Putnam sessions
This is the homepage for the DU study sessions for the 2019 William Lowell Putnam contest exam. It will contain links to the weekly problem sets we use, sites with old Putnam problems and solutions, and other study materials.

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Weekly problem sets
Frequently Asked Questions

Q: What is the Putnam exam?

A: The Putnam exam is the preeminent undergraduate mathematics exam in North America. It's organized by the Mathematical Association of America and is taken by over 4,000 participants at more than 500 colleges and universities. It is characterized by extremely interesting and challenging problems.

The exam is given on the first Saturday of December every year; it begins at 8 AM (on the west coast) and lasts until 4 PM. Participants work on one set of six problems for the first three hours, then there is a lunch break, and participants work on a second set of six problems for the last three hours.

Q: Am I eligible to take the Putnam exam?

A: Any student enrolled at DU who does yet not have an undergraduate degree can take the Putnam exam up to four times. So unless you've already taken the Putnam exam four times or have received a bachelor's degree, you are eligible. You don't have to have a particular major, grade point average, or anything.

Q: What goes on at the weekly sessions?

A: I hand out practice problems to students the week before (they are also available for download at this page). Students try to solve the problems at home, and then we have students present solutions to the problems in the practice sessions; participation from non-presenters is also important for making sure that solutions really are airtight.

Q: I've only taken math classes X, Y, and Z; can I still come to the meetings?

A: It would be a very good idea to have taken MATH 2200 (Proofs) before attempting these problems, since the entire Putnam exam is proof-based. In the past, students without this course have often had trouble formally expressing their ideas. If you want to attend the meetings and haven't had that course yet, feel free to send me an e-mail or come talk to me. That being said, the most important tools for the Putnam are problem solving, “thinking outside the box,” and the ability to clearly express mathematical thinking, and we will spend much of our time working on these skills. So, not having had any "high-level" math classes yet is not a problem at all.

Q: What topics do I need to know for the Putnam?

A: In theory, any mathematics which is covered in an undergraduate course could appear; this is part of what makes the exam so difficult. However, some topics appear more often than others. Here are some rough categories, ranked by usefulness for the exam.

  • Essential: All of high-school mathematics (algebra, geometry, trigonometry), some calculus (derivatives, integrals, limits, basic differential equations), and foundations (sets, induction, roots of polynomials, binomial coefficients, the AM/GM inequality)
  • Appear often: Number theory (primes, congruences/modular arithmetic), probability (either via counting or via integrals), linear algebra (matrix multiplication, determinants), recursively defined sequences
  • Appear occasionally, or can be helpful behind the scenes: Complex numbers, permutations, game theory, abstract algebra, generating functions
Q: How is the Putnam exam graded?

A: Each solution is graded on a 10-point scale. So, the maximum score on the exam is 120 points. However, very few people approach this score; in fact, the MEDIAN score on the Putnam exam is almost always 0 or 1 point! So, earning any points at all on the exam is an accomplishment.

The VAST majority of solutions are awarded either 0, 1, 2, 8, 9, or 10 points. Roughly speaking, only virtually perfect solutions receive substantial credit (8, 9, or 10 points); if a solution has any nontrivial flaws, it will almost certainly receive 0, 1, or 2 points. The grading is also quite strict; the Putnam graders' criteria for a mathematically rigorous proof may be harsher than what you've dealt with in mathematics courses.

Since writing clear and rigorous solutions is so crucial to earning points on the exam, I will be happy to provide feedback on any written solutions to our weekly problems; just submit any solution(s) you'd like to have graded to me at the weekly sessions and I'll return them by the next week.

Q: Where can I find some old Putnam problems to work on?

A: There are many sites which keep repositories of old problems (and solutions!). One of the largest is at the MAA American Mathematics Competition page.