PUMaC Preparation: Princeton Competition Guide

PUMaC preparation is unlike preparing for any other tournament, because one of its rounds gives you a week. That single difference changes what the competition rewards and what a team should practise.

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 — and the thing they are least prepared for.

Why a week is harder than an hour

PUMaC preparation has to account for this: students assume more time makes a round easier. In practice the Power Round exposes weaknesses that a 50-minute test conceals.

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 so that a grader follows it 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. A real share of PUMaC preparation goes to 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, carrying a triage habit into a round that does not want it.

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 as a strategy rather than an administrative detail. 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. Students whose strength is counting will find our combinatorics course the fastest route to making that a qualifying subject.

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 rather than discovering the rule at registration.

The route for teams abroad

Most students outside the United States 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. 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.

For an Indian team in particular, it is also excellent preparation for RMO and INMO, which are written-proof examinations that most students reach with no proof-writing practice at all.

What to cover

PUMaC preparation covers four subjects, the standard olympiad areas worked to tournament depth.

Algebra: polynomials, functional equations, inequalities, sequences. Number theory: modular arithmetic, orders, Diophantine work. Geometry: cyclic configurations, power of a point, and the coordinate and trigonometric methods that resolve a problem quickly when elegance is not available. Combinatorics: bijective counting, recursion, invariants and extremal arguments.

Because a student only sits two, there is a real argument for depth over breadth — but the Power Round draws on everything, so a team collectively needs all four covered.

How a team should run the Power week

This is where PUMaC preparation differs most from other tournaments. Seven days sounds like plenty and disappears quickly, so the teams that score well treat it as a project with a schedule rather than a deadline.

A workable pattern: read the whole paper together on day one and agree what each part is asking, because the later parts almost always depend on the earlier ones. Work individually for two or three days. Reconvene to compare, and crucially to find the parts nobody has cracked.

Leave the final two days entirely for writing. That allocation surprises teams who assume writing is a formality, and it is the single most common reason a strong team submits a weak document.

The last read should be done by whoever contributed least to a given section, because they are the only person who will notice the step that was obvious to the author and to nobody else.

How PUMaC compares

Against HMMT, PUMaC is slower in its individual rounds and far more demanding in its proof work. HMMT February has a proof-based team round of 60 minutes; PUMaC gives a week and expects correspondingly more.

Against the Stanford Math Tournament, the structures differ most in the Power Round length — SMT's is 80 minutes, PUMaC's is seven days — and those are genuinely different skills.

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 current eligibility and registration details.