Show me some pigeonhole problems I'm preparing myself to a combinatorics test. A part of it will concentrate on the pigeonhole principle. Thus, I need some hard to very hard problems in the subject to solve.
I would be thankful if you can send me links\books\or just a lone problem.

IF $\alpha>0$ is irrational, then the fractional parts of $n\alpha$ forms a dense subset of $[0,1]$. 
Given a positive integer $n>0$, Fibonacci numbers modulo $n$ is periodic. 
These  are very famous ones from number theory. There are proofs of these using pigeonhole principle. 
