Hem A Sphere. Topic: Geometry/Logic. Level: AIME.

Problem: (2002 Putnam - A2) Given any five points on a sphere, show that some four of them must lie on a closed hemisphere.

Solution: Well, that doesn't sound so bad. And, in fact, it really isn't. First, let's maximize the number of points on the boundary of a great circle; this is, in general, two. We can't arbitrarily choose three points and always draw a great circle through all three (think about it).

So now we have a great circle through two points. Well, maybe these hemispheres work. There are three points left, so by the Pigeonhole Principle one of the closed hemispheres has at least four points. Whoa, we're done. QED.

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Comment: That was almost too simple. Especially for a Putnam problem; shows you how knowing how to think can take you a long ways even if you don't know much math.

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Practice Problem: Can you generalize this to an $ n $-dimensional sphere?