Question 28
Let $ABC$ be an acute-angled triangle whose circumcircle is $\Gamma$. The tangents to $\Gamma$ at $B$ and $C$ meet at $P$. On the arc $AC$ not containing $B$ a point $M$, different from $A$ and $C$, is taken such that the line $AM$ meets the line $BC$ at $K$. Let $R$ be the reflection of $P$ in the line $AM$, and let $Q$ be the intersection point of the lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and let $L$ be the point where the line through $A$ parallel to the line $PR$ meets the line $PJ$. Prove that the points $L$, $J$, $A$, $Q$ and $K$ lie on one circle.