The Harvard–MIT Mathematics Tournament (HMMT) is an annual high school math competition started in 1998. The location of the tournament, in Cambridge, Massachusetts, alternates between Harvard University (even years) and MIT (odd years). The contest is written and staffed entirely by Harvard and MIT students, and is considered to be one of the most prestigious high school math competitions in the world.
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HMMT is attended by teams of eight students each. Teams can represent a single school, or a regional math team as large as a state. In recent years, teams have represented over 20 states, as well as Africa, Asia, Europe, and South America.
HMMT consists of three rounds: the Individual Round, the Team Round, and the Guts Round. No calculator or computational aids of any kind are allowed during the contest.
The Individual Round consists of exams in Algebra, Combinatorics, and Geometry. Each of the three exams is 50 minutes in length and contains 10 questions. The exams are open-answer; that is, the answers given will be numbers such as 7 or 11/20.
For the Team Round, the eight-person teams may decide to compete in the A Team Round or the B Team Round. Each test is a 60 minute collaborative round, consisting of 10-15 questions from any area of elementary mathematics . While the B Team Round is easier, it is weighted only about 50% of the A team test when calculating the overall tournament champions.
The A Team Round is a collaborative event with proof-style problems, arranged into groups of several problems on the same theme. Thorough justifications are required for full credit. The Team Round is worth a total of 400 points, and problems are weighted according to difficulty. The event is similar to an ARML Power Round, but the problems are easier and more numerous. This round is targeted at teams comfortable with rigorous mathematical proofs.
The B Team Round is a collaborative event with a mixture of proof-style and short answer questions. Thorough justifications are required for full credit on the proof questions.
The Guts Round is an 80-minute team event with 36 short-answer questions on an assortment of subjects, of varying difficulty and point values. Each team is seated in a predetermined spot, and the questions are divided into groups of four. At the starting signal, each team sends a runner to an assigned problem station to pick up copies of the first set of four problems for each team member. As soon as a team has answers for one problem set, the runner may bring the answers to the problem station and pick up the next set. It is not expected that students will finish all the problems. Grading is immediate and scores are posted in real time, resulting in an exciting atmosphere for the competitors. The Guts round is worth a total of approximately 400 points.
HMMT also features events on the Friday evening prior to the tournament. Some of these events include a dinner and social for students and coaches, and Mini-Events such as math talks about famous problems and math-related games.
HMMT uses a unique scoring algorithm to score the competitors on the Individual Rounds. While the problems on these tests are weighted according to difficulty, they are done so after the testing has completed. As explained here, this helps create a very fair method for weighting problems according to their actual difficulty (as determined by how often and by whom they were solved) as opposed to their perceived difficulty prior to the tournament. The weights assigned to each problem are calculated using a scoring algorithm that takes into account which problems were solved by which students. The weights of the problems on the Team and Guts Rounds are given on the tests.
Prizes are given to the ten highest-scoring individuals overall, the top ten scorers on each of the subject rounds, the ten highest-scoring teams on the Team Round (A and B), and the ten highest-scoring teams on the Guts Round. The top ten teams overall will be named the Sweepstakes winners. The calculation of Sweepstakes scores is roughly half individual round performance and half collaborative round performance.
The difficulty of the competition is compared to that of ARML, the AIME, or the Mandelbrot Competition, though it is considered to be a bit harder than these contests. The contest organizers state that, "HMMT, arguably one of the most difficult math competitions in the United States, is geared toward students who can comfortably and confidently solve 6 to 8 problems correctly on the American Invitational Mathematics Examination (AIME)." As with most high school competitions, knowledge of calculus is not required; however, calculus may be necessary to solve a select few of the more difficult problems on the Individual Rounds.
The results of HMMT can be seen below:
| Year | Overall Champion | Individual Champion | Team Round A Champion | Guts Round Champion |
|---|---|---|---|---|
| 2011 | Saratoga High School/SFBA | Xiaoyu He | North Carolina | Saratoga High School/SFBA |
| 2010 | Phillips Exeter Academy | Ben Gunby | TJHSST | AAST |
| 2009 | TJHSST | Ice Pasupat | Lehigh Valley ARML | Lehigh Valley ARML |
| 2008 | Phillips Exeter Academy | Brian Hamrick | New York City Math Team | Quagga |
| 2007 | The WOOTlings | Arnav Tripathy | The WOOTlings | TJHSST |
| 2006 | Phillips Exeter Academy | Nimish Ramanlal | TJHSST | AAST |
| 2005 | Phillips Exeter Academy | Thomas Mildorf | TJHSST | Florida |
| 2004 | TJHSST | Tiankai Liu | TJHSST | Phillips Exeter Academy |
| 2003 | TJHSST | Tony Zhang | TJHSST | AAST |
| 2002 | Newton South High School | Ricky Liu | Newton South High School | Lexington High School |
| 2001 | Lexington High School | Ricky Liu | Lexington High School | Newton South High School |
| 2000 | Newton South High School | Ricky Liu | Newton South High School | Newton South High School |
| 1999 | Newton South High School | n/a | Newton South High School | n/a |
| 1998 | Lexington High School | n/a | Lexington High School | n/a |
HMNT, the Harvard-MIT November Tournament, has been held since 2008 at MIT for teams of six students within a 125-mile radius of Cambridge, Massachusetts. The tournament is similar in style to HMMT, and is organized by the same Harvard and MIT students as HMMT. Instead of three topic tests, HMNT has two Individual Rounds: a General Test (ten questions from Algebra, Geometry, and Combinatorics) and a Theme Test (ten questions, many of which are tied together by a common theme). Additionally, the Team Round is entirely short answer, instead of proof-based. HMNT is considered to be an easier alternative to HMMT; while schools may compete at both tournaments, no students may compete at both HMNT and HMMT in the same school year as of 2011-12. The results of HMNT can be seen below:
| Year | Overall Champion | Individual Champion | Team Round Champion | Guts Round Champion |
|---|---|---|---|---|
| 2011 | Phillips Exeter Academy | Forest Tong | Lexington High School | Brookline High School |
| 2010 | Phillips Exeter Academy | Ravi Jagadeesan | Phillips Exeter Academy | Lexington High School |
| 2009 | ABRHS | Xiaoyu He | Phillips Exeter Academy | ABRHS |
| 2008 | Western Mass ARML | Sam Trabucco | Western Mass ARML | Westford Academy |
HMMT is currently sponsored by the MIT Mathematics Department, the Harvard Mathematics Department, Jane Street Capital, and Two Sigma.