Interesting Geometry

Problem

I heard this former Putnam problem from a friend. The solution requires no advanced math, just clever thinking.

Given a sphere with five points placed at random on its surface. How many points can you guarantee will fall within a well-chosen closed hemisphere?

Solution

The answer is four. One great circle, which defines the boarder of the hemisphere, contains two points. This divides the surface of the sphere in two, one side containing two points, the other containing one. Choose the side which contains two points, and thus the closed hemisphere captures four points.

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