PUMaC Preparation: Princeton University Mathematics Competition Training

A competition with a week-long round

The Princeton University Mathematics Competition has been run by Princeton students since 2006. Most of it looks like other tournaments: individual subject tests, a collaborative team test, proof-based finals for the leaders. One part does not.

The Power Round is released online about a week before the competition, and a team has roughly seven days to complete it with full written justification. There is no clock in the room, no triage, no deciding whether a problem is worth four more minutes. There is a structured proof-based paper and a week to do it properly.

For students whose entire competition experience has been timed sprints, this is the most valuable thing on the calendar. It is also the thing they are least prepared for.

Why a week is harder than an hour

Students assume more time makes a round easier. In practice the Power Round exposes weaknesses that a 50-minute test hides.

A timed test rewards recognising a problem type. A week-long proof-based paper rewards building an argument, noticing that part (d) needs the lemma from part (b), and writing in a way that a grader will follow on a first reading. There is no partial credit for having had the idea in conversation. Teams that treat it as homework to divide up and staple together score below teams of the same ability who treat it as one document with one voice.

Five weeks of this course go to Power Round work, including the unglamorous part: agreeing notation, reconciling sections written by different people, and reading the submission as a grader would before sending it.

Eight problems in sixty minutes

The individual tests are 8 problems in 60 minutes, short answer, and each student picks two subjects from Algebra, Combinatorics, Geometry and Number Theory. No collaboration.

Seven and a half minutes a problem is a slower pace than HMMT or SMT allow, and it changes the right approach. There is time to think, which means a problem abandoned at ninety seconds was probably abandoned too early. Students trained purely on sprint tournaments under-score here by reflex.

The scoring decides your subject choice

An individual score is 25 per cent from each of the two subject tests and 50 per cent from the Individual Finals, and the Finals are open to the top ten per subject per division.

Read that again as a strategy. Half a student's score comes from a round they must first qualify for, and qualification is per subject. So picking two subjects of middling strength to spread risk is usually worse than picking one genuine strength and giving it the best chance of a top-ten finish. We work this through with each student rather than letting them default to their two favourite topics.

Divisions, and the mixed-team rule

Division A is for experienced teams with harder problems; Division B is for newer teams. One rule surprises people: a team drawn from more than one school must compete in Division A, whatever its experience. A group of friends from different schools therefore faces the harder paper, and should prepare for it.

The route for teams abroad

Most of our students cannot travel to Princeton, and PUMaC provides for this directly. The Power-Only Competition lets a team submit the Power Round alone and be ranked separately.

This is a genuinely good option rather than a consolation one. The Power Round is the round that teaches the most, it needs no travel, and a week of structured proof work is the closest thing on the competition calendar to what mathematics is actually like.

Who this suits

Students working at AMC 12 and AIME level. Division B is the sensible entry point for a team new to the competition, subject to the mixed-school rule above.

The competition date is not published on the rules page as a fixed annual slot and has historically fallen in the autumn, so confirm it at pumac.princeton.edu along with the current eligibility and registration details.