21-295: Putnam Seminar (Fall 2012)

Po-Shen Loh

Last updated 1 December 2012.


We have doubled the number of official sections from 2 to 4, in order to maintain optimal class sizes. Meeting times and locations are:

Additional activities considered to be part of the course

Course description

Problem solving is an essential skill in every discipline, from mathematics to carpentry. This class seeks to develop that ability through challenging (but fun) problems which require some creativity to solve. These problems will generally come from mathematical competitions, and students will also have the opportunity to try their hands at two regional/national competitions, the VTRMC and the Putnam.

Yet although competitions are the title of this course, the syllabus will actually be constructed around general problem-solving techniques in mathematics. The instructor's aim is to use the competition problems to teach some mathematics which is not usually seen in ordinary classes, but is also beneficial to learn.

There is no official reference for this course, as much of the material will come from the instructor's experience in coaching the United States Math Olympiad team. Nevertheless, it should be mentioned that there will be some correlation with the book Putnam and Beyond, by Razvan Gelca and Titu Andreescu. Students are not required to purchase the book!

Levels and expectations

The aim of this class is to provide a comprehensive introduction to the various branches of mathematics which appear in the Putnam and related competitions. Each week will focus on a particular theme, and the teaching style will be midway between a pure lecture and a pure problem-solving session.

This year, we will offer 4 levels of the Putnam seminar, so that students can personalize their experience by selecting the one(s) that they benefit the most from. No prior math competition experience is required, and beginners are welcome. Indeed, this seminar seeks to complement the standard math curriculum by concentrating on the raw creative problem-solving skills which are essential for original work in almost every field. The levels are roughly divided as follows.


This year, we are very fortunate to have two graders for this course, who will provide feedback to students on weekly assignments. The main purpose of these assignments is to allow students to practice the art of writing mathematics efficiently. Since it can be difficult to solve problems on one's own, grading will be based on effort. Attendance at the weekly section meeting, together with submission of the assignment from the previous week (with either a full solution or ideas), will count for one weekly point. The final grade will be calculated through the following formula:

   Score = (number of weekly points) + (3 for the VTRMC) + (6 for the Putnam).

   Ratio = Score / MaximumPossible.

   MaximumPossible is 23 for Tue, 22 for Mon (due to Labor Day), and 22 for Thu/Fri (due to Thanksgiving).

Grades are then based on the Ratio, using the standard scale A = 90%+, B = 80%+, etc.

Detailed syllabus

Week 1 Proof by contradiction
Lecture / Wed
Week 2 Number theory
Sun / Lecture / Wed
Week 3 Polynomials
Sun / Lecture / Wed
Week 4 Calculus
Sun / Lecture / Wed
Week 5 Continuous functional equations
Sun / Lecture / Wed
Week 6 Inequalities
Sun / Lecture / Wed
Week 7 Convergence
Sun / Lecture / Wed
Week 8 Recursions
Sun / Lecture / Wed
Week 9 Linear algebra
Sun / Lecture / Wed
(Sat Oct 27)
Competition from 9:00am - 11:30am, in Scaife 125.
Week 10 Combinatorics
Lecture / Wed
Week 11 Geometry
Sun / Lecture / Wed
Week 12 Integer polynomials
Sun / Lecture / Wed
Week 13 Basic arguments
Sun / Lecture / Wed
Wed/Sun meetings will still run, with optional attendance, for students who stay at CMU for Thanksgiving.
Week 14 Higher math
Sun / Lecture / Wed
(Sat Dec 1)
Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Scaife 125.
Week 15 Beyond Putnam (optional, and Monday only)

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