IMO 2009 Problem 4

Let $\triangle ABC$ be a triangle with $AB = AC$. The angle bisectors of $\angle CAB$ and $\angle ABC$ meet the sides $BC$ and $CA$ at $D$ and $E$, respectively. Let $K$ be the incenter of triangle $ADC$. Suppose that $\angle BEK = 45^\circ$. Find all possible values of $\angle CAB$.

(This problem was suggested by Peter Vandendriessche and Jan Vonk from Belgium and by Hojoo Lee from South Korea.)