Mastering Mathematics at the Harvard-MIT Mathematics Tournament (HMMT)
Harvard-MIT Mathematics Tournament (HMMT) stands as a model of excellence in the world of high school mathematics competitions. Renowned for its rigorous challenges and attracting some of the brightest young minds from across the globe, HMMT has earned its place as one of the most prestigious competitions for mathematics in the world. The competition, hosted annually at Harvard University and the Massachusetts Institute of Technology (MIT), transcends typical math contests, pushing participants to showcase not only their problem-solving skills but also their creativity and deep understanding of foundational mathematical concepts. Read on if you want to learn about HMMT and what skills will be necessary for you to succeed.
What is the HMMT?
Established to foster a deep appreciation for mathematical problem-solving, the tournament serves as an annual gathering of some of the brightest young minds in mathematics. The HMMT has maintained three primary objectives: 1) to encourage excitement for mathematics, 2) to support students in moving beyond traditional math curricula, and 3) to demonstrate what a future in mathematics might look like at a practical level.
The competition, held jointly on campus at Harvard and MIT, spans a day at each institution with challenging individual, team, and guts rounds. Participants, often rising seniors with a passion for mathematics, engage in a rigorous series of problem sets that not only test their mathematical acumen but also encourage creative thinking and a deep understanding of mathematical concepts.
The HMMT individual test is ten problems, which students complete over the course of 50 minutes. These questions are drawn from topics in algebra, geometry, and combinatorics, and are worth 800 points. The team tests are proof-based and worth 400 points. Finally, the guts round, an 80-minute team event with 36 short answer questions, is completed piecewise as students progress through multiple sets of problems. This round is also worth 400 points.
Because of the contribution of raw individual scores, as well as the importance of diversity in the team-based rounds, it will be important to join the competition with a strong team behind you. The HMMT considers a full team to be six students for the November (Harvard) competition and eight for the February (MIT) tournament. No student may attend both the November and the February competitions for the same academic year.
Set Your Expectations for the HMMT
In order to do well at HMMT, the single most important thing you could do is practice similar problems! Learning how to see a problem and quickly determine the right approach is a skill that can only be learned through repeated exposure, so take some time to learn what to expect from the different types of problems featured on the exams.
Algebra Problems
Algebra problems are incredibly common on HMMT exams, so mastery over your algebra courses will be essential! You may be asked to do things like factor polynomials, solve systems of equations, or examine functions, equations, and inequalities. These problems will rely heavily on your problem-solving abilities and your confidence when approaching new problems.
Example Algebra Problem:
Larry can swim from Harvard to MIT (with the current of the Charles River) in 40 minutes, or back (against the current) in 45 minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in 15 minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss. (9th Annual HMMT)
Geometry Problems
Geometry problems are another staple of HMMT exams and usually require the creative use of lines, angles, and shapes to solve problems. You might need to solve for areas or volumes, use creative visualizations to imagine the scenario, or work in multiple dimensions.
Example Geometry Problem:
A cube of edge length s > 0 has the property that its surface area is equal to the sum of its volume and five times its edge length. Compute all possible values of s. (10th Annual HMMT)
Combinatorics Problems
These types of problems are highly creative and allow participants to focus heavily on improving their reasoning skills as they think through the patterns involved. You might employ concepts involving probability, statistics, permutations, and combinations to solve these problems on the HMMT.
Example Combinatorics Problem:
There are 25 red balls and 25 blue balls in a bag. Alice and Bob take turns drawing balls from the bag without replacement, with Alice going first. Interestingly, they notice that whenever a red ball was drawn, the next ball drawn (by any person) was never red. Suppose Bob drew m red balls. Given that the 33rd ball drawn was red, what is the sum of all possible values of m? (HMMT Problem of the Week S25W18)
Why Should You Enter?
The competitive nature of HMMT has made it a focal point for admissions officers at elite research universities worldwide. A stellar performance at HMMT can set a student apart, providing a distinctive edge when it comes time to apply to top-tier academic institutions.
For the top-performing teams and individuals at the HMMT, there are several different prizes available. In addition to the gold, silver, and bronze individual/team awards, there will also be awards for the top international participant/team, the top female participant/all-female team, the top public school team, the top public school team, the top underclassman, etc.
The HMMT competition also offers a range of scholarships to students with exceptional performance at the competition. These scholarships have helped talented young mathematicians continue pursuing their studies at prestigious summer programs like the Research Science Institute or Mathcamp!
If you are interested in competing at HMMT, consider applying to the 1:1 Mentorship Program at InspiritAI! The intersection of AI and mathematics is fascinating and ever-growing, and our mentors – hailing from the world's top research institutions and tech companies – will walk you through an entire research project from start to end. Whether you are a coding novice or a virtuoso, our individualized mentorship program will help you develop an exceptional research project. For more information, do not hesitate to contact InspiritAI with questions.
