International Mathematical Olympiad 2015 Problem 3

Let $ABC$ be an acute triangle with $AB > AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocentre, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HQA = 90°$, and let $K$ be the point on $\Gamma$ such that $\angle HKQ = 90°$. Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different, and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.