Math Olympiad training handouts
I have taught classes at various math olympiad training programs. Here are some of my handouts and training material.
- Integer Polynomials - MOP 2007 Black group
Integer polynomials, including various irreducibility criteria.
- Inequalities - Canadian 2008 Winter Training
Contains a short essay discussing the IMO 2001 inequality.
- Polynomials - Canadian 2008 Summer Training
Advanced techniques in polynomials. Roots of unity, integer divisibility, intermediate value theorem, Lagrange interpolation, Chebyshev polynomials, irreducibility criteria, and Rouché’s theorem.
- Determinants: Evaluation and Manipulation - MIT UMA Putnam Talk
- Linear algebra tricks for the Putnam - MIT UMA Putnam Talk
- Algebraic Techniques in Combinatorics - MOP 2007 Black Group
Applications of linear algebra and posets to olympiad-style combinatorics problems.
- Tiling - MOP 2007 Blue group
Discussion of tiling boxes with bricks. Contains many coloring and tiling problems.
- Counting in Two Ways - MOP 2007 Blue and Black group
- Combinatorics: bijections, catalan numbers, counting in two ways - Canadian 2008 Winter Training
- Combinatorics: pigeonhole principle, coloring, binomial coefficients, bijections - AwesomeMath 2007
- Combinatorics: counting in two ways, generating functions, algebraic combinatorics - AwesomeMath 2007
- Lemmas in Euclidean Geometry - Canadian 2007 Summer Training
A collection of commonly occuring configurations in geometry problems.
- Cyclic Quadrilaterals – The Big Picture - Canadian 2009 Winter Training
Explores many properties of the complete cyclic quadrilateral and its Miquel point, and also discusses several useful geometric techniques.
- Three Lemmas in Geometry (Solutions) - Canadian 2010 Winter Training
- Power of a Point (Solutions) - UK Trinity Training 2011 (Mint group)
- Circles - Canadian 2008 Summer Training
Contains a section on a particular tangent circle configuration, and another section on projective geometry, poles and polars. Here’s some additional food for thought.
- Similarity - Canadian 2007 Summer Training
Applications of similar triangles and spiral similarity.
- an ± 1 (Solutions) - UK Trinity Training 2011
Working with expression of the form an ± 1 and the exponent lifting lemma.
- Modular arithmetic: Divisibility, Fermat, Euler, Wilson, residue classes, order - AwesomeMath 2007
Here are some of my book recommendations for preparing for math competitions, in roughly increasing levels of difficulty.
- Lehoczky and Rusczyk, The Art of Problem Solving, Volume 1: the Basics
- Lehoczky and Rusczyk, The Art of Problem Solving, Volume 2: and Beyond
- Zeitz, The Art and Craft of Problem Solving