PL-OM 2007 Problem 1

In acute triangle $ABC$ point $O$ is circumcenter, segment $CD$ is a height, point $E$ lies on side $AB$ and point $M$ is a midpoint of $CE$ . Line through $M$ perpendicular to $OM$ cuts lines $AC$ and $BC$ respectively in $K$ , $L$ . Prove that $\frac{LM}{MK}=\frac{AD}{DB}$