BG-MO 1983 Problem 3

A regular triangular pyramid $ABCD$ with the base side $AB=a$ and the lateral edge $AD=b$ is given. Let $M$ and $N$ be the midpoints of $AB$ and $CD$ respectively. A line $\alpha$ through $MN$ intersects the edges $AD$ and $BC$ at $P$ and $Q$ , respectively.

(a) Prove that $AP/AD=BQ/BC$ . (b) Find the ratio $AP/AD$ which minimizes the area of $MQNP$ .