KMO 2012 Problem 1

Let $ABC$ be an obtuse triangle with $\angle A > 90^{\circ}$ . Let circle $O$ be the circumcircle of $ABC$ . $D$ is a point lying on segment $AB$ such that $AD = AC$ . Let $AK$ be the diameter of circle $O$ . Two lines $AK$ and $CD$ meet at $L$ . A circle passing through $D, K, L$ meets with circle $O$ at $P ( \ne K )$ . Given that $AK = 2, \angle BCD = \angle BAP = 10^{\circ}$ , prove that $DP = \sin ( \frac{ \angle A}{2} )$ .