New Zealand Maths Olympiad Committee online » Notes

For your amusement (integer sequences)

Michael — Tue, 15 Dec 2009 01:59:10 +0000

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.

Michael

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

Bertrand’s Postulate

Michael — Tue, 18 Aug 2009 19:43:33 +0000

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.

Michael

Transformational geometry

Heather — Tue, 21 Apr 2009 09:50:36 +0000

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 nzmathsolymp@gmail.com.

Geometric inequalities

Heather — Tue, 21 Apr 2009 05:20:18 +0000

These notes by Arkadii Slinko are a gentle introduction to geometric inequalities, with many nice examples.

Solutions to some of the problems are available, and can be obtained by writing to nzmathsolymp@gmail.com.

Limits, continuity and completeness

Heather — Wed, 08 Apr 2009 09:09:12 +0000

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.)

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.

Polynomials, pure mathematics, Princeton Companion

Heather — Thu, 02 Apr 2009 00:23:49 +0000

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.

Perhaps you know (at least in theory) the formulae for roots of general cubics and quartics. Then you can carry out the above algorithm even if initially you can only see how to break down your polynomial into linear, quadratic, cubic and quartic factors. That certainly covers most of the polynomials you’ll ever encounter at school.

But then problems arise. The Fundamental Theorem of Algebra is the aforementioned factorization theorem. More precisely, it states that every polynomial with integer, rational, real, and, importantly, even complex coefficients has a unique decomposition into linear complex-coefficiented factors. Its proof is one of the coolest things you’ll learn in university complex analysis. On the other hand, Galois theory, one of the coolest things you’ll learn in university abstract algebra, shows that there’s no general formula for the roots of greater-than-fourth degree polynomials. The result is the kind of paradox that logicians love to hate: polynomials with roots that you know exist, but which you have no algorithm for constructing.

The two links in the paragraph above are to sample articles of the Princeton Companion to Mathematics, a recently-published “encyclopedia of modern pure mathematics” edited by mathematician, Fields medallist, British 1981 IMO perfect scorer, and blogger Timothy Gowers. More sample articles available here (log in with username Guest and password PCM, and click the link to Resources). Two others which I especially like are: a quick tour of fundamental definitions and structures (Section Four is a great introduction to the notions of limit and continuity, which, as an Olympiad hopeful, you may well have struggled through lately); and a discussion of how professional mathematicians go about problem-solving.

Olympiad mathematics is full of beautiful ideas, which show up all the better for being restricted to within a tightly-contained syllabus. Modern pure mathematics, as the Companion does an exhilarating job of showing, takes the beautiful ideas and lets them loose.

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…

Collinearity and concurrence

Heather — Sun, 22 Mar 2009 07:18:53 +0000

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.