Olympiad problems

2024 Czech and Slovak Olympiad III A, 6

Find all right triangles with integer side lengths in which two congruent circles with prime radius can be inscribed such that they are externally tangent, both touch the hypotenuse, and each is tangent to another leg of the right triangle.

2005 All-Russian Olympiad, 4

Tags: geometry , incenter , linearity of power of a point , side bash , trigonometry

$w_B$ and $w_C$ are excircles of a triangle $ABC$. The circle $w_B'$ is symmetric to $w_B$ with respect to the midpoint of $AC$, the circle $w_C'$ is symmetric to $w_C$ with respect to the midpoint of $AB$. Prove that the radical axis of $w_B'$ and $w_C'$ halves the perimeter of $ABC$.

2021 Balkan MO Shortlist, G8

Tags: geometry , shortlist , side bash

Let $ABC$ be a scalene triangle and let $I$ be its incenter. The projections of $I$ on $BC, CA$, and $AB$ are $D, E$ and $F$ respectively. Let $K$ be the reflection of $D$ over the line $AI$, and let $L$ be the second point of intersection of the circumcircles of the triangles $BFK$ and $CEK$. If $\frac{1}{3} BC = AC - AB$, prove that $DE = 2KL$.

2008 Baltic Way, 20

Tags: incenter , geometry , side bash

Let $ M$ be a point on $ BC$ and $ N$ be a point on $ AB$ such that $ AM$ and $ CN$ are angle bisectors of the triangle $ ABC$. Given that $ \frac {\angle BNM}{\angle MNC} \equal{} \frac {\angle BMN}{\angle NMA}$, prove that the triangle $ ABC$ is isosceles.