IMO Shortlist 1960 problem 6

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Dodao/la: arhiva
2. travnja 2012.
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 be 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 diamter of the base of the cone at the vertex of the cone.
Izvor: Međunarodna matematička olimpijada, shortlist 1960