Question

Prove that $a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$

score 211 · Answer 1 · 2026-03-25 09:28:16

211

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Prove that $a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$

Published on 25 Mar 2026 - 9:28

score 108 · Answer 2 · 2019-03-19 04:31:22

108

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Given $a+b+c+d=4$, find the minimum value of $\Sigma_{cyc}\frac{a}{b^3+4}$

Published on 19 Mar 2019 - 4:31

score 295 · Answer 3 · 2019-03-20 15:34:08

295

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Difficult inequality with 4 variables

Published on 20 Mar 2019 - 15:34

score 118 · Answer 4 · 2026-03-26 10:45:13

118

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About the fact that $\frac{a^2}{b} + \frac{b^2}{a} + 7(a + b) \ge 8\sqrt{2(a^2 + b^2)}$

Published on 26 Mar 2026 - 10:45

score 283 · Answer 5 · 2026-03-26 10:45:12

283

Views

Prove that $\frac{1}{ab}+\frac{1}{cd} \geq \frac{a^2+b^2+c^2+d^2}{2}$

Published on 26 Mar 2026 - 10:45

score 72 · Answer 6 · 2019-03-26 16:22:24

72

Views

AM-GM inequality for functions

Published on 26 Mar 2019 - 16:22

score 113 · Answer 7 · 2026-03-26 12:08:25

113

Views

Prove that $\frac{2-2a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2} \leq \frac{9}{4}$

Published on 26 Mar 2026 - 12:08

score 296 · Answer 8 · 2019-03-30 15:03:14

296

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What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$

Published on 30 Mar 2019 - 15:03

score 365 · Answer 9 · 2026-03-25 06:30:21

365

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show this inequality to $\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $

Published on 25 Mar 2026 - 6:30

score 223 · Answer 10 · 2026-03-25 15:57:53

223

Views

A nice Nesbitt inequality from a strange inequality

Published on 25 Mar 2026 - 15:57

score 891 · Answer 11 · 2019-04-06 13:52:18

891

Views

Upper bound estimate $ab^3\leq c(a^2+b^2)^2$. Find optimal, or good, constants $c$

Published on 06 Apr 2019 - 13:52

score 181 · Answer 12 · 2019-04-08 14:02:48

181

Views

How to correctly find the value of theta for which $\frac{a}{\cos\theta}+\frac{b}{\sin\theta}$ is minimum?

Published on 08 Apr 2019 - 14:02

score 126 · Answer 13 · 2019-04-09 00:52:31

126

Views

prove this inequality with $x^3+y^3+z^3$

Published on 09 Apr 2019 - 0:52

score 217 · Answer 14 · 2026-03-26 13:44:15

217

Views

Inequality related with $abcd=(1-a)(1-b)(1-c)(1-d)$

Published on 26 Mar 2026 - 13:44

score 311 · Answer 15 · 2019-04-10 03:04:03

311

Views

Area of a rectangle inside a triangle with given coordinates

Published on 10 Apr 2019 - 3:04

List Question

Prove that $a\sqrt{b^3+1}+b\sqrt{c^3+1}+c\sqrt{a^3+1} \leq 5$

Given $a+b+c+d=4$, find the minimum value of $\Sigma_{cyc}\frac{a}{b^3+4}$

Difficult inequality with 4 variables

About the fact that $\frac{a^2}{b} + \frac{b^2}{a} + 7(a + b) \ge 8\sqrt{2(a^2 + b^2)}$

Prove that $\frac{1}{ab}+\frac{1}{cd} \geq \frac{a^2+b^2+c^2+d^2}{2}$

AM-GM inequality for functions

Prove that $\frac{2-2a^2}{1+a^2}+\frac{1-b^2}{1+b^2}+\frac{1-c^2}{1+c^2} \leq \frac{9}{4}$

What is wrong with this solution of find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$

show this inequality to $\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $

A nice Nesbitt inequality from a strange inequality

Upper bound estimate $ab^3\leq c(a^2+b^2)^2$. Find optimal, or good, constants $c$

How to correctly find the value of theta for which $\frac{a}{\cos\theta}+\frac{b}{\sin\theta}$ is minimum?

prove this inequality with $x^3+y^3+z^3$

Inequality related with $abcd=(1-a)(1-b)(1-c)(1-d)$

Area of a rectangle inside a triangle with given coordinates

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