TITLE: Interesting applications of the pigeonhole principle
QUESTION [51 upvotes]: I'm a little late in realizing it, but today is Pigeonhole Day. Festivities include thinking about awesome applications of the Pigeonhole Principle. So let's come up with some. As always with these kinds of questions, please only post one answer per post so that it's easy for people to vote on them.
Allow me to start with an example:

Brouwer's fixed point theorem can be proved with the Pigeonhole Principle via Sperner's lemma. There's a proof in Proofs from The Book (unfortunately, the google books preview is missing page 148)

By the way, if you happen to be in Evans at Berkeley today, come play musical chairs at tea!

REPLY [4 votes]: I think the solutions of these questions are very interesting (by using pigeon-hole principle), first question is easy, but second question is more advanced:
1) For any integer $n$, There are infinite integer numbers with digits only $0$ and $1$ where 
they are divisible to $n$. 
2) For any sequence $s=a_1a_2\cdots a_n$, there is at least one $k$, such that $2^k$ begin with $s$.