New Zealand Maths Olympiad Committee online » Combinatorics

Notes on discrete and polyhedral geometry

Michael — Thu, 27 Aug 2009 21:04:43 +0000

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.

Michael

Enumeration

Chris — Tue, 07 Apr 2009 23:56:40 +0000

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.

Induction

Chris — Fri, 27 Mar 2009 02:40:43 +0000

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…

The Principle of Inclusion-Exclusion

Chris — Mon, 02 Feb 2009 04:25:53 +0000

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.

Recurrence relations

Chris — Tue, 27 Jan 2009 04:06:35 +0000

Some notes and problems on finding and solving recurrence relations. Read these if you’ve ever wondered how to find a formula for the Fibonacci sequence!

The Seven Colour Theorem

Michael — Wed, 14 Jan 2009 04:29:34 +0000

Here are the slides from Chris Tuffley’s talk on the seven colour theorem at the 2009 camp. These weren’t really intended as a stand alone document, so you’ll have to fill in some of the gaps yourselves!

Basic Counting Principles

Michael — Mon, 12 Jan 2009 22:42:28 +0000

Notes on the basic rules of counting, used at various camps.

Basic graph theory

Michael — Mon, 12 Jan 2009 22:39:37 +0000

Introductory notes on graph theory handed out at the 2009 camp.

Pigeon Hole Principle

Michael — Mon, 12 Jan 2009 22:36:28 +0000

Some notes on the pigeonhole principle, one of the most fundamental and useful results from discrete mathematics.