39th Balkan Mathematical Olympiad | BMO 2022 | Problem 1 - 6 May 2022

Problem 1

Let $ABC$ be an acute triangle such that $CA \neq CB$ with circumcircle $\omega$ and circumcentre $O$ . Let $t_A$ and $t_B$ be the tangents to $\omega$ at $A$ and $B$ respectively, which meet at $X$ . Let $Y$ be the foot of the perpendicular from $O$ onto the line segment $CX$ . The line through $C$ parallel to line $AB$ meets $t_A$ at $Z$ . Prove that the line $YZ$ passes through the midpoint of the line segment $AC$ .