Question

Prove that $\sum \frac{x}{x^2+7}\le \frac{3}{8}$

score 204 · Answer 1 · 2026-03-24 21:53:36

204

Views

Prove that $\sum \frac{x}{x^2+7}\le \frac{3}{8}$

Published on 24 Mar 2026 - 21:53

score 80 · Answer 2 · 2019-11-11 07:02:49

80

Views

inequality under condition $x+y+z=3$

Published on 11 Nov 2019 - 7:02

score 104 · Answer 3 · 2019-11-17 02:00:40

104

Views

Inequality with 5 cyclic variables

Published on 17 Nov 2019 - 2:00

score 178 · Answer 4 · 2026-03-15 16:43:08

178

Views

Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$

Published on 15 Mar 2026 - 16:43

score 849 · Answer 5 · 2019-11-29 03:15:33

849

Views

Prove that the minimum values of $x^2+y^2+z^2$ is $27$ with given condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$.

Published on 29 Nov 2019 - 3:15

score 93 · Answer 6 · 2019-11-30 20:51:39

93

Views

Find all positive real solutions of the system

Published on 30 Nov 2019 - 20:51

score 73 · Answer 7 · 2019-12-18 10:05:05

73

Views

Inequality $\sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+c}{18}$

Published on 18 Dec 2019 - 10:05

score 169 · Answer 8 · 2019-12-29 09:14:45

169

Views

Inequality $\frac{x^3}{x^2+y^2}+\frac{y^3}{y^2+z^2}+\frac{z^3}{z^2+x^2} \geqslant \frac{x+y+z}{2}$

Published on 29 Dec 2019 - 9:14

score 102 · Answer 9 · 2020-01-02 16:57:00

102

Views

Prove that $\sum_{i=1}^n\frac{\sum_{i'=1}^na_{i'}^p - a_i^p}{\sum_{i'=1}^na_{i'}^q - a_i^q}\le n\cdot\frac{\sum_{i=1}^na_i^p}{\sum_{i=1}^na_i^q}$.

Published on 02 Jan 2020 - 16:57

score 185 · Answer 10 · 2020-01-14 14:30:30

185

Views

Given positives $a, b, c$, prove that $\frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$.

Published on 14 Jan 2020 - 14:30

score 149 · Answer 11 · 2020-01-16 05:10:37

149

Views

How to prove the harmonic-geometric mean inequality by solving an optimization?

Published on 16 Jan 2020 - 5:10

score 174 · Answer 12 · 2020-02-03 08:41:42

174

Views

Minimize $\frac{2}{1-a}+\frac{75}{10-b}$

Published on 03 Feb 2020 - 8:41

score 29 · Answer 13 · 2020-02-13 08:34:56

29

Views

How to prove $\sum_{i=1}^{k}x_i^2\ge k$ if $\sum_{i=1}^{k}\frac{1}{x_i}=k$ and $\min (x_i-1)^2$ is sufficiently small?

Published on 13 Feb 2020 - 8:34

score 122 · Answer 14 · 2020-03-22 14:38:07

122

Views

Cauchy-Schwarz inequality for $a_1^4 + a_2^4 + \cdots + a_n^4 \geqslant n$

Published on 22 Mar 2020 - 14:38

score 151 · Answer 15 · 2020-03-29 15:01:53

151

Views

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$: $\sum_{cyc}\frac{4}{a+3b}\geq \sum_{cyc}\frac{3}{a+2b}$

Published on 29 Mar 2020 - 15:01

List Question

Prove that $\sum \frac{x}{x^2+7}\le \frac{3}{8}$

inequality under condition $x+y+z=3$

Inequality with 5 cyclic variables

Proving $\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$ when $a+b+c=3$

Prove that the minimum values of $x^2+y^2+z^2$ is $27$ with given condition $\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$.

Find all positive real solutions of the system

Inequality $\sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+c}{18}$

Inequality $\frac{x^3}{x^2+y^2}+\frac{y^3}{y^2+z^2}+\frac{z^3}{z^2+x^2} \geqslant \frac{x+y+z}{2}$

Prove that $\sum_{i=1}^n\frac{\sum_{i'=1}^na_{i'}^p - a_i^p}{\sum_{i'=1}^na_{i'}^q - a_i^q}\le n\cdot\frac{\sum_{i=1}^na_i^p}{\sum_{i=1}^na_i^q}$.

Given positives $a, b, c$, prove that $\frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$.

How to prove the harmonic-geometric mean inequality by solving an optimization?

Minimize $\frac{2}{1-a}+\frac{75}{10-b}$

How to prove $\sum_{i=1}^{k}x_i^2\ge k$ if $\sum_{i=1}^{k}\frac{1}{x_i}=k$ and $\min (x_i-1)^2$ is sufficiently small?

Cauchy-Schwarz inequality for $a_1^4 + a_2^4 + \cdots + a_n^4 \geqslant n$

For $a,b,c\in\left[\frac{1}{\sqrt{6}}, 6\right]$: $\sum_{cyc}\frac{4}{a+3b}\geq \sum_{cyc}\frac{3}{a+2b}$

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