Question

Find the minimum value of $P=\left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)$

score 151 · Answer 1 · 2026-03-25 23:42:17

151

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Find the minimum value of $P=\left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)$

Published on 25 Mar 2026 - 23:42

score 81 · Answer 2 · 2017-03-28 00:37:20

81

Views

complex AM GM inequality, probably based on weighted means

Published on 28 Mar 2017 - 0:37

score 202 · Answer 3 · 2017-04-03 15:17:05

202

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How prove this inequality $\sum\limits_{cyc}\sqrt{\frac{yz}{x^2+2016}}\le\frac{3}{2}$

Published on 03 Apr 2017 - 15:17

score 92 · Answer 4 · 2017-04-05 18:54:40

92

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Finding Maximum value of consecutive positive integers with constraints

Published on 05 Apr 2017 - 18:54

score 132 · Answer 5 · 2017-04-06 03:19:51

132

Views

How to prove $\sqrt{x(3x+y)} + \sqrt{y(3y+z)} +\sqrt{z(3z+x)} \le 2(x+y+z)$

Published on 06 Apr 2017 - 3:19

score 3.2k · Answer 6 · 2026-03-25 22:05:02

3.2k

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Prove that $\frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2} \text{ when } abc=1$

Published on 25 Mar 2026 - 22:05

score 223 · Answer 7 · 2023-02-01 22:19:19

223

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Prove or disprove that the inequality is valid if $x,y,z,u$ are positive numbers and $x+y+z+u=2$.

Published on 01 Feb 2023 - 22:19

score 186 · Answer 8 · 2023-02-03 17:56:43

186

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Find the maximum value of $ab+bc+cd+de+ef+fa,$ given that $a+b+c+d+e+f=1$

Published on 03 Feb 2023 - 17:56

score 54 · Answer 9 · 2023-02-10 17:05:36

If $a$, $b$, and $c$ are positive numbers, then $\sqrt{a^2+ab+b^2}+\sqrt{a^2+ac+c^2}+\sqrt{b^2+bc +c^2}\ge\sqrt{3}(\sqrt{ab}+\sqrt{ac}+\sqrt{bc}).$

score 111 · Answer 10 · 2023-02-24 20:40:12

111

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Proving $4(x^4+y^4+z^4)\geq \sum_{cyc}xy(x+y)^2\geq 4xyz(x+y+z)$, for $x,y,z\geq0$

Published on 24 Feb 2023 - 20:40

score 36 · Answer 11 · 2026-02-23 06:25:52

36

Views

Is there any simple way of proving triangle inequality for $d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}, \,\,x,y\in \mathbb{R}$

Published on 23 Feb 2026 - 6:25

score 85 · Answer 12 · 2023-03-10 14:38:02

85

Views

Minimum value of f(θ)= a²sec²θ +b² cosec²θ using AM- GM inequality

Published on 10 Mar 2023 - 14:38

score 84 · Answer 13 · 2023-03-15 18:54:01

84

Views

Inequality proof with AM-GM

Published on 15 Mar 2023 - 18:54

score 37 · Answer 14 · 2023-03-16 18:26:24

37

Views

Another inequality related to (weighted) AM-GM ...

Published on 16 Mar 2023 - 18:26

score 185 · Answer 15 · 2026-03-26 04:34:54

185

Views

Finding non-trivial real numbers which satisfy all three of: $\sum_{i=1}^{n}a_i=0,\sum_{i=1}^{n}{a_i}^3=0$ and $\sum_{i=1}^{n}\lvert a_i\rvert=1.$

Published on 26 Mar 2026 - 4:34

List Question

Find the minimum value of $P=\left(3+\frac{1}{a}+\frac{1}{b}\right)\left(3+\frac{1}{b}+\frac{1}{c}\right)\left(3+\frac{1}{c}+\frac{1}{a}\right)$

complex AM GM inequality, probably based on weighted means

How prove this inequality $\sum\limits_{cyc}\sqrt{\frac{yz}{x^2+2016}}\le\frac{3}{2}$

Finding Maximum value of consecutive positive integers with constraints

How to prove $\sqrt{x(3x+y)} + \sqrt{y(3y+z)} +\sqrt{z(3z+x)} \le 2(x+y+z)$

Prove that $\frac{1}{a^3(b+c)} + \frac{1}{b^3(a+c)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2} \text{ when } abc=1$

Prove or disprove that the inequality is valid if $x,y,z,u$ are positive numbers and $x+y+z+u=2$.

Find the maximum value of $ab+bc+cd+de+ef+fa,$ given that $a+b+c+d+e+f=1$

If $a$, $b$, and $c$ are positive numbers, then $\sqrt{a^2+ab+b^2}+\sqrt{a^2+ac+c^2}+\sqrt{b^2+bc +c^2}\ge\sqrt{3}(\sqrt{ab}+\sqrt{ac}+\sqrt{bc}).$

Proving $4(x^4+y^4+z^4)\geq \sum_{cyc}xy(x+y)^2\geq 4xyz(x+y+z)$, for $x,y,z\geq0$

Is there any simple way of proving triangle inequality for $d(x,y)= \dfrac{2|x-y|}{\sqrt{1+x^2}+\sqrt{1+y^2}}, \,\,x,y\in \mathbb{R}$

Minimum value of f(θ)= a²sec²θ +b² cosec²θ using AM- GM inequality

Inequality proof with AM-GM

Another inequality related to (weighted) AM-GM ...

Finding non-trivial real numbers which satisfy all three of: $\sum_{i=1}^{n}a_i=0,\sum_{i=1}^{n}{a_i}^3=0$ and $\sum_{i=1}^{n}\lvert a_i\rvert=1.$

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