# Putnam Competition

#### What is the putnam competition?

The William Lowell Putnam Competition is the preeminent undergraduate mathematics competition in North America. It is organised by the Mathematical Association of America (MAA) and is taken by over 4,000 participants at more than 500 colleges and universities. The most common motivation for taking the Putnam Competition exam is the challenge of attacking extremely difficult math problems that would not be seen in the typical math curriculum.

The exam is given on the first Saturday of December every year from 8am to 4pm. UBC participants sit the exam on campus.

The competition consists of two 3-hour sessions, one in the morning and one in the afternoon. During each session, participants work on 6 challenging mathematical problems.

#### Prizes

Monetary prizes are awarded to the participants with the highest scores and to the departments of mathematics of the five institutions whose teams obtain the highest rankings. Prizes for the top five teams range from 5,000 - 25,000. Cash prizes for the top five students, honored as Putnam Fellows, are 2,500 each and other top students also receive prizes for their performance on the Putnam Mathematical Competition.

The Elizabeth Lowell Putnam Prize was established in 1992 to be "awarded periodically to a woman whose performance on the Competition has been deemed particularly meritorious." Recent winners of the Elizabeth Lowell Putnam Prize have received 1,000.

#### Eligibility

Any student enrolled at UBC, who does yet not have an undergraduate degree, can enter the competition up to four times. There are no other eligibility requirements, including a specific major or average grade point.

#### Apply to enter

https://personal.math.ubc.ca/~dghioca/Putnam/Putnam.html

#### FAQ

Enter as a team or an individual?

It is primarily an individual contest, and every participant solves the problems on their own, with no collaboration. After grading, the top 3 scorers from each university are selected by the
Putnam organizers automatically as being the team for that particular university.

How is the Putnam exam graded? What is the “Gap of Death”?

Each of the twelve problems is marked out of 10 points. Partial credit is given; however, the graders are instructed to assign only 0, 1, 2, 8, 9, or 10 as possible scores for each problem. As a result, only virtually perfect solutions receive 8 or more points; flawed solutions fall below the “Gap of Death” between 3 and 7 points, ending up with at most 2 points. Furthermore, what constitutes partial progress towards a solution is judged much more strictly than in most mathematics courses. To illustrate the difficulty of the problems and the strictness of the grading, note that the median score each year is usually 1 out of 120.

What topics do I need to know for the Putnam exam? What's the best way to prepare for it?

The Putnam exam doesn't test encyclopedic knowledge of advanced mathematics; rather, it tests problem solving, “thinking outside the box”, and the ability to find unexpected ways to interpret questions. Therefore the best way to prepare for the Putnam exam is to work on as many Putnam-like problems as possible. Problems from similar contests, like the Canadian/American/International Math Olympiads, can also be helpful. Be aware that every solution must be fully justified, and so Putnam problems are proof problems, as opposed to calculation problems.

As for topics, I would make the following lists:

• Essentials: 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)
• Often appear: Number theory (primes, congruence/modular arithmetic), how to set up probability problems (by counting or by integrals), linear algebra (matrix multiplication, determinants), recursively defined sequences, groups
• Occasionally appear, or can be helpful behind the scenes: Complex numbers, permutations, game theory, finite fields, abstract algebra, generating functions

Note that the topics that tend to be included on Putnam exams have drifted over the decades.

How can I get better at proofs?

Why do some people get better quickly when they work hard, while others don't seem to progress as fast? One answer is that deliberate practice is much more effective than going through the motions. From a Freakonomics blog post (boldface is my emphasis): “For example, in school and college, to develop mathematics and science expertise, we must somehow think deeply about the problems and reflect on what did and did not work. One method comes from the physicist John Wheeler (the PhD advisor of Richard Feynman). Wheeler recommended that, after we solve any problem, we think of one sentence that we could tell our earlier self that would have ‘cracked’ the problem. This kind of thinking turns each problem and its solution into an opportunity for reflection and for developing transferable reasoning tools.”

Tim Gowers, a Fields Medalist and world-class mathematical expositor as well, has written a series of essays on logic, mathematical foundations, and constructing proofs (the oldest entries are the most general and therefore probably the most helpful). Anyone who takes the trouble to thoughtfully read all these essays will definitely become better able to write and speak the language of mathematics, and their written solutions to Putnam problems will surely improve.

Where can I get old Putnam exams and solutions?

Old Putnam exams, solutions, and results, going back to about 1995, can be found at the privately maintained website of Prof. Kiran Kedlaya.

Three books that collect old Putnam problems and solutions and three books on mathematical problem solving in general are available in the Barber Learning Centre: