International Mathematical Olympiad 1960 Problem 6

Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let $V_{1}$ be the volume of the cone and $V_{2}$ the volume of the cylinder.

(a) Prove that $V_{1}\neq V_{2}$.

(b) Find the smallest number $k$ for which $V_{1}=kV_{2}$, for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.