Neil Sloane, who is the father of the encyclopedia of integer sequences, has produced a paper, which highlights seven of them. There are no problems to solve there (other than some hard open ones!), but lots of interesting stuff. For example, you might like to try and prove the following result before reading it:
Consider the sequence defined as follows a(1) = 1, a(2) = 2, and for n >= 3, a(n) is the smallest positive integer not yet in the sequence such that gcd(a(n), a(n-1)) > 1. Prove that every positive integer eventually appears in the sequence.
Problems in discrete geometry (i.e. the border between combinatorics and geometry) have made several recent appearances on the IMO (typically, granted as Q6). A nice book by Igor Pak covers much of the important material in this area — the book is aimed at undergaduate and graduate students in maths, but the first few chapters in particular are suitable for Olympiad level students.
As in many cases, the important thing is not the results themselves (though Helly’s theorem is a useful tool in lots of setting) but the “style” of proofs in this area.
The fact that, for every positive integer n, there is a prime between n and 2n is known as Bertrand’s postulate (which is a bit odd, as it’s a theorem, but anyhow …) It arises occasionally in Olympiad style problems (usually with the note “You may assume Bertrand’s Postulate that …”) Michael Nielsen has a nice post giving an elementary proof at the Polymath wiki.
Here are the notes from Arkadii Slinko’s Auckland squad lecture last weekend. They present solutions to ten geometric problems — some from contests, some classical. The common theme is the use of geometric transformations.
Solutions to some of the problems are available, and can be obtained by writing to email@example.com.
Occasionally, in contest problems, it helps to have a careful understanding of real numbers and real-valued functions. But what, exactly, is a real number? These Auckland squad training lecture notes by Heather Macbeth outline some basics.
(Update, 19/4/09: several errors fixed.)