BY-MO 2005 Problem 2

A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$ , so that $CN/BN = AC/BC = 2/1$ . The segments $CM$ and $AN$ meet at $O$ . Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$ . The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$ . Determine $\angle MTB$ .