OIM 2009 Problem 3

Let $C_1$ and $C_2$ be two congruent circles centered at $O_1$ and $O_2$ , which intersect at $A$ and $B$ . Take a point $P$ on the arc $AB$ of $C_2$ which is contained in $C_1$ . $AP$ meets $C_1$ at $C$ , $CB$ meets $C_2$ at $D$ and the bisector of $\angle CAD$ intersects $C_1$ and $C_2$ at $E$ and $L$ , respectively. Let $F$ be the symmetric point of $D$ with respect to the midpoint of $PE$ . Prove that there exists a point $X$ satisfying $\angle XFL = \angle XDC = 30^\circ$ and $CX = O_1O_2$ . Author: Arnoldo Aguilar (El Salvador)