Question

For real $a$, $b$, $c$ in $(0,1)$, if $a+b+c=2$, then $\frac{a}{1-a}\times\frac{b}{1-b}\times\frac{c}{1-c}\geq 8$

score 95 · Answer 1 · 2026-03-27 09:13:40

95

Views

For real $a$, $b$, $c$ in $(0,1)$, if $a+b+c=2$, then $\frac{a}{1-a}\times\frac{b}{1-b}\times\frac{c}{1-c}\geq 8$

Published on 27 Mar 2026 - 9:13

score 52 · Answer 2 · 2026-03-26 08:14:44

52

Views

Solving Inequality Relation:

Published on 26 Mar 2026 - 8:14

score 111 · Answer 3 · 2020-01-28 18:22:18

111

Views

Simple inequality proof problem of my authorship with fixed sum $a+b+c=3$.

Published on 28 Jan 2020 - 18:22

score 98 · Answer 4 · 2020-01-31 10:15:23

98

Views

Prove $\int_{0}^{1} \sqrt{x^n+1 }dx \geq \frac{2}{n+2}$ for any $n \in \mathbb{N}$

Published on 31 Jan 2020 - 10:15

score 96 · Answer 5 · 2020-02-06 10:43:44

96

Views

$ \sin{x}\ +\ \frac{1}{2}\sin{2x}\ +\ \frac{1}{3}\sin{3x}\ +\ \frac{1}{4}\sin{4x}\ =\ \frac{2}{3}\left(\cos{x}+1\right)\left(\sin^5x\ +\ 4\right)$

Published on 06 Feb 2020 - 10:43

score 49 · Answer 6 · 2026-03-25 14:28:54

49

Views

Solving an Inequality problem

Published on 25 Mar 2026 - 14:28

score 153 · Answer 7 · 2026-03-25 12:41:26

153

Views

Given positives $x, y , z$ such that $x + y + z = xyz$. Calculate the minimum value of $\frac{x - 1}{y^2} + \frac{y - 1}{z^2} + \frac{z - 1}{x^2}$.

Published on 25 Mar 2026 - 12:41

score 130 · Answer 8 · 2026-03-25 07:45:40

130

Views

Inequality with x,y,z fractions $\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$

Published on 25 Mar 2026 - 7:45

score 213 · Answer 9 · 2020-02-14 14:23:22

213

Views

If $h(x) = \dfrac{f(x)}{g(x)}$, then find the local minimum value of $h(x)$

Published on 14 Feb 2020 - 14:23

score 526 · Answer 10 · 2020-02-16 20:17:31

526

Views

Proof of $(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$ without use of derivatives

Published on 16 Feb 2020 - 20:17

score 191 · Answer 11 · 2020-02-20 20:13:21

191

Views

Kullback-Leibler divergence implies AM-GM inequality

Published on 20 Feb 2020 - 20:13

score 470 · Answer 12 · 2026-03-26 08:14:54

470

Views

Hard inequality problem (if $(a_2-a_1)^2 + (a_3-a_2)^2 + \ldots + (a_{2n}-a_{2n-1})^2 = 1$ ...)

Published on 26 Mar 2026 - 8:14

score 139 · Answer 13 · 2026-03-25 16:08:26

139

Views

Prove that $\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2$

Published on 25 Mar 2026 - 16:08

score 75 · Answer 14 · 2020-02-26 07:42:27

75

Views

An inequality: $ab=cd=1$, $(a,b,c,d \geq 0)$, prove that $(a+b)(c+d)+4 \geq 2(a+b+c+d)$

Published on 26 Feb 2020 - 7:42

score 117 · Answer 15 · 2020-02-27 20:05:16

117

Views

Find minimum of $a^2+b^2+\frac{a^2b^2}{(a+b)^2}-\frac{2(a^2+ab+b^2)}{a+b}$

Published on 27 Feb 2020 - 20:05

List Question

For real $a$, $b$, $c$ in $(0,1)$, if $a+b+c=2$, then $\frac{a}{1-a}\times\frac{b}{1-b}\times\frac{c}{1-c}\geq 8$

Solving Inequality Relation:

Simple inequality proof problem of my authorship with fixed sum $a+b+c=3$.

Prove $\int_{0}^{1} \sqrt{x^n+1 }dx \geq \frac{2}{n+2}$ for any $n \in \mathbb{N}$

$ \sin{x}\ +\ \frac{1}{2}\sin{2x}\ +\ \frac{1}{3}\sin{3x}\ +\ \frac{1}{4}\sin{4x}\ =\ \frac{2}{3}\left(\cos{x}+1\right)\left(\sin^5x\ +\ 4\right)$

Solving an Inequality problem

Given positives $x, y , z$ such that $x + y + z = xyz$. Calculate the minimum value of $\frac{x - 1}{y^2} + \frac{y - 1}{z^2} + \frac{z - 1}{x^2}$.

Inequality with x,y,z fractions $\frac{x}{y}+\frac{y}{z+x}+\frac{z}{x}\ge 2$

If $h(x) = \dfrac{f(x)}{g(x)}$, then find the local minimum value of $h(x)$

Proof of $(1-x)x^n \leq \frac{n^n}{\left(n+1\right)^{n+1}}$ without use of derivatives

Kullback-Leibler divergence implies AM-GM inequality

Hard inequality problem (if $(a_2-a_1)^2 + (a_3-a_2)^2 + \ldots + (a_{2n}-a_{2n-1})^2 = 1$ ...)

Prove that $\frac{1}{\sqrt{a+b+2}}+\frac{1}{\sqrt{b+c+2}}+\frac{1}{\sqrt{c+d+2}}+\frac{1}{\sqrt{d+a+2}}\le 2$

An inequality: $ab=cd=1$, $(a,b,c,d \geq 0)$, prove that $(a+b)(c+d)+4 \geq 2(a+b+c+d)$

Find minimum of $a^2+b^2+\frac{a^2b^2}{(a+b)^2}-\frac{2(a^2+ab+b^2)}{a+b}$

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