Putnam Exam Practice

OWA STATE UNIVERSITY
DEPARTMENT OF MATHEMATICS
Putnam Mathematics Exam Problem Solving Sessions
What is the Putnam Exam?
The William Lowell Putnam Mathematical Competition is the preeminent mathematics
competition for undergraduate college students in the United States and Canada.
Who can attend?
The problem sessions and the exam are open to any undergraduate student in any
major, but having some experience with writing mathematical proofs is helpful. Problem
sessions are a chance to meet some like-minded students who enjoy solving hard math
problems.
When/where are the sessions?
We will meet weekly on Thursdays at 3:20pm in Carver 400 (the undergraduate
lounge), beginning with Thursday, September 11, 2025. No need to sign up – just
show up! There will be pizza at our first meeting!
That sounds like fun. Where can I find more information?
Contact Dr. Jason McCullough (Email: jmccullo@iastate.edu) for more information,
Or visit the following webpages:
https://www.maa.org/math-competitions/putnam-competition
https://faculty.sites.iastate.edu/jmccullo/putnam-exam
What is the test like?
The exam consists of 12 proof-style questions given in two 3-hour sessions and will occur
on Saturday, December 6, 2025. The problems are more a test of mathematical creativity
than specific knowledge, but experience with calculus, linear algebra, and proofs is very
helpful. The median score in most years is 0, so getting one problem correct is a great
achievement. While it is an individual exam, the top 3 scores become the team score for
ISU. There are prizes for the top individuals and schools each year. Here are two sample
problems:
• In Determinant Tic-Tac-Toe, Player 1 enters a 1 in an empty 3 × 3 matrix. Player
0 counters with a 0 in a vacant position, and play continues in turn until the 3 × 3
matrix is completed with five 1’s and four 0’s. Player 0 wins if the determinant is 0
and player 1 wins otherwise. Assuming both players pursue optimal strategies, who
will win and how?
• Let f be a three times differentiable function (defined on R and real-valued) such
that f has at least five distinct real zeros. Prove that f + 6f
0 + 12f
00 + 8f
000 has at
least two distinct real zeros.
How can I prepare?
Taking lots of proof-based math courses is a good start. At the problem-solving sessions,
we focus on a different topic each week and discuss past Putnam problems on that topic
and how to solve them. The first problem set for Week 1 is available here:
https://faculty.sites.iastate.edu/jmccullo/putnam-exam