Yufei Zhao

Olympiad training handouts

Here are some math olympiad training handouts that I've made.

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

• 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 $a^n \pm 1$ and the exponent lifting lemma.
• Modular arithmetic: Divisibility, Fermat, Euler, Wilson, residue classes, order - AwesomeMath 2007

IMO 2008

In 2008, I was the deputy leader of the Canadian team for the 49th International Mathematical Olympiad (IMO) in Madrid, Spain. Here are the webpages dedicated to training camps and team that year:

• Winter Camp: the week-long training camp at York University in January, 2008.
• Summer Camp: the two-week long training camp prior to the IMO.
• IMO Team: dedicated to the Canadian IMO 2008 team.

Canadian IMO Training website

My competition records as a participant

... in case you're curious.

• Putnam: 2006 Fellow; 2007 Seventh place; 2008 Fellow; 2009 Fellow
• IMO: 2004 Bronze; 2005 Gold; 2006 Silver
• USAMO: 2004 HM; 2005 Third Place; 2006 Third Place
• Canadian MO: 2004 First Place; 2005 Third Place; 2006 Second Place