math problems written on a piece of paper within a notebook

Math students achieve excellence at 2021 Putnam Competition

In December of 2021, ten Northeastern students competed at the William Lowell Putnam Mathematical Competition, a university-level mathematics competition held annually on campuses across the United States and Canada. Northeastern’s participants included Yash Bhora, Devin Brown, Shuqi Lin, Siddhant Mane, Noble Mushtak, Harshal Nawade, Vedant Rautela, Jaynam Shah, Sanat Shajan, and Andrei Veliche. 

Highly qualified undergraduate math students from the United States and Canada participate in this preeminent competition. The exam consists of two three-hour sessions, each containing six problems. The problems are labeled in order of difficulty, the first set being A1 through A6 and the second set B1 through B6. Each problem is worth up to ten points, making the maximum possible score 120.

Achieving a score close to this caliber is rare; the problems are so challenging that even professional mathematicians have difficulty solving them. This challenge is evident as the median score is zero in most years, meaning that more than half of the competitors receive no points. Nevertheless, the top five scorers of the entire competition earn the title of “Putnam Fellows,” and this year’s fellows earned scores of 119, 110, 99, 90, and 89. Despite the rarity of achieving a positive score, all ten of our competing Northeastern students did just that. 

Although participants complete the exam independently, having a team still serves a purpose; the top three scorers of each team are considered that institution’s “Putnam Team,” a smaller group of top scorers within the entire team that compete. Northeastern’s top three scorers that formed the 2021 Putnam Team were Brown, Mushtak, and Veliche.  

“Based on previous years, I thought I would need at least a 40 or high 30s to rank in the top-200,” says Noble. “So, going into the contest, my goal was to solve four to five problems to do just that.” 

Noble and many of his teammates were well-prepared, thanks to their membership in the Northeastern Putnam Club. The club, run by Peter Crooks, Harm Derksen, Evan Dummit, Iva Halacheva, and Robin Walters, helps prepare students for the competition by practicing similar problems on the exam. Derksen has also served on the Putnam problems committee, responsible for drafting and editing the competition, a few years before coming to Northeastern. Although the competition occurs in December, the club runs year-round, even during the summer. Besides being great practice and preparation for the Putnam, the club functions mainly on its members’ affinity for solving problems rather than desiring to score highly within the competition. Furthermore, students do not have to compete to participate in the club.  

“We welcome participation from any students interested in coming to think about interesting math problems!” says Dummit. “Additionally, quite a lot of the fundamental problem-solving techniques that help in doing well on the Putnam are also valuable skills for solving mathematics problems in general (e.g., like the ones that students do on their homework assignments), so I think there’s an overall benefit to students who participate regardless of whether they take the Putnam itself.” 

Dummit is quite familiar with the competition, not only serving as a co-organizer of Northeastern’s club for the past two years, but also because he competed as an undergraduate alongside his team at Caltech University.  

“I competed all four years, with an honorable mention twice (at the time, that was top 70-80 nationally), and I’ve been interested in mathematics competitions for a long time,” he says. 

Dummit has an influential role in helping prepare his students for the complex problems on the exam. “I try to bring this ethos to the Putnam Club — we’re very informal, but the goal is to help our students get exposed to the kinds of problems in the competition so that they are better prepared for the event itself.” 

The role of the Putnam club organizers is to select problems for students to work on each week, attend club meetings, and discuss the problems with attendees.  

“I often try to come up with a theme when it’s my turn to put problems together – sometimes the problems are all organized around a particular topic, or a particular solving concept, or (sometimes) something more eclectic, like the topic I’m putting together for this week: pairs of problems from decades apart that share a similar idea,” Dummit says.

The dedicated work of organizers such as Dummit does not go unnoticed by the club’s members. “I have to thank Professor Dummit and Professor Derksen for organizing weekly Putnam problem sets for us in the club; that has been my main preparation for the competition,” says Noble. 

Congratulations to the students of this year’s competition and their high achievements! The following are a few examples of past problems. Can you solve them? 

  • (1989-A1) How many prime numbers are in the sequence 1, 101, 10101, 1010101, 101010101, …? 
  • (2002-A3) Given any five points on a sphere, show that some four of them must lie on a closed hemisphere. 
  • (1988-B1) A composite positive integer is a product ab with a and b integers greater than 1.  Show that every composite positive integer can be written as xy+xz+yz+1 for some positive integers x,y, and z. 
  • (2010-B4) Find all pairs of polynomials p(x) and q(x) with real coefficients such that p(x) q(x+1) – p(x+1) q(x) = 1. 
  • (1989-B4) Can a countably infinite set have an uncountable collection of non-empty subsets such that the intersection of any two of them is finite?