Competitions · Mathematics · PUMaC
PUMaC, the Princeton University Mathematics Competition.
Run since 2006 entirely by Princeton’s undergraduate Math Club, and sat by around six hundred students a year in teams of eight. Its individual round prices each question by how many people actually solved it, which means nobody - including the organisers - knows what a question is worth until the papers are marked.
At a glance
What the competition is, and who can enter.
PUMaC is organised and run by Princeton students. There are two ways to enter - an online Power Competition, or the Main Competition on the Princeton campus - and a team registers for one or the other, not both.
- Organiser
- The Princeton University Math Club, which has run the competition since 2006
- Scale
- Around 600 high-school students from the United States and other countries each year
- Team size
- Eight. Half-teams were permitted historically but since 2017 only full teams and individual entrants may apply
- Age limit
- Team members must be under 20 on the competition date, and must not be enrolled full time at a post-secondary institution
- Where a team comes from
- Normally a single high school. Teams drawn from a defined geographic region across several schools are permitted where distance makes that impractical, at the director’s discretion
- Entering alone
- Individuals may apply and are grouped into teams of eight. They are selected on merit in the same way as full teams and may record their mathematical achievements on the form
- Two competitions
- The Power Competition is the Power Round alone, taken online. The Main Competition is held at Princeton and includes the Power, Individual and Team rounds
- Two divisions
- A for experienced teams, B for newer ones. Only the individual tests differ; the Power, Team and Live rounds are identical, and awards are calculated separately
- International entrants
- Welcome. Where the timing is impossible, it is possible to enter the Power Round only; the remaining problems go online after the competition
- Hanlin’s record
- Hanlin students took first place on team total in Division A of the China region in 2026, and placed tenth globally in the US region in 2025
The rounds
Four rounds, and only two of them look like an exam.
A team sits all of these in one day at Princeton, alongside an evening of informal mathematical events.
Individual Round
Each student picks two of four subjects - algebra, combinatorics, geometry and number theory - and sits 50 minutes on each. Subject weightings change from year to year, and only the top five scores in a subject count toward the team.
Individual Finals
The top ten in each subject in each division advance to a separate paper: three proof questions of rising difficulty, seven marks each, over ninety minutes. Ties are settled on how completely the proofs are written.
Team Round
Fifteen fill-in questions at five marks each, worked collaboratively in around thirty minutes. Alongside the conventional problems it carries a "game" segment offering bonus marks, whose format is redesigned each year.
Power Round
A proof-based team round of eighty minutes on site. Some questions are released a week ahead for teams to submit online, and the on-site proofs are added to those. Outside resources are barred, though known theorems may be cited.
Scoring
The two rules that change how a team should be organised.
PUMaC’s scoring is more unusual than its format, and both of its quirks reward teams that have thought about them in advance.
Questions are priced by the field
Individual-round questions carry no pre-assigned marks. Each is valued from how many entrants solved it against how many attempted it, so a question’s worth is decided by the field on the day rather than by the setters beforehand.
Spreading the team counts more scores
A team generates sixteen individual scores, but only the top five in each subject count toward the team total. A team that piles into two subjects wastes results; one that spreads across all four has more of them counted.
Pick the subject you like
The organisers state plainly that choosing a subject for strategic reasons - because fewer people sit it, say - gains very little. Their advice is to enter the tests you are actually good at.
Individual Finals do not help the team
The overall team ranking is built from the Power, Individual, Live and Team rounds. A student’s Individual Finals result affects only their own standing, where it is worth half the individual overall score.
Divisions
A and B, and how a team ends up in one.
A team states a preference at registration, but the final decision rests with the competition director, who may place a team in a division other than the one it asked for.
| Division A | Division B | |
|---|---|---|
| Intended for | Experienced teams | Teams newer to the competition |
| Individual tests | More challenging | Pitched for less experienced entrants |
| Power, Team and Live rounds | Identical in both divisions | Identical in both divisions |
| Awards | Calculated separately | Calculated separately |
| Multi-school teams | Required to compete here, though an exception may be requested | Not available to teams drawn from more than one school |
Individual applicants may also state a division preference, but the director may move them based on apparent strength and on how many applications arrive. Applicants are contacted before any change is made.
Registration
The autumn calendar.
Registration opens in September for a competition in late November, and the deadlines are enforced strictly - nobody is admitted late.
Registration opens
Both team and general individual registration open on the same day.
Registration closes
At 5pm Eastern. Teams are then notified of acceptance or placed on a waiting list.
Power Round released
The advance portion of the Power Round is published for teams to work on and submit online.
Team roster finalised
Teams confirm which individuals are competing before the roster closes.
Power Round submissions due
Advance answers must be submitted online before the on-site rounds.
Competition day
All rounds are sat at Princeton, followed by an awards ceremony.
Awards
Team and individual, ten or fifteen deep.
Top-ten placings for the Power Round, the Team test, the overall team ranking, each individual subject and the individual overall are announced at the ceremony and published afterwards.
Preparation
How Hanlin prepares a PUMaC team.
The mathematics sits at or above AIME level, but the decisions that shape a team result are made weeks before anyone sits a paper.
Assigning subjects across eight people
Because only the top five scores in each subject count, who sits which two tests is a real decision with a real cost. It is planned deliberately rather than left to individual preference on the day.
Proof writing for two rounds
The Power Round and the Individual Finals are both proof-based, and the Individual Finals are marked out of seven a question in the olympiad manner. Writing a complete argument is trained separately from solving.
The advance Power Round
A week is given for the released portion, which makes it the one part of the competition where a team can genuinely work at its own pace. Teams practise dividing and drafting it rather than treating it as homework.
The Team Round pace
Fifteen questions in about thirty minutes across eight people is fast enough that a team without a division-of-labour habit will not finish. That habit is built by rehearsal, not by explanation.
Why Hanlin
Preparation for PUMaC at Hanlin is taught by full-time subject tutors, on a course tier chosen by placement paper rather than by year group.
Programme formats
Five ways this is taught.
Group size is the whole difference between these. It decides how much of the tutor's attention a student gets and how far the course can bend to them, so it is worth choosing rather than accepting.
| Format | Group size | What it suits |
|---|---|---|
| One-to-one | One student | Content, pace and emphasis set by what this student actually needs rather than by a syllabus. The only format that can be rebuilt mid-course. |
| Small group | Three to eight | Opens at three. Frequent discussion, and the tutor can slow down or move on according to what the group has absorbed. Most students chasing a higher award are taught in this one. |
| Class | Eight to twelve | Taught by the subject lead or a medal coach. Less back-and-forth, so it suits a student who will say when something is unclear. |
| Large class | Ten to twenty | For students in the middle of the placement range, and a way to try a competition before committing. Students often move to a small group or to one-to-one afterwards. |
| Past-paper class | No cap | Past papers worked through in sequence in the weeks before the sitting. Free to students already enrolled. |
Course length for PUMaC is set after a placement paper. The gap between where a student is and what the paper asks for is what the course has to close, and that gap is not the same for two students in the same year group.
How it runs
Six steps, four people on your side.
The same process for every student, so that nothing depends on one tutor remembering to do it.
Step 1Pre-course assessment
A placement paper before anything is booked, so the course tier is chosen on evidence rather than on a year group.
Step 2Tutor matching
A subject tutor is matched to the result, and their full background is provided before you agree to it.
Step 3Plan and group
A study group opens with four people on one student: the tutor, a planner, a supervisor and an academic manager.
Step 4Scheduled teaching
The teaching office sets the timetable around the competition date and school term, and teaching begins.
Step 5After-class reports
What was covered, and how the student handled it, reported after every lesson.
Step 6Homework follow-up
The teaching assistant and the form teacher both check that homework is done, which is where most preparation quietly fails.
Free resource pack
Past papers and preparation pack
The same material our own PUMaC students work from. Scan the code, say what you need, and an advisor sends it.
Tell an advisor which competition and year group, and the material comes back the same day. Free, and you do not have to enrol in anything. If we do not already hold what you need, we will go and find it.
- PDFPast papers, every year the organiser has released
- PDFWorked solutions, where the organiser publishes them
- PDFThe syllabus on one page - every topic, and how heavily it is examined
- PDFEntry checklist and the dates for the current cycle
Teaching team
Who would teach it.
Three of the twenty-two tutors on the published Hanlin roster - the ones whose subject this is.
Dr Zhang
PhD in theoretical mathematics from the University of Rochester and a postdoctoral fellow at the Shanghai Center for Mathematical Sciences at Fudan. An AMC-accredited coach, ASDAN-accredited for DMM and ITCCC, and an accredited UKMT BMO2 and PUMaC coach. His students have gone on to read mathematics and physics at Harvard, MIT, Princeton, Caltech and Cambridge.
Dr Zhuang
PhD in mathematics from the University of Denver, working in mathematical logic, after a first degree in applied mathematics at Southeast University. A repeat conference speaker on his research, with three years lecturing and assisting on undergraduate mathematics at Denver.
He
A Cambridge master's after a computer science degree with a mathematics minor at the University of Nebraska-Lincoln, and a US high school GPA of 4.13. Six years overseas and teaches entirely in English, across AMC, Math League, AP Computer Science and AP Calculus.
Where it is taught
Hanlin learning centres.
Teaching runs from Hanlin's own centres in Shanghai, Shenzhen, Chengdu and Hangzhou, and online for students elsewhere.
Talk to an advisor
Everything Hanlin does for PUMaC, in one conversation.
Entry, planning, coaching and the past papers. An advisor will tell you which of these a student actually needs, including when the answer is none of them.
Competition entry and planning
Which competitions suit this student, in what order, and by when - then we handle the registration, including the ones that can only be entered through a school or a centre.
Competition coaching
Taught by full-time subject tutors, on a course tier chosen by placement paper rather than by year group.
International curriculum tutoring
IB, AP, A-Level, IGCSE and the US high school curriculum, taught alongside school and timed around the exam calendar.
Competition past papers, free
A large library of past papers, mark schemes and syllabus breakdowns across every competition on this site. No charge and no enrolment.
Scan to add an advisor
Say which competition or course you are asking about. Replies in Chinese or English.
Or call +86 21 6352 6630FAQ
PUMaC, briefly answered.
What students and families ask before entering a team.
Who can enter PUMaC?
Students under 20 on the competition date who are not enrolled full time at a post-secondary institution. Teams are eight students, normally from one high school. Individuals may also apply and are grouped into teams of eight.
Can international students take part?
Yes. The organisers try to accommodate different time zones. Where the timing is genuinely impossible, a team can enter the Power Round only, and the remaining problems are made available online after the competition.
What is the difference between Division A and Division B?
Only the individual tests: Division A is for experienced teams and its individual papers are harder. The Power, Team and Live rounds are identical, and awards are calculated separately. Teams drawn from more than one school must enter Division A.
How are individual-round questions marked?
They carry no pre-assigned values. Each question is priced according to how many entrants solved it against how many attempted it, so its worth is determined by the field rather than set in advance.
Which two subjects should a student choose?
The ones they are strongest at. PUMaC staff state that choosing strategically - picking a subject because fewer people take it - gains very little, and only the top five scores in each subject count toward the team.
How is the team score built?
From the Power, Individual, Live and Team rounds. A team produces sixteen individual scores but only the top five in each subject count, so spreading students across all four subjects gets more of them included.
Do the Individual Finals affect the team result?
No. They affect only that student’s individual standing, where they are worth half the overall individual score - the two individual tests contributing a quarter each.
What happens if a team arrives late?
Contact a PUMaC representative immediately. A late team may only join between rounds, so a team arriving after the individual tests have started can still compete but forfeits the round already in progress.
Related
Other university-run team mathematics.
Contests students commonly enter in the same season as PUMaC.
PUMaC enquiries
Who sits which subject is a team decision.
Only the top five scores in each subject count, so assigning eight students across four papers changes the team total before anyone solves anything. Tell us who you have, and an advisor will work through the entry and the September deadline.