New Zealand Maths Olympiad Committee online

Here are some scanned notes by Michael on enumeration – the subtle art of counting. They were originally used for a course at Carnegie Mellon, and expand on the ideas in Basic Counting Principles.

The contents pages list a chapter on graph theory; unfortunately this chapter is not available at present.

Round about sixth form one learns that every polynomial can be factorized, as a product of linear factors.  Why?  Well, here’s a polynomial, see.  It’s probably a cubic with integer coefficients — after all, most nontrivial polynomials that one encounters are.  You play with it until you discover a root, likely by looking at integer factors of the highest and lowest coefficients.  Then you polynomial-divide through by the linear factor which that root gives you, and get a quadratic, whose roots there’s a formula for finding.  Tada!

Of course, there’s a problem with this algorithm:  it depends on figuring out how to break down your polynomial into only linear and quadratic factors.

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Induction is a powerful tool for proving that some result or formula is true for all natural numbers n, without resorting to handwaving or saying “and so on.” These notes by Chris Tuffley outline induction in its various forms – and explain just what it has to do with dominoes…

These geometry notes by Heather Macbeth come from the last month’s Auckland olympiad squad training.  They cover some techniques for proving collinearity and concurrence.  Along the way they prove all your favourite triangle geometry theorems, and do some cool things with homotheties and reflections.

(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.

In Basic Counting Principles you learnt how to find the size of a union of two sets. The powerful
Principle of Inclusion-Exclusion tells us how to generalise this formula to a union of arbitrarily many sets.