Past NZMO Problems
Problems in discrete geometry (i.e. the border between combinatorics and geometry) have made several appearances on the IMO. 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. It arises occasionally in Olympiad style problems (usually with the note “You may assume Bertrand’s Postulate …”). Michael Nielsen has a nice post giving an elementary proof at the Polymath wiki.
At least a couple of good, comprehensive introductions to elementary number theory are available online. These notes by Jim Hefferon and W. Edwin Clark are nicely written and gently-paced. These ones by Naoki Sato are a bit more Olympiad-focused.
A classic (pre 1900) textbook on algebra by Chrystal has been scanned and made available electronically. Volume 1 is not likely to be of much interest for training purposes but chapters 23 and 24 (on combinatorics), 32 through 34 (continued fractions), and 35 (number theory), of volume 2 are.
The Princeton Companion to Mathematics, an “encyclopedia of modern pure mathematics” is a fantastic book, edited by mathematician Timothy Gowers. It touches on a lot of topics, some of which are quite advanced, however section 1.3 on fundamental definitions and structures is very useful, in particular the parts about limits and continuity, and section 5.13 is a good overview of the Fundamental Theorem of Algebra. You may also find some useful notes on how professional mathematicians go about problem-solving.
Having the right mental habits is one of the keys to being a successful problem solver. A really good list was posted at a math education blog. The list is supposedly for sixth grade (i.e. year six) students, but is much more universal than that.
The site Abstract Math by Charles Wells presents some useful insights on writing and doing proofs. The author’s opinions in some areas are arguable (so you should read and think about them, rather than follow the advice verbatum). For instance, I wouldn’t recommend the “Languages of Math” section which is far too detailed at present. However, there are little gems to be found everywhere, and from within that section comes the link to Timothy Gowers article, “The language and grammar of mathematics.”. By the way, the Tricki is alive, although small and incomplete, but it is still a great resource.
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.