2016 Putnam Seminar

Po-Shen Loh

Carnegie Mellon


The Mathematical Association of America's William Lowell Putnam Mathematical 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 Carnegie Mellon University 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.

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. 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. Much of the material will come from the instructor's experience in coaching the United States Math Olympiad team (also organized by the Mathematical Association of America).

Course Number

  • 21-295 Putnam Seminar, Main Campus, Fall 2016.

Course Assistants


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:

  • Mon (introductory 1) meets at 3:30 in Porter 125C, for 1 hour. This is officially Section A.
  • Tue (introductory 2) meets at 4:30 in Baker 136A, for 1 hour. This is officially Section B.
  • Wed (problems 1-3) meets at 4:30 in Wean 5310 for 1 hour. This is not a substitute for any of the four main sections of the course.
  • Thu (intermediate) meets at 4:30 in Wean 5403, for 1 hour. This is officially Section C.
  • Fri (advanced) meets at 4:30 in Wean 5310, for 1 hour. This is officially Section D.
  • Sun (problems 4-6) meets at 3:30 in Porter A19C, for 3 hours. The first meeting is Sun Sep 6.
  • Office hours: Any time by appointment.

Additional activities considered to be part of the course

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.

  • Mon or Tue: For students who are new to problem solving. This is meant to provide a broad introduction to the area, and to inspire excitement. It will be taught in lecture format, with participation encouraged.

    Learning goals: By the end of the course, students should develop fundamental problem solving skills, and become accustomed to concentrating on a problem for an extended period of time. Students should also be able to recognize when proofs are written with sufficient rigor, and should gather a greater appreciation for mathematics as a broad field.

  • Thu: For students who have prior experience, and would like to sharpen their ability to discover new multi-step arguments. The content will overlap substantially with the other sections, but the style will be student-driven. Specifically, instead of following a traditional lecture-based format, these groups will advance by having students collectively brainstorm solutions with the instructor's guidance. Consequently, active participation is required.

    Learning goals: to increase ability to solve Putnam problems, and to communicate solutions effectively.

  • Fri: Similar to Thursday, but at ludicrous speed (we'll go to plaid).

  • Wed: Independent problem-solving session, with personalized hints. The instructor constantly moves around the room, observing students' progress, and essentially conducts a number of independent, parallel tutoring sessions at once. (This is most similar to the style he uses at the Math Olympiad Summer Program.) Students do not write up solutions during the hour-long meeting, focusing instead on raw problem solving.

    Learning goals: to master questions at the Putnam 1-3 level.

  • Sun: Independent problem-solving session, without hints. Students will write up solutions during the session.

    Learning goals: to master questions at the Putnam 4-6 level.


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
Thu (C) and Fri (D) meet together at 4:30 Thu Sep 15 in Hammerschlag B131.
Week 4 Calculus
Sun / Lecture / Wed
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 22)
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 Probability
Sun / Lecture / Wed
Week 13 Bare hands
Sun / Lecture
Week 14 Geometry
Sun / Lecture / Wed
(Sat Dec 3)
Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Scaife 125.
Week 15 No class

You are visitor number since 27 August 2016.