OIM 2005 Problem 5

Let $O$ be the circumcenter of acutangle triangle $ABC$ and let $A_1$ be some point in the smallest arc $BC$ of the circumcircle of $ABC$ . Let $A_2$ and $A_3$ points on sides $AB$ and $AC$ , respectively, such that $\angle BA_1A_2 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$ .

Prove that the line $A_2A_3$ passes through the orthocenter of $ABC$ .