User River Li

27 votes

Accepted

Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$

answered May 16 at 5:10

18 votes

Without calculator prove that $9^{\sqrt{2}} < \sqrt{2}^9$

answered Jan 4, 2023 at 14:26

14 votes

Find the real root of the almost symmetric polynomial $x^7+7x^5+14x^3+7x-1$

answered Oct 23, 2022 at 1:37

14 votes

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

answered Feb 20, 2021 at 4:59

13 votes

Show $\frac{\sin x_1\sin x_2\cdots\sin x_n}{\sin(x_1+x_2)\sin(x_2+x_3)\cdots\sin(x_n+x_1)}\le\frac{\sin^n(\pi/n)}{\sin^n(2\pi/n)}$, for $\sum x_i=\pi$

answered Jun 19, 2019 at 17:37

11 votes

A closed form for $\int_0^\pi \lvert \sin(m t) \cos(n t) \rvert \, \mathrm{d} t$

answered May 6, 2020 at 16:05

11 votes

Prove that $\ln x\leq\frac{x^{x+1/x}-1}{2}$ [ not solved ]

answered Aug 24, 2023 at 3:01

10 votes

Accepted

prove that for any $|x|\leq 1$, $|f(x)|\leq 5/4$

answered Oct 23, 2023 at 6:23

10 votes

Prove $a^{\sqrt{\log_ba}}+b^{\sqrt{\log_cb}}+c^{\sqrt{\log_ac}}\geqslant a+b+c$ for $a, b, c > 1$

answered Mar 9, 2023 at 7:38

10 votes

Prove that $\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+d^2}}+\frac{d}{\sqrt{d^2+a^2}}\leq3$

answered May 26, 2022 at 7:38

10 votes

Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$

answered Jun 8, 2019 at 4:28

10 votes

Accepted

show this inequality $\sqrt{\frac{a^b}{b}}+\sqrt{\frac{b^a}{a}}\ge 2$

answered Oct 14, 2021 at 15:37

10 votes

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

answered Feb 22, 2022 at 2:56

10 votes

Accepted

Inequality involving an absolute value

answered Apr 7 at 13:35

9 votes

Accepted

Prove/disprove that $a^{2m} + b^{2m} + c^{2m} > 2^{1-m}$ subject to $a + b + c= 0$ and $a^2 + b^2 + c^2 = 1$

answered Jul 9 at 0:41

9 votes

Prove $ \frac{x_1-x_2}{x_n+x_1} + \frac{x_2-x_3}{x_1+x_2}+\cdots+ \frac{x_n-x_1}{x_{n-1} +x_n}\le 0$ s.t. $x_1+\cdots+x_n=1$

answered May 1, 2023 at 10:54

9 votes

Accepted

Prove that $x^xy^y \geq \dfrac{x^2+y^2}{2}$

answered Sep 16, 2023 at 2:48

8 votes

Accepted

Inequality $(1-e^x)\ln(1-xe^{-x})\leq x^2$

answered Jul 20, 2022 at 3:50

8 votes

Accepted

Prove that $\sum \frac{a^3}{a^2+b^2}\le \frac12 \sum \frac{b^2}{a}$

answered Feb 3, 2022 at 2:27

8 votes

Accepted

Integral $\lim _{n\to \infty}\int _0^1\sqrt{\frac{1}{x}+n^2x^{2n}}\,dx$

answered May 8, 2020 at 14:06

8 votes

Accepted

How to prove inequality $2r^2(4+3r-6r^2-2r^4) <3$ for all $r \in [0,1]$

answered Sep 12 at 9:29

7 votes

Accepted

Convexity of $x \mapsto \frac{\|Ax-b\|^2_2}{1-\|x\|^2_2}$

answered Mar 29, 2020 at 16:27

7 votes

Evaluate $\lim_{n\to\infty} \prod_{k=1}^n \frac{2k}{2k-1}\int_{-1}^{\infty} \frac{{\left(\cos{x}\right)}^{2n}}{2^x} \; dx$

answered May 24, 2020 at 16:39

7 votes

How do I evaluate $\sum_{k = 1}^{\infty}\big[\frac{(-1)^{k - 1}}{k}\sum_{n = 0}^{\infty}\big\{\frac{1}{k(2^n) + 1}\big\}\big]$?

answered Jun 20, 2020 at 8:00

7 votes

A hard inequality indian olympiad problem

answered Sep 14, 2019 at 16:07

7 votes

compute the integral $\int_0^1 \int_0^1 \int_0^1 \frac{1}{(1+x^2+y^2+z^2)^2} dxdydz$

answered Oct 21, 2021 at 0:48

7 votes

Accepted

Prove that $(1+\frac{a^2+b^2+c^2}{ab+bc+ca})^{\frac{(a+b+c)^2}{a^2+b^2+c^2}} \leq (1+\frac{a}{b})(1+\frac{b}{c})(1+\frac{c}{a})$

answered Jun 8, 2022 at 2:50

7 votes

Prove $\frac{ab^2+2}{a+c} +\frac{bc^2+2}{b+a} +\frac{ca^2+2}{c+b} \geq \frac{9}{2}$ for $a,b,c\geq1$

answered Jan 30, 2023 at 15:00

7 votes

Accepted

A Weighted Gaussian Inequality: $E[\frac{\sigma_n^2 x_n^2}{\sum_{i=1}^n \sigma_i^2x_i^2} ] \ge \frac{\sigma_n^2}{\sum_{i=1}^n \sigma_i^2}$

answered Oct 4, 2022 at 6:13

7 votes

Prove $ \frac{x_1-x_2}{x_n+x_1} + \frac{x_2-x_3}{x_1+x_2}+\cdots+ \frac{x_n-x_1}{x_{n-1} +x_n}\le 0$ s.t. $x_1+\cdots+x_n=1$

answered May 1, 2023 at 7:11

Stack Exchange Network

1,311 Answers

Prove that $|z^2+1|\le 2$ implies $|z^3+3z+2|\le 6$

Without calculator prove that $9^{\sqrt{2}} < \sqrt{2}^9$

Find the real root of the almost symmetric polynomial $x^7+7x^5+14x^3+7x-1$

If a two variable smooth function has two global minima, will it necessarily have a third critical point?

Show $\frac{\sin x_1\sin x_2\cdots\sin x_n}{\sin(x_1+x_2)\sin(x_2+x_3)\cdots\sin(x_n+x_1)}\le\frac{\sin^n(\pi/n)}{\sin^n(2\pi/n)}$, for $\sum x_i=\pi$

A closed form for $\int_0^\pi \lvert \sin(m t) \cos(n t) \rvert \, \mathrm{d} t$

Prove that $\ln x\leq\frac{x^{x+1/x}-1}{2}$ [ not solved ]

prove that for any $|x|\leq 1$, $|f(x)|\leq 5/4$

Prove $a^{\sqrt{\log_ba}}+b^{\sqrt{\log_cb}}+c^{\sqrt{\log_ac}}\geqslant a+b+c$ for $a, b, c > 1$

Prove that $\frac{a}{\sqrt{a^2+b^2}}+\frac{b}{\sqrt{b^2+c^2}}+\frac{c}{\sqrt{c^2+d^2}}+\frac{d}{\sqrt{d^2+a^2}}\leq3$

Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$

show this inequality $\sqrt{\frac{a^b}{b}}+\sqrt{\frac{b^a}{a}}\ge 2$

Is each of $\int_0^\infty\frac{dx}{x^x},\int_0^\infty\frac{dx}{x^{x^{x^x}}},\int_0^\infty\frac{dx}{x^{x^{x^{x^{x^x}}}}},\cdots$ less than $2$?

Inequality involving an absolute value

Prove/disprove that $a^{2m} + b^{2m} + c^{2m} > 2^{1-m}$ subject to $a + b + c= 0$ and $a^2 + b^2 + c^2 = 1$

Prove $ \frac{x_1-x_2}{x_n+x_1} + \frac{x_2-x_3}{x_1+x_2}+\cdots+ \frac{x_n-x_1}{x_{n-1} +x_n}\le 0$ s.t. $x_1+\cdots+x_n=1$

Prove that $x^xy^y \geq \dfrac{x^2+y^2}{2}$

Inequality $(1-e^x)\ln(1-xe^{-x})\leq x^2$

Prove that $\sum \frac{a^3}{a^2+b^2}\le \frac12 \sum \frac{b^2}{a}$

Integral $\lim _{n\to \infty}\int _0^1\sqrt{\frac{1}{x}+n^2x^{2n}}\,dx$

How to prove inequality $2r^2(4+3r-6r^2-2r^4) <3$ for all $r \in [0,1]$

Convexity of $x \mapsto \frac{\|Ax-b\|^2_2}{1-\|x\|^2_2}$

Evaluate $\lim_{n\to\infty} \prod_{k=1}^n \frac{2k}{2k-1}\int_{-1}^{\infty} \frac{{\left(\cos{x}\right)}^{2n}}{2^x} \; dx$

How do I evaluate $\sum_{k = 1}^{\infty}\big[\frac{(-1)^{k - 1}}{k}\sum_{n = 0}^{\infty}\big\{\frac{1}{k(2^n) + 1}\big\}\big]$?

A hard inequality indian olympiad problem

compute the integral $\int_0^1 \int_0^1 \int_0^1 \frac{1}{(1+x^2+y^2+z^2)^2} dxdydz$

Prove that $(1+\frac{a^2+b^2+c^2}{ab+bc+ca})^{\frac{(a+b+c)^2}{a^2+b^2+c^2}} \leq (1+\frac{a}{b})(1+\frac{b}{c})(1+\frac{c}{a})$

Prove $\frac{ab^2+2}{a+c} +\frac{bc^2+2}{b+a} +\frac{ca^2+2}{c+b} \geq \frac{9}{2}$ for $a,b,c\geq1$

A Weighted Gaussian Inequality: $E[\frac{\sigma_n^2 x_n^2}{\sum_{i=1}^n \sigma_i^2x_i^2} ] \ge \frac{\sigma_n^2}{\sum_{i=1}^n \sigma_i^2}$

Prove $ \frac{x_1-x_2}{x_n+x_1} + \frac{x_2-x_3}{x_1+x_2}+\cdots+ \frac{x_n-x_1}{x_{n-1} +x_n}\le 0$ s.t. $x_1+\cdots+x_n=1$