VMO 2008 Problem 2

Given a triangle with acute angle $\angle BEC,$ let $E$ be the midpoint of $AB.$ Point $M$ is chosen on the opposite ray of $EC$ such that $\angle BME = \angle ECA.$ Denote by $\theta$ the measure of angle $\angle BEC.$ Evaluate $\frac{MC}{AB}$ in terms of $\theta.$