Olympiad problems

2003 Nordic, 3

The point ${D}$ inside the equilateral triangle ${\triangle ABC}$ satisfies ${\angle ADC = 150^o}$. Prove that a triangle with side lengths ${|AD|, |BD|, |CD|}$ is necessarily a right-angled triangle.

2019 Singapore Junior Math Olympiad, 1

Tags: geometry , right triangles , Isosceles Triangle , isosceles

In the triangle $ABC, AC=BC, \angle C=90^o, D$ is the midpoint of $BC, E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.

Estonia Open Senior - geometry, 1999.2.3

Tags: incircle , circumcircle , geometric inequality , area of a triangle , areas , geometry , right triangles

Two right triangles are given, of which the incircle of the first triangle is the circumcircle of the second triangle. Let the areas of the triangles be $S$ and $S'$ respectively. Prove that $\frac{S}{S'} \ge 3 +2\sqrt2$

2017 Germany, Landesrunde - Grade 11/12, 5

Tags: inequalities , geometry , geometric inequality , inradius , right triangles

In a right-angled triangle let $r$ be the inradius and $s_a,s_b$ be the lengths of the medians of the legs $a,b$. Prove the inequality \[ \frac{r^2}{s_a^2+s_b^2} \leq \frac{3-2 \sqrt2}{5}. \]

1993 Mexico National Olympiad, 1

Tags: geometry , area of a triangle , right triangles

$ABC$ is a triangle with $\angle A = 90^o$. Take $E$ such that the triangle $AEC$ is outside $ABC$ and $AE = CE$ and $\angle AEC = 90^o$. Similarly, take $D$ so that $ADB$ is outside $ABC$ and similar to $AEC$. $O$ is the midpoint of $BC$. Let the lines $OD$ and $EC$ meet at $D'$, and the lines $OE$ and $BD$ meet at $E'$. Find area $DED'E'$ in terms of the sides of $ABC$.

2008 Hanoi Open Mathematics Competitions, 9

Tags: geometry , right triangles

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.

2020 Malaysia IMONST 1, 13

Tags: area of a triangle , right triangles

Given a right-angled triangle with perimeter $18$. The sum of the squares of the three side lengths is $128$. What is the area of the triangle?

2011 Hanoi Open Mathematics Competitions, 10

Tags: geometry , right triangles

Consider a right -angle triangle $ABC$ with $A=90^{o}$, $AB=c$ and $AC=b$. Let $P\in AC$ and $Q\in AB$ such that $\angle APQ=\angle ABC$ and $\angle AQP = \angle ACB$. Calculate $PQ+PE+QF$, where $E$ and $F$ are the projections of $B$ and $Q$ onto $BC$, respectively.