Question

$a+b+c=1$, prove $(1+a)(1+b)(1+c) \ge 8(1-a)(1-b)(1-c) $, without AM-GM

score 146 · Answer 1 · 2022-07-12 10:03:13

$a+b+c=1$, prove $(1+a)(1+b)(1+c) \ge 8(1-a)(1-b)(1-c) $, without AM-GM

Published on 12 Jul 2022 - 10:03

score 256 · Answer 2 · 2022-07-27 16:14:29

If a, b, c are positive real numbers such that $a^2+ b^2+ c^2 = 1$ $( \frac{1}{a} +\frac{1}{b} +\frac{1}{c}) +a +b +c \geq 4\sqrt{3}$

score 86 · Answer 3 · 2026-03-28 00:56:16

86

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Geometric-arithmetic mean inequality applied to eigenvalues

Published on 28 Mar 2026 - 0:56

score 313 · Answer 4 · 2022-08-11 08:43:43

313

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Finding the minimum value of the expression $xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$

Published on 11 Aug 2022 - 8:43

score 104 · Answer 5 · 2022-08-15 02:42:38

104

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Prove that: $\sum_{1\le j\le n}{\frac{x_j}{\sum_{1\le i\le n} x_i-x_j}}\ge\frac{n}{n-1}$

Published on 15 Aug 2022 - 2:42

score 114 · Answer 6 · 2022-08-20 15:23:31

114

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min of $c$ in such that $\frac{\sum^n_{i=1}a_i}{n\sqrt[n]{\prod^n_{i=1}a_i}}\leq \left(\frac{\sqrt{a/b}+\sqrt{b/a}}2\right)^c$ for all $a_i\in [a,b]$.

Published on 20 Aug 2022 - 15:23

score 118 · Answer 7 · 2022-08-22 00:25:37

118

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Proving $30x + 3y^2 + \frac{2z^3}{9} + 36 (\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx}) \ge 84$

Published on 22 Aug 2022 - 0:25

score 78 · Answer 8 · 2022-08-26 18:08:18

78

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I can't come up with the intended solution to $\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab>18$

Published on 26 Aug 2022 - 18:08

score 79 · Answer 9 · 2026-03-14 15:50:53

79

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Show an inequality in three variables

Published on 14 Mar 2026 - 15:50

score 169 · Answer 10 · 2022-09-04 14:54:35

169

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Find the minimum of $xy+yz+zx+\frac1x+\frac2y+\frac5z$ for $x, y, z > 0$

Published on 04 Sep 2022 - 14:54

score 131 · Answer 11 · 2022-09-08 16:53:30

131

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Possible flaw in AM GM inequality

Published on 08 Sep 2022 - 16:53

score 28 · Answer 12 · 2026-03-28 02:23:47

28

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$ab+bc+ca \le 4{\sqrt 3}\Delta$ for a triangle with sides $a$, $b$ and $c$ with area $\Delta$

Published on 28 Mar 2026 - 2:23

score 289 · Answer 13 · 2022-10-16 02:09:47

289

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Prove: $\sum\frac{\sqrt{a+3}}{a+\sqrt{bc}}\ge\frac{2(\sqrt{a}+\sqrt{b}+\sqrt{c})}{\sqrt{a+b+c+1}}$

Published on 16 Oct 2022 - 2:09

score 87 · Answer 14 · 2022-10-17 13:03:33

Let $x,y,z\in\mathbb{R}^{+}$ such that $x+y+z = 1.$ T/F: $\frac{x^2 y^2}{(x+y)^2}+\frac{y^2 z^2}{(y+z)^2}+\frac{z^2 x^2}{(z+x)^2}\geq\frac{9}{4}xyz$

score 113 · Answer 15 · 2022-10-23 08:57:58

113

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Prove inequality for positive real numbers $x,y,z$

Published on 23 Oct 2022 - 8:57

List Question

$a+b+c=1$, prove $(1+a)(1+b)(1+c) \ge 8(1-a)(1-b)(1-c) $, without AM-GM

If a, b, c are positive real numbers such that $a^2+ b^2+ c^2 = 1$ $( \frac{1}{a} +\frac{1}{b} +\frac{1}{c}) +a +b +c \geq 4\sqrt{3}$

Geometric-arithmetic mean inequality applied to eigenvalues

Finding the minimum value of the expression $xy(x+y)^2 + yz(y+z)^2 + zx(z+x)^2$

Prove that: $\sum_{1\le j\le n}{\frac{x_j}{\sum_{1\le i\le n} x_i-x_j}}\ge\frac{n}{n-1}$

min of $c$ in such that $\frac{\sum^n_{i=1}a_i}{n\sqrt[n]{\prod^n_{i=1}a_i}}\leq \left(\frac{\sqrt{a/b}+\sqrt{b/a}}2\right)^c$ for all $a_i\in [a,b]$.

Proving $30x + 3y^2 + \frac{2z^3}{9} + 36 (\frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx}) \ge 84$

I can't come up with the intended solution to $\frac{a^2+b^2}{2a^5b^5} + \frac{81a^2b^2}{4} + 9ab>18$

Show an inequality in three variables

Find the minimum of $xy+yz+zx+\frac1x+\frac2y+\frac5z$ for $x, y, z > 0$

Possible flaw in AM GM inequality

$ab+bc+ca \le 4{\sqrt 3}\Delta$ for a triangle with sides $a$, $b$ and $c$ with area $\Delta$

Prove: $\sum\frac{\sqrt{a+3}}{a+\sqrt{bc}}\ge\frac{2(\sqrt{a}+\sqrt{b}+\sqrt{c})}{\sqrt{a+b+c+1}}$

Let $x,y,z\in\mathbb{R}^{+}$ such that $x+y+z = 1.$ T/F: $\frac{x^2 y^2}{(x+y)^2}+\frac{y^2 z^2}{(y+z)^2}+\frac{z^2 x^2}{(z+x)^2}\geq\frac{9}{4}xyz$

Prove inequality for positive real numbers $x,y,z$

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