List questions from geometric-inequalities

Published on 12 Apr 2026 - 14:41

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INMO 1998 question

Published on 30 Apr 2026 - 9:03

935

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

Published on 17 Apr 2026 - 0:20

107

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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 16 Apr 2026 - 18:50

143

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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 10 May 2026 - 15:26

325

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Difficult inequality with three real variables

Published on 13 Apr 2026 - 14:21

556

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Find at least one solution to matrix inequality

Published on 13 Apr 2026 - 21:03

114

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Proof using AM-GM inequality

Published on 14 Apr 2026 - 12:05

91

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geometry inequality

Published on 14 Apr 2026 - 23:14

96

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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 10 May 2026 - 16:36

201

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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 10 May 2026 - 13:51

119

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

Published on 16 Apr 2026 - 12:37

268

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

Published on 13 Apr 2026 - 2:38

224

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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 10 May 2026 - 14:36

71

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

Published on 14 Apr 2026 - 21:52

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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