Problems - page 3
IMO Training, Problem 1
Let $ABC$ be a triangle inscribed in the circle $(O)$, with the altitudes $AD$ and $CF$ intersecting at the orthocenter $H$. Let $M$ be the midpoint of $AH$. The line $OM$ intersects $BC$ at $P$. Let $Q$ be a point such that $AQ \parallel BC$ and $MQ \perp AB$. Point $K$ lies on $DF$ such that $AK \perp AC$. Points $S$ and $T$ are the reflections of $A$ over the lines $QK$ and $QF$, respectively. Let $L$ be the projection of $A$ onto $PQ$. Prove that $Q, L, S, T$ are concyclic.
