½ = 1/x² + 1/y² +...+ 1/z²

All variables must be different, positive integers, and there must be a finite number of terms.

I posed this problem to a bright freshman I met on campus, who found the following infinite solution in less than a minute:

Take all powers of 2, 3, and 5.

Note that the infinite geometric sequence 1/a^2, 1/a^4, 1/a^6, ... has sum 1/(a^2-1).

Thus the 2's have sum 1/3, the 3's have sum 1/8, and the 5's have sum 1/24.  The total sum is 1/3+1/8+1/24=1/2.

I was very impressed.