ÖMO 2011 Problem 6

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$ . The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$ . We choose two points $A$ and $B$ , one on each of the arcs $PQ$ of $k_1$ . ( $PBQA$ is a convex quadrangle.) Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$ . The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ . Show that there is a point $Z$ that lies on all of these lines $UV$ .