Problem Solving Seminar / Putnam Exam Workshop Math 491A (Prof. Jenia Tevelev)

The Putnam Exam is the most prestigious and difficult mathematical competition for undergraduate students in the United States and Canada. It is administered simultaneously on the first Saturday in December at about 500 participating colleges. There is no fee to take the exam and its results are kept confidential. The test consists of two three-hour sessions separated by a lunch break. If you enjoy problem solving, want to challenge yourself and meet like-minded people, think about a career of a mathematician, consider taking the Putnam Exam.

To participate in the Exam, you have to reserve a seat in SEPTEMBER (i.e. very early!) by simply e-mailing Prof. Tevelev that you want to take it.

We also invite you to attend a weekly Putnam Exam preparation seminar Math 491A. The goal of this very informal workshop is to improve your problem-solving skills. The worksheets are distributed weekly, and in class we have (sometimes very heated!) discussions of various approaches to these problems. We learn how to start solving a problem, how to separate worthy ideas from dead ends, how to use standard proof techniques and numerous tricks.

If you have never taken this class before, you will have to enroll on SPIRE (for 1 credit). If you have taken this class before, you can participate without registering. The class is very informal and run mostly by students themselves. Sometimes we have joint events with the Math Club. The grade in 491A is assigned based on class participation: to get an A, it suffices to attend every week, be active in class, and take the Putnam Exam in December.

The (optional) textbook for this class is The Art and Craft of Problem Solving by Paul Zeitz (any edition will work).

Worksheets from 2014

On September 3 we are going to start brainstorming some problems
Mathematical Induction, due September 10
Games and Invariants, due September 17
Geometry, Vectors, and Complex Numbers, due September 24
Graph Theory, due October 1
Probability, due October 8
On October 15 we are going to re-examine previous worksheets. Focus on harder problems.
Linear Algebra, due October 22
Number Theory, due October 29
Real Analysis, due November 5
Polynomials, due November 19 (no class on November 12 - Tuesday schedule)
Combinatorics, due December 3 (no class on November 26 - Thanksgiving)

We will have a mock Putnam exam at 6:30pm on Monday, Nov 24 in LGRT 1234. There will be 4 problems, the exam will last 2 hours.

The Putnam competition will take place on Saturday Dec 6 in LGRT 1234. There will be two sessions: Session A 10am-1pm and Session B 3pm-6pm. All students should take both sessions. Each session will feature 6 problems. I suggest that you arrive to both sessions 10 minutes early to fill the paperwork.

Bonus worksheets

Inequalities (2011)
Real Analysis (2008)
Number Theory (2011)
Number Theory (2007)
Algebra and Groups (2009)
Polynomials (2010)
Algebraic Techniques (2007)
Geometry, Vectors, Complex Numbers (2009)
Games (2010)
Invariants and Games (2008)
Invariants and Games (2009)
Fibonacci and other sequences (2010)
Induction and Pigeonhole principle (2010)
Pigeonhole principle (2007)
Brainstorming problems (for the first class) (2010)
Generating Functions (2009)

You can find out more about the history of the Putnam Competition at
Problems, solutions, and winners of recent competitions are available at

Tips for the prof.

I was assigned to coordinate the Putnam exam and teach Math 491A. What do I do?

  • Choose a 75 minute time slot for the class meeting.
  • Ask your colleagues to advertise the Putnam competition and Math 491A in their classes.
  • You will receive a letter from the Putnam organizers very early (in September). By this time you should have a list of students who are going to take the exam. You should also select three of them to represent "Team UMass". I don't disclose to students whether they are on the team or not.
  • A fun thing to do is to have a joint party with other Putnam coordinators at Five Colleges after the exam.
  • Make arrangements for a mock Putnam exam.
  • Don't forget to reserve a classroom for the Putnam Exam.
  • Here's how I would run Math 491A. Give students a week to solve worksheet problems, then discuss them in class (for the first class I would usually prepare a short "brainstorming" worksheet). I almost never go to the blackboard myself and let students volunteer. Most students enjoy going to the blackboard. To make the experience less intimidating, arrange the chairs in a semi-circle. I often encourage students to present incomplete solutions or ideas for solutions so that we have something to "brainstorm". I would moderate the discussion, provide brief background if necessary (rarely), cut discussion which clearly goes nowhere (but very reluctantly, I prefer to wait until the discussion dies out on its own), and sometimes suggest hints. I also try to get as many students as possible to the blackboard in every class.
  • Try to include a mixture of easier and harder problems in every worksheet to keep seminars entertaining to all students. Most of the problems in my worksheets are not "Putnam-hard" but some of them are. Like in real math, there is no expectation that we are going to solve every problem every time. I prefer not to discuss problems unless at least some of the students worked on them.
  • Most of the problems in these worksheets are not original. Many of them are taken from the old Putnam exams, from various books (including the textbook by Paul Zeitz), from folklore, and from (with Ravi Vakil's permission). Feel free to use these worksheets in your class but please provide a similar copyright disclaimer.