Math Olympiad teaching notes

Last updated 29 August 2013.


Note to high-school students

Three years ago, I and my Associate Department Head John Mackey, with the direct support of Carnegie Mellon President Jared Cohon and two significant alumni donors, launched a new “ultra-honors” talent incubator program for extremely motivated and advanced students. The curriculum is individually customized to suit each scholar's background and aspiration, and all students are personally mentored by myself and John.

If you enjoyed the high-school Olympiad competitions, and are interested in the possibility of joining our program, please feel free to contact me for further discussion. For example, if you were in my classes at the Math Olympiad Summer Program, or if I met you at an international competition, please let me know if you send us an application for undergraduate admission. I will try my best to help you.


2014 United States Math Olympiad Program

Starting this season, I will be the Leader for the United States delegation to the International Mathematical Olympiad, which will be held in 2014 in South Africa.

2013 United States Math Olympiad Program

I was the Deputy Team Leader for the United States at the 2013 International Mathematical Olympiad, in Santa Marta, Colombia. At the Math Olympiad Summer Program, I led an NSF-supported initiative to bridge the gap between Olympiad training and research mathematics, supervising fast-paced undergraduate research projects in combinatorics, in addition to teaching several courses to high-school students. Our research produced a paper, joint with Ronald Graham, Linus Hamilton, and Ariel Levavi, which we have submitted for publication. Lecture notes for the courses are below:

Topic Level
Graph theory    Red
Graph theory    Blue
Combinatorics of sets    Blue/Red
Extremal combinatorics    Red/Blue
Convexity    Red/Blue
Advanced methods in combinatorics    Black/Blue

2012 United States Math Olympiad Program

I was the Deputy Team Leader for the United States at the 2012 International Mathematical Olympiad, in Mar del Plata, Argentina. At the Math Olympiad Summer Program, I led an NSF-supported initiative to bridge the gap between Olympiad training and research mathematics, supervising fast-paced undergraduate research projects in combinatorics, in addition to teaching several courses to high-school students. Our research produced a paper, joint with Jenny Iglesias and Nate Ince, which we have submitted for publication. Lecture notes for the courses are below:

Topic Level
Pigeonhole principle    Red
Introductory graph theory    Red
Graph theory and designs    Blue
Applications of probabilistic and algebraic methods in combinatorics    Black

A strong combinatorics background came in handy on problem 3 of the IMO, which was the most challenging problem on Day 1 (and highlighted by Terry Tao on his blog). Team USA built up a substantial lead over all other countries on this problem, but lost the lead through the Euclidean geometry problem on Day 2.

2011 United States Math Olympiad Program

I was the Deputy Team Leader for the United States at the 2011 International Mathematical Olympiad, in Amsterdam, Netherlands. I returned to the Math Olympiad Summer Program for two weeks. This time, in addition to teaching several courses in Combinatorics, I also directed a new initiative (sponsored by a new grant from the National Science Foundation) to connect Olympiad mathematics with research mathematics. Lecture notes are below:

Topic Level
Graph theory    Red, Green, Blue
Algebraic methods in combinatorics    Black
Combinatorics of sets    Red, Green, Blue, Black
Probabilistic methods in combinatorics    Black

2010 United States Math Olympiad Program

I was the Deputy Team Leader for the United States at the 2010 International Mathematical Olympiad, in Astana, Kazakhstan, and the Team Leader at the 2010 Romanian Masters in Mathematics in Bucharest. I returned to the Math Olympiad Summer Program for a week, teaching several courses in Combinatorics. Lecture notes are below.

Topic Level
Pigeonhole principle    Red
Trees    Red, Green, Blue
Extremal arguments    Blue
Matching and planarity    Red, Green, Blue
Probabilistic methods in combinatorics    Blue


2009 United States Math Olympiad Summer Program

I returned as an Instructor for the week of June 14, to teach several courses in Combinatorics. Lecture notes are below. The three highlighted lectures include topics that I encountered during graduate school, which also illustrate techniques relevant to Olympiad problem solving.

Topic Level
Probabilistic methods in combinatorics    Blue, Black
Graph theory: introduction Red, Blue
Graph theory II Red, Blue
Algebraic methods in combinatorics Black
Extremal graph theory Red, Blue
Combinatorial gems Blue, Black

2008 United States Math Olympiad Summer Program

I returned as an Instructor for the week of June 23. Unfortunately, I did not have time time to stay for the entire program because I was concentrating on my Ph.D. research.

The two highlighted lectures introduce concepts and methods that I learned through my Ph.D. research with Benny Sudakov, and illustrate how these beautiful techniques from research mathematics are also useful in the context of Olympiad problem solving.

Topic Level
Collinearity and concurrence Red
Graph theory Red, Blue
Probabilistic methods in combinatorics    Black
Convexity (inequalities) Red, Blue
Algebraic methods in combinatorics Black

Useful references for some of the above topics:


2004 United States Math Olympiad Summer Program

I was the Deputy Team Leader for the United States at the 2004 International Mathematical Olympiad (Athens, Greece), and an Instructor at the Summer Program.

(I prepared fewer handouts compared to 2003 because I mostly lectured from the book A Path to Combinatorics for Undergraduates: Counting Strategies, by Titu Andreescu and Zuming Feng.)

2003 United States Math Olympiad Summer Program

This was the first year that I did a significant amount of teaching; I was a Junior Instructor. My notes are below.

2002 United States Math Olympiad Summer Program

Akamai made a very substantial gift to the national Math Olympiad program in 2002, enabling the centralized USAMO, and a vastly enlarged MOP (up to about 180 students, compared to around 30 the previous year). Many IMO alumni returned as first-time staff members that year. I came as a grader, but I volunteered several lectures because teaching was exhilarating.

2001 International Mathematical Olympiad (Washington, D.C., United States of America)

Former members of the USA IMO team were invited back to assist at IMO 2001, which was hosted in Washington, D.C. Older alumni served as coordinators. Several of us were still in college at the time, and came back as tour guides. I was the guide for Singapore.


1999 United States Math Olympiad Summer Program

I was wrong in 1998, and returned to MOP again. Since the USA Team Leader, Titu Andreescu, had close connections with the Romanian Math Olympiad movement, we also spent a week training with the Romanian team in Sinaia before the IMO, which was in Bucharest, Romania that year.

1998 United States Math Olympiad Summer Program

I returned as a veteran this year. Back then, students sometimes contributed lectures for fun, and I gave the last lecture on the last day, on an unusual technique that I had come up with to solve geometry problems (my favorite subject at the time). The handout is unedited, hence the high-schoolish humor.

(To put the first sentence on the handout in context: in the previous millenium, the “three-year rule” stated that if a student had already been to MOP twice, then he or she would only be invited to MOP again if he or she made the IMO team. I thought it would be a long shot to make the IMO team, and so I expected that this would be my first, last, and only lecture at the program.)

1997 United States Math Olympiad Summer Program

This was my introduction to the MOP world.


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