USAMO 2005 — Days 1 and 2 Question 3

Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on side $BC$. Construct a point $C_1$ in such a way that the convex quadrilateral $APBC_1$ is cyclic, $\overline{QC_1} \parallel \overline{CA}$, and $C_1$ and $Q$ lie on opposite sides of line $AB$. Construct a point $B_1$ in such a way that the convex quadrilateral $APCB_1$ is cyclic, $\overline{QB_1} \parallel \overline{BA}$, and $B_1$ and $Q$ lie on opposite sides of line $AC$. Prove that the points $B_1$, $C_1$, $P$, and $Q$ lie on a circle.

Proposed by Zuming Feng (credited in the supplied source).