New Zealand Maths Olympiad Committee online

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

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.

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.

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.

These notes by Heather Macbeth discuss a useful criterion for the orthogonality of two line segments.  They’ll also give you some practice using vectors for geometry.