Junior Balkan Mathematical Olympiad 2021 - Problem 3 - 1 July 2021

Problem 3. Let $ABC$ be an acute scalene triangle with circumcenter $O$ . Let $D$ be the foot of the altitude from $A$ to the side $BC$ . The lines $BC$ and $AO$ intersect at $E$ . Let $s$ be the line through $E$ perpendicular to $AO$ . The line $s$ intersects $AB$ and $AC$ at $K$ and $L$ , respectively. Denote by $\omega$ the circumcircle of triangle $AKL$ . Line $AD$ intersects $\omega$ again at $X$ .
Prove that $\omega$ and the circumcircles of triangles $ABC$ and $DEX$ have a common point.

proposed by Bosnia and Herzegovina