2005 OIM Problems/Problem 5

Problem

Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ be a point on the smallest arc $BC$ of the circumcircle of the triangle $ABC$ . Let $A_2$ and $A_3$ be points on the sides $AB$ and $AC$ respectively such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$ . Show that The line $A_2A_3$ passes through the orthocenter of the triangle $ABC$ .