Question

How to prove $x^{x/(1+x)}/(1+x)\geq1/2$

score 144 · Answer 1 · 2026-03-26 09:17:45

144

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How to prove $x^{x/(1+x)}/(1+x)\geq1/2$

Published on 26 Mar 2026 - 9:17

score 84 · Answer 2 · 2023-12-01 05:12:41

84

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Understanding an inequality in the proof

Published on 01 Dec 2023 - 5:12

score 82 · Answer 3 · 2026-03-26 08:02:32

82

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

Published on 26 Mar 2026 - 8:02

score 81 · Answer 4 · 2026-03-26 09:18:04

81

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Can anyone help prove or disprove that $\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{N+1}$, where $x_i>0$

Published on 26 Mar 2026 - 9:18

score 48 · Answer 5 · 2023-12-06 04:46:04

48

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Inequality about $\sum_{cyc}\dfrac{a^3}{(bc)^2+1}$ and $a+b+c=ab+bc+ca.$

Published on 06 Dec 2023 - 4:46

score 89 · Answer 6 · 2026-03-25 16:13:52

89

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Help to prove $\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$.

Published on 25 Mar 2026 - 16:13

score 56 · Answer 7 · 2023-12-12 09:48:55

56

Views

Reversing AM-GM Inequality When We Have Bounded Variable

Published on 12 Dec 2023 - 9:48

score 68 · Answer 8 · 2023-12-12 15:47:51

68

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Prove that $a^2(b^2+4)+b^2(c^2+4)+c^2(a^2+4) \geq 15$

Published on 12 Dec 2023 - 15:47

score 65 · Answer 9 · 2023-12-14 09:27:52

65

Views

A nice 3 variable inhomogeneous asymmetric inequality by TATA box.

Published on 14 Dec 2023 - 9:27

score 36 · Answer 10 · 2023-12-16 16:57:26

36

Views

Proving Inequality Involving Sums and Products of Variables in a Probabilistic Analysis

Published on 16 Dec 2023 - 16:57

score 161 · Answer 11 · 2023-12-18 16:08:33

161

Views

Prove $\sum\limits_{\mathrm{cyc}}\frac{a}{a^2+bc+4} \leq \frac{1}{2}$ for $a,b,c>0$ with $a+b+c=6$

Published on 18 Dec 2023 - 16:08

score 80 · Answer 12 · 2023-12-20 17:45:22

80

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How to check that the arithmetic mean of any subset $S\ne\{1\}$ of $A={1,2,...,n}$ is at least $\frac 32$?

Published on 20 Dec 2023 - 17:45

score 92 · Answer 13 · 2023-12-21 03:33:03

92

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Prove that $\sum\limits_{\mathrm{cyc}} \frac{a^5 + b^5}{(b^2 + c^2)(c^2 + a^2)} \geq \frac{3}{2} (abc)^{\frac{2}{3}}$

Published on 21 Dec 2023 - 3:33

score 88 · Answer 14 · 2023-12-22 16:14:34

88

Views

Prove the inequality $(1+2^{x-1}+2^y)(1+2^{y-1}+2^z)(1+2^{z-1}+2^x)\leq 2^{x^2+y^2+z^2+3}$

Published on 22 Dec 2023 - 16:14

score 72 · Answer 15 · 2023-12-27 03:54:27

72

Views

Prove that $\prod_{i=1}^n(1-x_i)\ge 1/2$.

Published on 27 Dec 2023 - 3:54

List Question

How to prove $x^{x/(1+x)}/(1+x)\geq1/2$

Understanding an inequality in the proof

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

Can anyone help prove or disprove that $\frac{\Pi_i x_i ^\frac{x_i}{1+\sum_i x_i}}{1+\sum_i x_i} \geq \frac{1}{N+1}$, where $x_i>0$

Inequality about $\sum_{cyc}\dfrac{a^3}{(bc)^2+1}$ and $a+b+c=ab+bc+ca.$

Help to prove $\frac{a^3b+b^3c+c^3a}{a+b+c}+1\ge 2\sqrt{\frac{a^2b^2+b^2c^2+c^2a^2}{a+b+c}}$.

Reversing AM-GM Inequality When We Have Bounded Variable

Prove that $a^2(b^2+4)+b^2(c^2+4)+c^2(a^2+4) \geq 15$

A nice 3 variable inhomogeneous asymmetric inequality by TATA box.

Proving Inequality Involving Sums and Products of Variables in a Probabilistic Analysis

Prove $\sum\limits_{\mathrm{cyc}}\frac{a}{a^2+bc+4} \leq \frac{1}{2}$ for $a,b,c>0$ with $a+b+c=6$

How to check that the arithmetic mean of any subset $S\ne\{1\}$ of $A={1,2,...,n}$ is at least $\frac 32$?

Prove that $\sum\limits_{\mathrm{cyc}} \frac{a^5 + b^5}{(b^2 + c^2)(c^2 + a^2)} \geq \frac{3}{2} (abc)^{\frac{2}{3}}$

Prove the inequality $(1+2^{x-1}+2^y)(1+2^{y-1}+2^z)(1+2^{z-1}+2^x)\leq 2^{x^2+y^2+z^2+3}$

Prove that $\prod_{i=1}^n(1-x_i)\ge 1/2$.

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