2016 OIM Problems/Problem 3

Problem

Let $ABC$ be an acute triangle whose circumcircle is $\Gamma$ . The tangents to $\Gamma$ through $B$ and $C$ intersect at $P$ . On the arc $AC$ that does not contain $B$ , we get a point $M$ , different from $A$ and $C$ , such that the line $AM$ cuts the line $BC$ at $K$ . Let $R$ be the symmetric point of $P$ with respect to the line $AM$ , and $Q$ the point of intersection of the lines $RA$ and $PM$ . Let $J$ be the midpoint of $BC$ and $L$ be the point where the line parallel by $A$ to the line $PR$ intersects the line $PJ$ . Show that the points $L, J, A, Q$ and $K$ are on the same circle.