The course is intended for students with a strong interest and considerable experience in problem solving. Participation is by invitation only; most students will be selected on the basis of their applications to the seminar. Experience in problem-solving competitions is highly desirable, such as the Mathematical Olympiad from the student's native country. All students taking the seminar are required to take the William Lowell Putnam Mathematical Competition on the first Saturday in December. Much (but not all) of the seminar will be geared toward preparing for this Competition.

Schedule and Expectations

The seminar will hold two one-hour meetings each week (except for Institute holidays). In general, the second class each week will be devoted to a lecture on a topic useful for mathematical problem-solving. Two sets of problems will be passed out - a set directly related to the lectures, and a set of supplementary problems on a variety of topics. Students must hand in written solutions to six of these problems at the next class meeting. At least four of these solutions should come from the lecture-based problem set. During the first class of each week, students will present their solutions to the problems they are handing in that day.


Problem sets will be graded and returned. The class is Pass-Fail only, and anyone making a reasonable effort will pass. There will be problems at all levels of difficulty, so no student should be unable to solve close to six problems each week. Some class participation is required of every student; in particular, every student must occasionally present a solution to the rest of the class.


1 The Pigeonhole principle
2 Probability theory
3 Congruences and divisibility
4 Generating functions
5 Integer part
6 Inequalities
7 Zeros of polynomials
8 Putnam practice
9 Hidden independence
10 Putnam practice (cont.)
11 Limit problems

Putnam practice (cont.)

Linear algebra problems