Problem Solving Seminar / Putnam Exam Workshop Math 491A

I taught the Putnam Seminar in 2007, 2008, 2009, 2010, 2011, and 2014. Although I no longer teach it, I hope this webpage from 2014 will be useful for future participants of this seminar.

The Putnam Exam is the most prestigious and challenging mathematical competition for undergraduate students in the United States and Canada. It is administered simultaneously on the first Saturday in December at approximately 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, meet like-minded people, and are considering a career as a mathematician, then consider taking the Putnam Exam..

To participate in the exam, you must reserve a seat in September, which is very early! Simply send an email to the professor scheduled to teach the weekly Putnam Exam preparation seminar, Math 491A, indicating your intention to take the exam.

We also invite you to attend Math 491A. The goal of this very informal workshop is to improve your problem-solving skills. 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, and how to use standard proof techniques and numerous tricks.

If you have never taken this class before, you will need 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 is mostly run by students themselves. Sometimes, we have joint events with the Math Club. Grades in 491A are assigned based on class participation. To earn an A, it is sufficient 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 started by 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 re-examined previous worksheets and focussed 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 had a mock Putnam exam at 6:30pm on Monday, Nov 24 in LGRT 1234. There were 4 problems, the exam lasted 2 hours.

The Putnam competition took place on Saturday Dec 6 in LGRT 1234. There were two sessions: Session A 10am-1pm and Session B 3pm-6pm. All students took both sessions. Each session featured 6 problems. Students arrived 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 http://math.scu.edu/putnam/index.html.
Problems, solutions, and winners of recent competitions are available at http://kskedlaya.org/putnam-archive/.



Tips for the professor assigned to teach Math 491A.

You were assigned to coordinate the Putnam exam and teach Math 491A. Here's what you need to do:

  • Choose a 75-minute time slot for the class meeting.
  • Ask your colleagues to help advertise the Putnam competition and Math 491A in their classes.
  • You will receive a letter from the Putnam organizers early in September. By this time, you should have a list of students who are going to take the exam. Select three of them to represent 'Team UMass.' It's a good idea not to disclose to students whether they are on the team or not.
  • Consider organizing 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 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. Encourage students to present incomplete solutions or ideas for solutions so that you have something to `brainstorm.' Moderate the discussion, provide brief background if necessary (rarely), cut discussion that clearly goes nowhere (but very reluctantly, prefer to wait until the discussion dies out on its own), and sometimes suggest hints. Try to get as many students as possible to the blackboard in every class.
  • Include a mixture of easier and harder problems in every worksheet to keep seminars entertaining for all students. While most problems in the worksheets are not 'Putnam-hard,' some of them are. There is no expectation that you will 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 are taken from old Putnam exams, various books (including the textbook by Paul Zeitz), folklore, and from http://math.stanford.edu/~vakil/putnam07/ (with Ravi Vakil's permission). Most of the problems in these worksheets are not original. Many are taken from old Putnam exams, various books (including the textbook by Paul Zeitz), folklore, and from