Question

Does the opposite inequality of A.M ≥ G.M holds?

score 69 · Answer 1 · 2019-04-10 17:31:34

69

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Does the opposite inequality of A.M ≥ G.M holds?

Published on 10 Apr 2019 - 17:31

score 74 · Answer 2 · 2019-04-11 23:28:09

74

Views

A tough inequality with a constraint?

Published on 11 Apr 2019 - 23:28

score 122 · Answer 3 · 2019-04-15 02:46:03

122

Views

Proving $\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca}$ for positive $a$, $b$, $c$

Published on 15 Apr 2019 - 2:46

score 142 · Answer 4 · 2026-03-25 09:32:00

142

Views

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Published on 25 Mar 2026 - 9:32

score 190 · Answer 5 · 2019-04-17 06:45:29

190

Views

Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$

Published on 17 Apr 2019 - 6:45

score 78 · Answer 6 · 2026-03-26 12:46:23

78

Views

Proving the AM-GM Inequality with a given fact

Published on 26 Mar 2026 - 12:46

score 295 · Answer 7 · 2026-03-25 06:06:19

295

Views

Given three non-negatve numbers $a,b,c$. Prove that $1+a^{2}+b^{2}+c^{2}+4abc\geqq a+b+c+ab+bc+ca$ .

Published on 25 Mar 2026 - 6:06

score 117 · Answer 8 · 2019-04-23 16:19:10

117

Views

How to show that $ab+bc+ca\le \frac34$

Published on 23 Apr 2019 - 16:19

score 177 · Answer 9 · 2026-03-24 23:46:38

177

Views

Given three non-negative numbers $a,b,c$ so that $a+b+c=3$. Prove that $\prod\limits_{cyc}(\!2+ a^{2}\!)+abc\geqq 28$ .

Published on 24 Mar 2026 - 23:46

score 103 · Answer 10 · 2019-04-29 05:37:16

103

Views

Solve the equation $\frac{7x^2 - x + 4}{\sqrt{3x^2 - 1} + \sqrt{x^2 - x} - x\sqrt{x^2 + 1}} = 2\sqrt 2$.

Published on 29 Apr 2019 - 5:37

score 85 · Answer 11 · 2019-04-30 08:04:52

85

Views

Minima and maxima of $ab+\frac{1}{ab}$

Published on 30 Apr 2019 - 8:04

score 36 · Answer 12 · 2026-03-26 12:42:30

36

Views

AM-GM-HM inequality problem

Published on 26 Mar 2026 - 12:42

score 46 · Answer 13 · 2019-05-06 13:35:05

46

Views

Show that for reals $>0$ $a_1,a_2,...,a_n$ : $a_1a_2...a_n < 1 \iff$ $\exists k \in \mathbb{N}$ $\geq 2$ such that $\sum_{j=1}^n \sqrt[k]{a_i} < n$

Published on 06 May 2019 - 13:35

score 61 · Answer 14 · 2019-05-11 09:29:53

61

Views

Inequalities in mathematics

Published on 11 May 2019 - 9:29

score 219 · Answer 15 · 2026-03-25 06:13:08

219

Views

Prove that $k=0$ is the only $constant$ so $(\sum\limits_{cyc}a^{3}-\sum\limits_{cyc}a^{2}b)-k(a-b)(a-c)(b+c)\geqq0$ .

Published on 25 Mar 2026 - 6:13

List Question

Does the opposite inequality of A.M ≥ G.M holds?

A tough inequality with a constraint?

Proving $\frac{1}{6}a+\frac{1}{3}b+\frac{1}{2}c \geq \frac{6abc}{3ab+bc+2ca}$ for positive $a$, $b$, $c$

Find the minimum and maximum values of $P=\left( 6-a^2-b^2-c^2\right)\left(2-abc\right)$

Prove the inequality $\frac{1}{1+x^2}+\frac{1}{1+y^2}+\frac{1}{1+xy} \geq \frac{3}{1+\frac{(x+y)^2}{4}}$ when $x^2+y^2=1$

Proving the AM-GM Inequality with a given fact

Given three non-negatve numbers $a,b,c$. Prove that $1+a^{2}+b^{2}+c^{2}+4abc\geqq a+b+c+ab+bc+ca$ .

How to show that $ab+bc+ca\le \frac34$

Given three non-negative numbers $a,b,c$ so that $a+b+c=3$. Prove that $\prod\limits_{cyc}(\!2+ a^{2}\!)+abc\geqq 28$ .

Solve the equation $\frac{7x^2 - x + 4}{\sqrt{3x^2 - 1} + \sqrt{x^2 - x} - x\sqrt{x^2 + 1}} = 2\sqrt 2$.

Minima and maxima of $ab+\frac{1}{ab}$

AM-GM-HM inequality problem

Show that for reals $>0$ $a_1,a_2,...,a_n$ : $a_1a_2...a_n < 1 \iff$ $\exists k \in \mathbb{N}$ $\geq 2$ such that $\sum_{j=1}^n \sqrt[k]{a_i} < n$

Inequalities in mathematics

Prove that $k=0$ is the only $constant$ so $(\sum\limits_{cyc}a^{3}-\sum\limits_{cyc}a^{2}b)-k(a-b)(a-c)(b+c)\geqq0$ .

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