Question 91
Let $\triangle ABC$ be an acute triangle with circumcenter $O$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABO$ and $ACO$, respectively. Suppose that the circumcircle of triangle $AO_1O_2$ intersects line segment $BC$ at two distinct points $P$ and $Q$ (excluding the endpoints), with the four points $B, P, Q, C$ appearing in this order along the segment. Let $O_3$ be the circumcenter of triangle $OPQ$. Prove that the three points $A, O, $ and $O_3$ lie on a straight line.