2023 Putnam Seminar

Po-Shen Loh

Carnegie Mellon

Attendance Form

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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 the Putnam Competition.

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 2023.

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:

  • Wed (introductory 1) meets at 4:00 in Doherty A302, for 50 minutes. This is officially Section B.
  • Thu (intermediate) meets at 4:00 in Posner 146, for 50 minutes. This is officially Section C.
  • Fri (introductory 2) meets at 3:00 in Posner A35, for 50 minutes. This is officially Section A.
  • Fri (advanced) meets at 4:00 in GHC 4301, for 50 minutes. This is officially Section D.
  • Tue (problems 1-3) meets at 5:00 in Wean 5403 for 50 minutes. This is not a substitute for any of the four main sections of the course. The first meeting is Tue Sep 5.
  • Sun (problems 4-6) meets at 3:30 in Wean 4709, for 3 hours. The first meeting is Sun Sep 10.
  • 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 21-127/128 Concepts of Mathematics or 15-151 Mathematical Foundations for Computer Science are 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.

  • A (Fri 3:00) or B (Wed 4:00): 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.

  • C (Thu 4:00): 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.

  • D (Fri 4:00): Similar to C, but at ludicrous speed (we'll go to plaid).

  • Tue: 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) + (6 total for both halves of the Putnam).

   Ratio = Score / MaximumPossible.

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

Detailed syllabus

Week 1 Introduction
Week 2 Polynomials
No Sun / Lecture / Tue
Week 3 Number theory
Sun / Lecture / Tue
Week 4 Calculus
Sun / Lecture / Tue
No class Fri. Come to Wed/Thu sections if you'd like to see the material. Fri sections are not responsible for handing in future homework problems corresponding to this week's handout.
Week 5 Functional equations
Sun / Lecture / Tue
Sun Putnam is moving to Wean 4709 for rest of semester
Week 6 Inequalities
Sun / Lecture / Tue
Week 7 Convergence
Sun / Lecture / Tue
Break Fall Break
Week 8 Recursions
Sun / Lecture / Tue
Week 9 Linear algebra
Sun / Lecture / Tue
Week 10 Combinatorics
Sun / Lecture / Tue
Week 11 Integer polynomials
Sun / Lecture / Tue
Week 12 Thanksgiving Break
Week 13 Geometry
Sun / Lecture / Tue
(Sat Dec 2)
Competition from 10:00am - 1:00pm and 3:00pm - 6:00pm, in Baker A51.
Week 14 Beyond Putnam
No Sun / Lecture / No Tue

Course Policies

Late Work. There is no penalty for late homework, except that all homework must be submitted by two weeks after its original due date, but no later than the final class day of the semester. Any homework submitted after that date receives zero credit. However, students are strongly encouraged to submit homework on time, because this keeps the class in sync for healthy collaboration on homework, and homework is very helpful for exam preparation.

Re-grade Requests. Re-grade requests must be submitted in writing, within one week of receiving the graded assignment. They will be honored, but please note that it is possible for scores to decrease after a re-grade.

Accommodations for students with disabilities. If you have a disability and require accommodations, please contact Catherine Getchell, Director of Disability Resources, 412-268-6121, getchell@cmu.edu. If you have an accommodations letter from the Disability Resources office, I encourage you to discuss your accommodations and needs with me as early in the semester as possible. I will work with you to ensure that accommodations are provided as appropriate.

Statement on Student Wellness. As a student, you may experience a range of challenges that can interfere with learning, such as strained relationships, increased anxiety, substance use, feeling down, difficulty concentrating and/or lack of motivation. These mental health concerns or stressful events may diminish your academic performance and/or reduce your ability to participate in daily activities. CMU services are available, and treatment does work. You can learn more about confidential mental health services available on campus at: http://www.cmu.edu/counseling/. Support is always available (24/7) from Counseling and Psychological Services: 412-268-2922.

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