TW-TST 2001 Problem 4

Let $\Gamma$ be the circumcircle of a fixed triangle $ABC$ , and let $M$ and $N$ be the midpoints of the arcs $BC$ and $CA$ , respectively. For any point $X$ on the arc $AB$ , let $O_1$ and $O_2$ be the incenters of $\vartriangle XAC$ and $\vartriangle XBC$ , and let the circumcircle of $\vartriangle XO_1O_2$ intersect $\Gamma$ at $X$ and $Q$ . Prove that triangles $QNO_1$ and $QMO_2$ are similar, and find all possible locations of point $Q$ .