ILL 1972 Problem 17

A solid right circular cylinder with height $h$ and base-radius $r$ has a solid hemisphere of radius $r$ resting upon it. The center of the hemisphere $O$ is on the axis of the cylinder. Let $P$ be any point on the surface of the hemisphere and $Q$ the point on the base circle of the cylinder that is furthest from $P$ (measuring along the surface of the combined solid). A string is stretched over the surface from $P$ to $Q$ so as to be as short as possible. Show that if the string is not in a plane, the straight line $PO$ when produced cuts the curved surface of the cylinder.