Math Olympiad training handouts

• Handouts from Canadian IMO Training camps
• Book recommendations

I have taught classes at various math olympiad training programs. Here are some of my handouts and training material.

If you don’t know where to start, I recommend Cyclic Quadrilaterals—The Big Picture and Three Lemmas in Geometry.

Algebra

• 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

Combinatorics

• Bijections
• 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

Geometry

• 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.

Number theory

• 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

Book recommendations

Here are some of my book recommendations for preparing for math competitions, in roughly increasing levels of difficulty.

Introductory

• 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

Advanced

• Engel, Problem Solving Strategies
• Andreescu and Enescu, Mathematical Olympiad Treasures
• Andreescu and Gelca, Mathematical Olympiad Challenges
• Andreescu and Dospinescu, Problems from the Book
• Andreescu and Dospinescu, Straight from the Book
• Djukić et al., The IMO Compendium (complete collection of IMO shortlist problems)