17th Turkish Mathematical Olympiad

Let $H$ be the orthocenter of an acute triangle $ABC$, and let $A_1,B_1,C_1$ be the feet of the altitudes belonging to the vertices $A,B,C$, respectively. Let $K$ be a point on the smaller $A{B}_1$ arc of the circle with diameter $AB$ satisfying the condition $\angle HKB=\angle C_{1}KB$. Let $M$ be the point of intersection of the line segment $A{A}_1$ and the circle with center $C$ and radius $CL$ where $KB\cap C{C}_1=\{L\}$. Let $P$ and $Q$ be the points of intersection of the line $C{C}_1$ and the circle with center $B$ and radius $BM$. Show that $A,K,P,Q$ are concyclic.