21-295: Putnam Seminar (Fall 2015)

Po-Shen Loh

Last updated 29 October 2015.

Course Assistants


The Putnam competition provides the excuse to run a problem-guided tour of mathematics, while also developing core problem-solving skills that enhance the ability to learn and use higher mathematics. The questions which appear on the handouts are specially selected to spark discussions about famous mathematical results.

Although this course is named after a contest, its purpose is not to breed competition. Rather, we seek to develop a collaborative team spirit among the community of sharp and motivated students who self-identify themselves by joining this course. The CMU Putnam seminar currently involves about 3% of the undergraduate student body, and provides a venue for this fellowship of scholars to gather on a weekly basis, with topics of discussion ranging from mathematics to career advice.

Finally, our scope is not limited to CMU. Long-term, we aim to develop the talent base in the Greater Pittsburgh Area through partnerships with other institutions. This year, we are happy to include a spectacular new University of Pittsburgh student in our meetings.


There will be 6 meeting days per week, to provide optimal class sizes for the many CMU students who are on our extended Putnam Team. 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 carefully-selected problems which simultaneously develop sophisticated problem-solving techniques and inspire discussions about more advanced mathematics. The instructor's aim is to use the competition problems to provide a tour through many interesting topics, and to expose a bridge to higher mathematics.

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 happen to 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 5 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, although familiarity with 21-127 Concepts of Mathematics is assumed. 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.


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 counts for 1/3 of a weekly point, and the other 2/3 comes from the assignment turned in that week (gaining credit from either full solutions or ideas). 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), 22 for Thu (due to Thanksgiving), and 21 for Fri (due to Mid-Semester Break and Thanksgiving).

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

Detailed syllabus

Week 1 Introduction
Lecture / Wed
Week 2 Polynomials
Sun / Lecture / Wed
Week 3 Number theory
Sun / Lecture / Wed
Wed problem session meets at 4:30 Mon Sep 14 in Wean 5310.
Week 4 Calculus
Sun / Lecture / Wed
Sun problem session meets at usual time and place.
Mon section (A) meets at 3:30 Fri Sep 18 in Porter 125C.
Tue section (B) meets at 5:30 Fri Sep 18 in Gates 4307.
Wed problem session meets at usual time and place.
Thu (C) and Fri (D) sections meet at 4:30 Mon Sep 28 in Wean 5403.
Week 5 Functional equations
Sun / Lecture / Wed
Week 6 Inequalities
Sun / Lecture / Wed
Week 7 Convergence
Sun / Lecture / Wed
Week 8 Recursions
Sun / Lecture / Wed
(Sat Oct 24)
Competition from 9:00am - 11:30am, in Scaife 125.
Week 9 Linear algebra
Lecture / Wed
Week 10 Combinatorics
Sun / Lecture / Wed
Week 11 Integer polynomials
Sun / Lecture / Wed
Week 12 Analysis
Sun / Lecture / Wed
Week 13 Bare hands
Sun / Lecture
Week 14 Geometry
Sun / Lecture / Wed
(Sat Dec 5)
Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Scaife 125.
Week 15 Beyond Putnam
Sun / Lecture / Wed

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