Question

If AM-GM fails for some list of $n_0\ge 2$ numbers, then for all $n\ge n_0$, there's a list of $n$ numbers such that AM-GM fails

score 67 · Answer 1 · 2026-03-29 07:40:07

67

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If AM-GM fails for some list of $n_0\ge 2$ numbers, then for all $n\ge n_0$, there's a list of $n$ numbers such that AM-GM fails

Published on 29 Mar 2026 - 7:40

score 130 · Answer 2 · 2026-03-26 06:08:10

130

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Show that $\sqrt[n]{x_1x_2\dots x_n}+\sqrt[n]{(1-x_1)(1-x_2)\dots(1-x_n)}\leq 1$

Published on 26 Mar 2026 - 6:08

score 84 · Answer 3 · 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 94 · Answer 4 · 2023-07-12 16:02:11

94

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Using AM-GM inequality to find the minima of a function

Published on 12 Jul 2023 - 16:02

score 162 · Answer 5 · 2023-07-13 00:04:47

162

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Failure of AM, GM inequality while trying find the minimum value of a function.

Published on 13 Jul 2023 - 0:04

score 158 · Answer 6 · 2023-07-19 09:55:01

158

Views

Prove that $\sum\limits_{cyc}{\frac{1}{2(a-b)^2+ab}} \geq \frac{16(ab+bc+ca)^3(abc-1)}{\sqrt{a}+\sqrt{b}+\sqrt{c}}+3$

Published on 19 Jul 2023 - 9:55

score 146 · Answer 7 · 2026-03-26 06:05:38

146

Views

Find the maximum of the expression, given that $a+b+c+d=100$

Published on 26 Mar 2026 - 6:05

score 212 · Answer 8 · 2023-07-21 16:21:04

212

Views

Prove $\frac{1}{\sqrt{5a+4}}+\frac{1}{\sqrt{5b+4}}+\frac{1}{\sqrt{5c+4}}\geq1$ for $ab + bc + ca = 3$

Published on 21 Jul 2023 - 16:21

score 123 · Answer 9 · 2026-03-26 06:04:15

123

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Find the image of function $f$ (without calculus)

Published on 26 Mar 2026 - 6:04

score 72 · Answer 10 · 2023-07-30 13:42:34

72

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For positive real $a,b$ Prove that $\frac{a^3b}{(a+b)^4}\le \frac{27}{256}$

Published on 30 Jul 2023 - 13:42

score 64 · Answer 11 · 2026-03-25 18:53:38

64

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Inequality from Riasat Inequalitus

Published on 25 Mar 2026 - 18:53

score 252 · Answer 12 · 2026-03-25 14:36:44

252

Views

If $a,b,c\ge0:a+b+c=3,$ prove $\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$

Published on 25 Mar 2026 - 14:36

score 210 · Answer 13 · 2023-07-31 13:11:29

210

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If $a,b,c>0$ and $abc≥1$, prove that $ \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Published on 31 Jul 2023 - 13:11

score 86 · Answer 14 · 2023-08-01 06:48:55

86

Views

Inequality from Riasat book

Published on 01 Aug 2023 - 6:48

score 74 · Answer 15 · 2023-08-01 15:54:18

74

Views

Inequality $\frac{a^3}{bc} + \frac{b^3}{ac} + \frac{c^3}{ab} + \frac{1}{abc} \ge min$ with $a+b+c = \frac{3}{2}$

Published on 01 Aug 2023 - 15:54

List Question

If AM-GM fails for some list of $n_0\ge 2$ numbers, then for all $n\ge n_0$, there's a list of $n$ numbers such that AM-GM fails

Show that $\sqrt[n]{x_1x_2\dots x_n}+\sqrt[n]{(1-x_1)(1-x_2)\dots(1-x_n)}\leq 1$

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

Using AM-GM inequality to find the minima of a function

Failure of AM, GM inequality while trying find the minimum value of a function.

Prove that $\sum\limits_{cyc}{\frac{1}{2(a-b)^2+ab}} \geq \frac{16(ab+bc+ca)^3(abc-1)}{\sqrt{a}+\sqrt{b}+\sqrt{c}}+3$

Find the maximum of the expression, given that $a+b+c+d=100$

Prove $\frac{1}{\sqrt{5a+4}}+\frac{1}{\sqrt{5b+4}}+\frac{1}{\sqrt{5c+4}}\geq1$ for $ab + bc + ca = 3$

Find the image of function $f$ (without calculus)

For positive real $a,b$ Prove that $\frac{a^3b}{(a+b)^4}\le \frac{27}{256}$

Inequality from Riasat Inequalitus

If $a,b,c\ge0:a+b+c=3,$ prove $\sqrt{a+b+b^2}+\sqrt{b+c+c^2}+\sqrt{c+a+a^2}\ge 3\sqrt{3}.$

If $a,b,c>0$ and $abc≥1$, prove that $ \frac{a^2}{b} + \frac{b^2}{c} + \frac{c^2}{a} \geq \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Inequality from Riasat book

Inequality $\frac{a^3}{bc} + \frac{b^3}{ac} + \frac{c^3}{ab} + \frac{1}{abc} \ge min$ with $a+b+c = \frac{3}{2}$

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