I recently stumbled upon some math questions and spent the next several hours fiddling with them. My favorite of the set was the following:
Suppose f(n) is the number of 1's in the binary representation of the integer n. Compute the f(1)/(1*2) + f(2)/(2*3) + f(3)/(3*4) + ....

While that question is pitched at advanced undergraduate math majors, here's another that's a bit easier:
Suppose five distinct points are placed on the surface of a sphere. Show that four of them lie on some hemisphere.

One more:
Suppose you are given an ordered sequence of n whole numbers. Show that some consecutive subsequence has sum divisible by n. (Eg. given 1, 4, 2, we see that 4+2 = 6 is divisible by 3.)

Now my question: does anyone else have any similarly interesting problems they'd like to share?