Problem 3 : The Problem on 53rd Putnam Competition (1992)

Problem Description :

Four points are chosen at random on the surface of a sphere. What is the probability that the center of the sphere lies inside the tetrahedron whose vertices are at the four points? (It is understood that each point is in- dependently chosen relative to a uniform distribution on the sphere.)

Solution :

Well, I did get close to solving this in a good way, but a far more elegant mathematical solution with powerful and profound visualization is already there.