# Math Olympiad training handouts

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

*a*± 1 (Solutions) - UK Trinity Training 2011^{n}

Working with expression of the form*a*± 1 and the exponent lifting lemma.^{n}- 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
- Djukić et al., The IMO Compendium (complete collection of IMO shortlist problems)