EGMO 2026 Problem 5

Let $ABC$ be an acute triangle with $AC > AB$. Denote by $\omega$ its circumcircle and by $O$ its circumcentre. Let $K$ be the intersection of the tangents to $\omega$ at $B$ and $C$. Circle $ABK$ intersects line $BC$ again at $Z \neq B$. Let $L$ be the midpoint of $KZ$. Let $X$ be the intersection of lines $KZ$ and $AB$. Let $V$ be the point on circle $ABL$ on the same side of $BC$ as $A$ such that $OV$ is perpendicular to $KZ$. Prove that $LV$ is perpendicular to $CX$.