Resources

Problem Solving Tactics

Every aspiring olympiad mathematician should own a copy of PST.

Derek Holton's Mathematical Olympiad Series

This book series could be very useful to some of you - I would recommend getting the one at the right level for you rather than all of them. A few libraries in New Zealand hold these books - just search in the national library catalogue.

Evan Chen's Notes

Evan has amassed a nice collection of lecture notes aimed at students training for the IMO.

A Mathematical Olympiad Primer

Geoff Smith was the president of the IMO board for 10 years, so is intimately familiar with the IMO, and has some excellent advice.

Number theory

At least a couple of good, comprehensive introductions to elementary number theory are available online. These ones by Naoki Sato are a bit more Olympiad-focused.

Chrystal’s Algebra

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.

Discrete and polyhedral geometry

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 Princeton Companion to Mathematics

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.

Useful list for 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.

Recommended Mathematics Literature

This site is now fairly old, but still has some good recommendations for mathematics textbooks, puzzle books and other good reading.

Integer sequences

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.

The Geometry Junkyard

David Eppstein has put together an interesting collection of random things related to geometry. I guarentee you will find something interesting.

Gusty Games

Our very own Ross Atkins has an archive of programming workshops on this website. There are also some fun and interesting mathematical challenges in the "apps" section, as well as a couple of puzzle apps published to the Android and iOS stores.