MathNet 2011 Question 65

Let $ABC$ be acute. Choose $P,Q$ on the extensions of altitudes $BB_1,CC_1$ beyond $B_1,C_1$, respectively, so that $\angle PAQ=90^\circ$. Let $AF$ be the altitude of triangle $APQ$. Prove $\angle BFC=90^\circ$. (A. Polyanskiy)