List questions from geometric-inequalities

Published on 25 Mar 2026 - 9:44

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INMO 1998 question

Published on 15 Oct 2018 - 3:48

932

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Minimizing $\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$

Published on 26 Oct 2018 - 11:11

105

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Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$.

Published on 25 Mar 2026 - 6:07

142

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$M$ is a point in an equaliateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$.

Published on 25 Mar 2026 - 12:50

322

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Difficult inequality with three real variables

Published on 18 Nov 2018 - 18:00

553

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Find at least one solution to matrix inequality

Published on 21 Nov 2018 - 17:16

110

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Proof using AM-GM inequality

Published on 30 Nov 2018 - 16:36

86

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geometry inequality

Published on 09 Dec 2018 - 9:42

94

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A simple proof for $\prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$?

Published on 25 Mar 2026 - 3:21

200

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A triangle has sides $a$, $b$, $c$ and medians $m_a$, $m_b$, $m_c$. Show $(ab+bc+ca)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 2\sqrt{3}(m_a+m_b+m_c)$

Published on 25 Mar 2026 - 7:45

116

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Write in a concise form a set in 4-dimensional space.

Published on 19 Dec 2018 - 14:38

265

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Generalizing Heron's Formula for Cyclic $n$-gons

Published on 02 Jan 2019 - 17:54

221

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Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$

Published on 25 Mar 2026 - 4:59

68

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Is there a right triangle with angles $A$, $B$, $C$ such that $A^2+B^2=C^2$?

Published on 15 Jan 2019 - 8:28

List Question

Complex number inequalities

INMO 1998 question

Minimizing $\sqrt{(x+3)^2 + 49} + \sqrt{(x-5)^2 + 64}$

Prove that $|A_1A_2|^2+|A_2A_3|^2+\ldots+|A_{n-1}A_n|^2+|A_nA_1|^2\leq 9R^2$.

$M$ is a point in an equaliateral $ABC$ of area $S$. $S'$ is the area of the triangle with sides $MA,MB,MC$. Prove that $S'\leq \frac{1}{3}S$.

Difficult inequality with three real variables

Find at least one solution to matrix inequality

Proof using AM-GM inequality

geometry inequality

A simple proof for $\prod_{i=1}^d a_i+\prod_{i=1}^d b_i \le \prod_{i=1}^d (a_i^d+b_i^d)^{1/d}$?

A triangle has sides $a$, $b$, $c$ and medians $m_a$, $m_b$, $m_c$. Show $(ab+bc+ca)(\frac{1}{a}+\frac{1}{b}+\frac{1}{c})\geq 2\sqrt{3}(m_a+m_b+m_c)$

Write in a concise form a set in 4-dimensional space.

Generalizing Heron's Formula for Cyclic $n$-gons

Prove that $\frac{a}{c\sqrt{a^2+1}}+\frac{b}{a\sqrt{b^2+1}}+\frac{c}{b\sqrt{c^2+1}}\ge \frac{3}{2}$

Is there a right triangle with angles $A$, $B$, $C$ such that $A^2+B^2=C^2$?

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