Question 22
Let $O$ be the circumcenter of an acute triangle $ABC$ and $A_1$ be a point on the minor 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 $\angle BAA_1 = \angle OAC$ and $\angle CA_1A_3 = \angle OAB$. Show that the line $A_2A_3$ passes through the orthocenter of the triangle $ABC$.