Here's another one:

Say you have a set S of sets, where each subset has size N. What is the maximum size of S for a given N such that for all pairs of sets in S, they share exactly one element? My brother and I have figured out that the upper bound is N * (N - 1) + 1, and have verified that that is the case for N = 1, 2, and 3, but haven't found a way to show that that is the answer.

EDIT: Read below, this isn't quite perfect.