List questions from a-m-g-m-inequality

Published on 25 Mar 2026 - 6:04

57

Views

Prove $\sum \sqrt{3a+\frac{1}{b}} \geq 6$

Published on 23 Feb 2026 - 4:21

110

Views

$AM-GM$ for $(a + \frac{1}{a})^2 + (b + \frac{1}{b})^2 \geqslant \frac{25}{2}$

Published on 17 May 2020 - 19:22

138

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Prove $(a+b+c)^3 (a+b-c)(b+c-a)(c+a-b) \leqq 27a^2 b^ 2 c^2$

Published on 25 Mar 2026 - 16:02

199

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Prove $\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geqq 1$

Published on 25 Mar 2026 - 14:25

153

Views

Olympiad Inequality Question

Published on 22 May 2020 - 8:09

82

Views

Using AM-GM to show that if $\{a_i\},\{f_i\}$ are positive sequences s.t. $\sum a_i=\infty$ and $f_i\to f>0$, then $(\sum f_ia_i)/(\sum a_i)\to f$

Published on 23 May 2020 - 2:06

50

Views

Solve extremal problem with two constraints using AM-GM

Published on 24 May 2020 - 12:15

30

Views

Maximize area of isosceles triangle with given median

Published on 24 May 2020 - 23:10

70

Views

Refinement inequality of : $\sqrt{x}+x^{\frac{x}{x+1}}\geq x+1$

Published on 25 May 2020 - 14:25

151

Views

A strange inequality

Published on 25 May 2020 - 16:31

193

Views

Prove that $\left(\dfrac{b}{a}+\dfrac{d}{c}\right)\cdot\left(\dfrac{a}{b}+\dfrac{c}{d}\right)\geq4$ with $a>0, b>0 , c> 0$ and $d>0.$

Published on 25 May 2020 - 23:50

91

Views

Show inequality $a\sqrt{b - 1} + b\sqrt{a - 1} \le ab$

Published on 26 May 2020 - 15:04

236

Views

Prove $\sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ac + a^2} \ge \sqrt{3}(a + b + c)$

Published on 25 Mar 2026 - 14:25

58

Views

How to prove inequality $(2n-1)x^{-1} + (2n+1)x^{2n-1} > \frac{(2n+1)^2}{2n}, \quad n > 2$ and $ 0 < x < 1$

Published on 28 May 2020 - 8:30

List Question

Inequality $\sum_{cyc}\frac{a}{2a^2+a+1}\leq \frac{3}{4}$

Prove $\sum \sqrt{3a+\frac{1}{b}} \geq 6$

$AM-GM$ for $(a + \frac{1}{a})^2 + (b + \frac{1}{b})^2 \geqslant \frac{25}{2}$

Prove $(a+b+c)^3 (a+b-c)(b+c-a)(c+a-b) \leqq 27a^2 b^ 2 c^2$

Prove $\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geqq 1$

Olympiad Inequality Question

Using AM-GM to show that if $\{a_i\},\{f_i\}$ are positive sequences s.t. $\sum a_i=\infty$ and $f_i\to f>0$, then $(\sum f_ia_i)/(\sum a_i)\to f$

Solve extremal problem with two constraints using AM-GM

Maximize area of isosceles triangle with given median

Refinement inequality of : $\sqrt{x}+x^{\frac{x}{x+1}}\geq x+1$

A strange inequality

Prove that $\left(\dfrac{b}{a}+\dfrac{d}{c}\right)\cdot\left(\dfrac{a}{b}+\dfrac{c}{d}\right)\geq4$ with $a>0, b>0 , c> 0$ and $d>0.$

Show inequality $a\sqrt{b - 1} + b\sqrt{a - 1} \le ab$

Prove $\sqrt{a^2 + ab + b^2} + \sqrt{b^2 + bc + c^2} + \sqrt{c^2 + ac + a^2} \ge \sqrt{3}(a + b + c)$

How to prove inequality $(2n-1)x^{-1} + (2n+1)x^{2n-1} > \frac{(2n+1)^2}{2n}, \quad n > 2$ and $ 0 < x < 1$

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