Question

Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus

score 94 · Answer 1 · 2026-03-25 11:15:00

94

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Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus

Published on 25 Mar 2026 - 11:15

score 104 · Answer 2 · 2026-03-25 18:50:37

104

Views

Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.

Published on 25 Mar 2026 - 18:50

score 94 · Answer 3 · 2022-05-17 21:09:48

94

Views

Find bounds of $\prod_{i=1}^k (x_i-x_{i+1})$ where $\sum_{i=1}^k x^2_i=1$

Published on 17 May 2022 - 21:09

score 124 · Answer 4 · 2022-05-18 16:53:58

124

Views

A conditional negative definite quadratic form involving $\ln$ function

Published on 18 May 2022 - 16:53

score 68 · Answer 5 · 2022-05-19 13:31:33

68

Views

Let m be the smallest number among: $(x-y)^2, (y-z)^2, (z-x)^2$ Prove $m \le \frac{1}{2}(x^2+y^2+z^2)$

Published on 19 May 2022 - 13:31

score 41 · Answer 6 · 2022-05-22 07:33:33

41

Views

How can i find minimum value of this inequality

Published on 22 May 2022 - 7:33

score 375 · Answer 7 · 2022-05-25 00:30:40

375

Views

Let $x,y,z\in[0,1]$. Find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$.

Published on 25 May 2022 - 0:30

score 224 · Answer 8 · 2026-03-30 16:46:45

224

Views

Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$

Published on 30 Mar 2026 - 16:46

score 87 · Answer 9 · 2022-05-27 08:56:40

87

Views

Finding a limit using AM-GM?

Published on 27 May 2022 - 8:56

score 83 · Answer 10 · 2022-05-29 04:17:07

83

Views

An Application of Cauchy Schwarz - AM-GM in Discrete Probability Measures

Published on 29 May 2022 - 4:17

score 97 · Answer 11 · 2022-06-05 06:40:07

97

Views

Prove $ \frac{(c-a)^{2}}{6c} \le \frac{a+b+c}{3} - \frac{3}{ \frac{1}{a} + \frac{1}{b}+ \frac{1}{c}}$

Published on 05 Jun 2022 - 6:40

score 101 · Answer 12 · 2022-06-16 03:55:03

101

Views

Let $f:\mathbb R^+\to\mathbb R$ be a continuous function satisfied $f(a)+f(b)\ge f(2\sqrt{ab})$ for all $a,b>0$ , is $f$ differentiable?

Published on 16 Jun 2022 - 3:55

score 83 · Answer 13 · 2022-06-17 02:47:22

83

Views

prove that :$(1+\frac{1}{n})^k<1+\frac{k}{n}+\frac{k^2}{2n^2}$.

Published on 17 Jun 2022 - 2:47

score 2.7k · Answer 14 · 2022-07-01 03:17:24

2.7k

Views

Weighted AM-GM Inequality

Published on 01 Jul 2022 - 3:17

score 44 · Answer 15 · 2022-07-03 17:08:15

44

Views

if a,b,c>0 and $abc=1$ prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq a+b+c$

Published on 03 Jul 2022 - 17:08

List Question

Prove that $(1+x)^k/k + (1-x)^m/m\geq 1/k +1/m$ without calculus

Prove that for positive reals $x,y,z$, $x^6+y^6+z^6 + 6x^2y^2z^2 \geq 3xyz(x^3+y^3+z^3)$.

Find bounds of $\prod_{i=1}^k (x_i-x_{i+1})$ where $\sum_{i=1}^k x^2_i=1$

A conditional negative definite quadratic form involving $\ln$ function

Let m be the smallest number among: $(x-y)^2, (y-z)^2, (z-x)^2$ Prove $m \le \frac{1}{2}(x^2+y^2+z^2)$

How can i find minimum value of this inequality

Let $x,y,z\in[0,1]$. Find the maximum value of $\sqrt{|x-y|}+\sqrt{|y-z|}+\sqrt{|z-x|}$.

Find number of continuous functions satisfying the equation $4\int_{0}^{\frac{3}{2}}f(x)dx+125\int_{0}^{\frac{3}{2}}\frac{dx}{\sqrt{f(x)+x^2}}=108$

Finding a limit using AM-GM?

An Application of Cauchy Schwarz - AM-GM in Discrete Probability Measures

Prove $ \frac{(c-a)^{2}}{6c} \le \frac{a+b+c}{3} - \frac{3}{ \frac{1}{a} + \frac{1}{b}+ \frac{1}{c}}$

Let $f:\mathbb R^+\to\mathbb R$ be a continuous function satisfied $f(a)+f(b)\ge f(2\sqrt{ab})$ for all $a,b>0$ , is $f$ differentiable?

prove that :$(1+\frac{1}{n})^k<1+\frac{k}{n}+\frac{k^2}{2n^2}$.

Weighted AM-GM Inequality

if a,b,c>0 and $abc=1$ prove that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\geq a+b+c$

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