 03:59:23

237

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

Published on 23 Mar 2026 - 3:59

score 129 · Answer 2 · 2026-03-22 21:30:02

129

Views

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 22 Mar 2026 - 21:30

score 80 · Answer 3 · 2026-03-23 12:39:44

80

Views

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

Published on 23 Mar 2026 - 12:39

score 147 · Answer 4 · 2026-03-23 06:29:57

147

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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 23 Mar 2026 - 6:29

score 149 · Answer 5 · 2023-03-12 20:38:22

149

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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 12 Mar 2023 - 20:38

score 42 · Answer 6 · 2023-06-12 12:26:37

42

Views

Simple Combinatorial Optimization problem, analytical solution.

Published on 12 Jun 2023 - 12:26

score 93 · Answer 7 · 2023-07-04 11:47:23

93

Views

$a,b,c\in\mathbb R^+$, 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 04 Jul 2023 - 11:47

score 83 · Answer 8 · 2023-07-11 12:54:35

83

Views

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

Published on 11 Jul 2023 - 12:54

score 41 · Answer 9 · 2023-07-25 08:15:31

41

Views

Isoperimetric inequality using rearrangements

Published on 25 Jul 2023 - 8:15

score 63 · Answer 10 · 2023-07-30 14:10:22

63

Views

Inequality from Riasat Inequalitus

Published on 30 Jul 2023 - 14:10

score 95 · Answer 11 · 2023-08-07 04:09:46

95

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 07 Aug 2023 - 4:09

score 603 · Answer 12 · 2023-09-28 15:40:59

603

Views

How to prove the equations have only one real solution?

Published on 28 Sep 2023 - 15:40

score 72 · Answer 13 · 2023-11-15 17:43:44

72

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 15 Nov 2023 - 17:43

score 48 · Answer 14 · 2023-11-29 14:02:12

48

Views

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

Published on 29 Nov 2023 - 14:02

score 128 · Answer 15 · 2023-12-29 11:45:11

128

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 29 Dec 2023 - 11:45

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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