USAMO 1987 Problem 4

$\mathrm{M}$ is the midpoint of $\mathrm{XY}$. The points $\mathrm{P}$ and $\mathrm{Q}$ lie on a line through $\mathrm{Y}$ on opposite sides of $\mathrm{Y}$, such that $|\mathrm{XQ}|=$ $2|\mathrm{MP}|$ and $|\mathrm{XY}| / 2<|\mathrm{MP}|<3|\mathrm{XY}| / 2$. For what value of $|\mathrm{PY}| /|\mathrm{QY}|$ is $|\mathrm{PQ}|$ a minimum?