List questions from tangent-line-method

Published on 24 Mar 2026 - 19:51

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$ \sin^2x_1+\dots \sin^2x_{10}=1$ implies $ 3(\sin x_1+\dots \sin x_{10})\leq \cos x_1 +\dots +\cos x_{10}. $

Published on 08 Jul 2017 - 18:02

126

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Prove this inequality $2(a+b+c)\ge\sqrt{a^2+3}+\sqrt{b^2+3}+\sqrt{c^2+3}$

Published on 01 Aug 2017 - 15:07

110

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Chebyshev Inequality toughnut

Published on 14 Aug 2017 - 0:24

137

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Inequality : $\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)^4 \geq \frac{1}{27}\left(\sum_{cyc}\sqrt{ab}\right)^4$

Published on 10 Sep 2017 - 8:39

152

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Inequality on tangent and secant function

Published on 21 Sep 2017 - 9:18

486

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How to prove $(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$?

Published on 06 Oct 2017 - 15:35

272

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Let $a_1, a_2,\ldots, a_{100}$ be non-zero real numbers such that$a_1+ a_2+\cdots+ a_{100}=0$. Then $\sum_{i=1}^{100} a_i 2^{a_i}$

Published on 31 Oct 2017 - 6:05

600

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How to prove $|\sum_{i=1}^n a_i|\le \sqrt{n} \sqrt{\sum_{i=1}^n a_i^2}$

Published on 19 Dec 2017 - 2:45

152

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Find the minimum of expression: $\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$

Published on 23 Dec 2017 - 22:44

81

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Find max value of $a$

Published on 01 Jan 2018 - 13:08

64

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Prove an inequality $\frac{x^2}{1+y}+\frac{y^2}{1+x}\ge 1$ where $x,y \ge 0$ and $x+y=2$.

Published on 07 Jan 2018 - 18:36

588

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$\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y} \ge \frac{1}{2}(x+y+z)$

Published on 08 Jan 2018 - 3:00

1.8k

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A inequality proposed at Zhautykov Olympiad 2008

Published on 25 Sep 2012 - 7:06

190

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$\frac{(b+c-a)^2}{(b+c)^2+a^2}+ \frac{(c+a-b)^2}{(c+a)^2+b^2}+ \frac{(a+b-c)^2}{(a+b)^2+c^2} \ge \frac{3}{5}$

Published on 20 Jan 2013 - 3:44

List Question

Prove the inequality

$ \sin^2x_1+\dots \sin^2x_{10}=1$ implies $ 3(\sin x_1+\dots \sin x_{10})\leq \cos x_1 +\dots +\cos x_{10}. $

Prove this inequality $2(a+b+c)\ge\sqrt{a^2+3}+\sqrt{b^2+3}+\sqrt{c^2+3}$

Chebyshev Inequality toughnut

Inequality : $\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)^4 \geq \frac{1}{27}\left(\sum_{cyc}\sqrt{ab}\right)^4$

Inequality on tangent and secant function

How to prove $(a_1+a_2+\dots a_n)\left(\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\right)\ge n^2$?

Let $a_1, a_2,\ldots, a_{100}$ be non-zero real numbers such that$a_1+ a_2+\cdots+ a_{100}=0$. Then $\sum_{i=1}^{100} a_i 2^{a_i}$

How to prove $|\sum_{i=1}^n a_i|\le \sqrt{n} \sqrt{\sum_{i=1}^n a_i^2}$

Find the minimum of expression: $\frac{2-x}{3+x}+\frac{2-y}{3+y}+\frac{2-z}{3+z}$

Find max value of $a$

Prove an inequality $\frac{x^2}{1+y}+\frac{y^2}{1+x}\ge 1$ where $x,y \ge 0$ and $x+y=2$.

$\frac{x^2}{y+z}+\frac{y^2}{x+z}+\frac{z^2}{x+y} \ge \frac{1}{2}(x+y+z)$

A inequality proposed at Zhautykov Olympiad 2008

$\frac{(b+c-a)^2}{(b+c)^2+a^2}+ \frac{(c+a-b)^2}{(c+a)^2+b^2}+ \frac{(a+b-c)^2}{(a+b)^2+c^2} \ge \frac{3}{5}$

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