## Official Main Competition Rules

Welcome to the PUMaC Main Competition Rules. Unless otherwise stated, all of the rules apply to both division A and division B.

## Eligibility

- A full team will consist of 8 members.
- A team member must not have reached his or her 20th birthday before the PUMaC competition date.
- A team member must not have been enrolled full-time in a post-secondary institution before the PUMaC competition date.
- Half teams of 4 students are allowed, and will, if possible, be paired with another half team. Partial teams not consisting of exactly 4 students will not be allowed to register.
- See Important Dates for registration deadlines. Teams that register after the late registration deadline, or after the contest’s capacity is reached, will be put on a waiting list, and may be allowed to participate in the competition if space allows, or if previously registered teams withdraw.
- Generally, teams should come from single high schools, but teams from geographical regions (comprising members from several schools) coming from a long distance will be allowed for their own convenience. The current Director of the Princeton University Mathematics Competition will have the final word on the legitimacy of teams that do not come from an individual school.
- Teams may register for the A or the B Division. The A Division is designed for the more experienced teams, and the B division is recommended for teams newer to competitions (Division B will be slightly less challenging than division A). Divisions will be given different tests on the day of the competition. The Power Test will be identical for both divisions, and teams in different divisions will be held to different grading standards. Awards for the two divisions will be separately calculated.
- Teams that do not come from a single high school must compete in the A Division.
- The Director of the Princeton University Mathematics Competition reserves the right to ask high-scoring teams in the B Division to compete in the A Division for the following three years.

## Format of Test Questions

Any answers and solutions submitted for grading on the Power, Team, and Individual Tests should follow the formatting guidelines indicated below. Answers and solutions submitted in a format that contradicts what is described below are subject to being graded as incorrect. Should an irresolvable appeal arise concerning grading, the current Director the Princeton University Mathematics Competition will have the final word.

The following is a list of conventions. A PDF version of the conventions is available here.

- All Individual Test answers are integers.
- Some Individual Test questions ask you to concatenate the numerator and denominator of a fraction to produce an integer answer. For instance, if your answer is 101/1746, you should submit the integer 1011746 as your answer.
- The words “compute,” “find,” or “evaluate” always call for an answer in simplest form, according to the usual mathematical consensus. (For instance, 9/6, 4 + 3, and 4 sin(30◦) are unacceptable; 3/4, 11, and sin(17◦) are acceptable.) Justification is not necessary. When there’s no mathematical consensus about which of several answers is “most simplified,” any of them will be accepted: for example, 3/2, 11, and 1.5 are all acceptable.
- When a question calls for an “ordered pair (a, b),” the answer must be given precisely in that form, including the parentheses and the comma. The same applies for other ordered n-tuples.
- When a polygon is named by letters, the letters are vertices occurring in their given order around the polygon. (For example, a polygon named ABCDE is understood as a pentagon with vertices A, B, C, D, and E occurring in that order.) Unless otherwise specified, all polygons are non-degenerate (no angles of 0 or π) and non-self-intersecting, but not necessarily convex.
- Written numbers and logarithms are base 10 unless indicated otherwise by a subscript. The use of log(x) also implies that x is positive. For example, log3 81 = 4. Exception: ln(x) refers to log
_{e}(x). - The letter i is used for complex numbers, where i
^{2}= −1. - Divisors (or factors) of an integer refer to positive integer divisors only. Proper divisors of an integer are divisors other than the integer itself.
- Prime numbers refers to positive primes only.
- Some problems refer to the digits of a number. In these cases the digits are usually underlined. For instance, in the question “Find the missing digits A and B if k = A 2 5 B and k is a multiple of 72” (k = 1000A+200+50+B where A is an integer between 1 and 9 and B is an integer between 0 and 9, and k is not (necessarily) the product of A, 2, 5, and B).
- Diagrams are not necessarily drawn to scale.
- The greatest lower bound of a set is the largest number which is less than or equal to all elements of the set. Thus 2 is the greatest lower bound for both {x: 2<x} and {x: 2≤x}. Similarly, 3 is the least upper bound for both {x: x<3} and {x: x≤3}.
- The open interval bounded by real numbers a and b, a<b, will be denoted by (a, b), and the closed interval bounded by a and b, a≤b, will be denoted by [a, b]. Semi-open/semi-closed intervals will be denoted by (a, b] and [a, b).
- The sum of all elements of the empty set is 0. Likewise, the product of all elements of the empty set is 1.

## The Power Test

The Power Test rules in the Main Competition are identical to those given in the Power Competition Rules page. They are duplicated here for convenience. Note: although the test rules are identical, the rules for submissions differ.

The Power Test is a way for PUMaC to expose a team to what real mathematics is about – questions that may take more than an hour or two and require creative thinking for solutions.

## Test Rules

- The Power Test will present an interesting definition or advanced concept. It will be made available online for teams to upload a week before the competition date.
- The entire team may collaborate on this test. There is no time limit, except teams must submit their solutions at the registration desk on the day of the competition. As such, the team will have approximately a week to work on the exam. Teams may not collaborate with each other.
- No outside resources may be used for the competition.
- Students will be required to submit written solutions with justification. Solutions are to be submitted at registration on the day of the competition.
- The Power Test will have several parts. Results from previous parts may be assumed true in later parts, even if the team is unable to prove these previous results. Students should cite these results when they use them. For example, “From Part II, Question 1, …”
- Solutions should be written on one side of the paper only. Each page should also have on it the team ID number and problem number.
- Although not necessary by any means, typed solutions are welcome. (For help with this, see the LaTeX Wikibook )
- More instructions will be given on the Power Test. If instructions here and on the power test itself conflict with one another, the instructions on the power test should be followed.

## Submission

Teams must submit their Power Test solutions at the check-in table on the day of the competition. Teams will not be able to submit their solutions to the Power Test at any later time. Any discrepancies will be dealt with by the current Director of PUMaC.

## The Individual Tests

Each participant of PUMaC will take two individual tests from a choice of four. The individual round exams will have eight problems in the subject areas chosen by the participants (either Algebra, Combinatorics, Geometry, or Number Theory). Each individual test is 60 minutes. The maximum score on any individual round is 40 points, and the eight questions are weighted as follows: 3-3-4-4-5-6-7-8.

- No tools such as calculators, books, protractors, compasses, or rulers will be allowed on any individual test.
- Participants must choose
**two**tests from a choice of four. More information on the tests can be found on the Competition page and in the FAQ. - Participants will be allowed to change what competitions they will be taking up until Late Registration ends. Afterwards, any changes in individual tests must be approved by PUMaC. No individual will be able to change what tests he or she wil be taking when they get to the exam room.
- Participants are expected to record their answers on an official PUMaC answer sheet. Then, proctors will call up participants individually to enter scores into the computer grading system.
- Up to 6 participants in a team can take the same test. However, if more than 5 people take the same test, the team will be put into a disadvantage.
- Participants are NOT allowed to collaborate on this part of the competition.
- All answers will be integers.

## The Team Test

The Team Test is the only on-site test in which collaboration is allowed (and is encouraged). Each team will be given a set of instructions and problems, and they must do their best to finish the exam within the time limit given. **This round is not proof-based.**

- No tools such as calculators, books, protractors, compasses, or rulers will be allowed on the team test.
- Team tests will have a time limit of 30 minutes
- Participants are allowed to collaborate on this part of the competition.
- Teams are expected to submit an official answer sheet with their answers, team number, and division.

## The Individual Finals

The top 10 individuals in each test, in each division, will be asked to take the Individual Finals exam. **This competition is proof-based**.

- The Individual Finals exam has a time limit of one hour.
- Participants are NOT allowed to collaborate on this part of the competition
- Participants will NOT be allowed to leave early from the exam room.

## Scoring

- Final Team standings will be based upon their performance in the Power, Individual, and Team rounds. Individual Finals scores have no bearing on a team’s final score.
**In the end, each team will have 16 Individual Round scores. Only the top 5 scores in each subject will count towards a team’s final score.**For example, if a team only takes Algebra and Geometry exams, then only the top five scores in those tests will count toward that team’s final score. However, if a team distributes its team members more evenly among individual tests, then more of their scores can count.- Individual rankings will be calculated as follows:
**Each of the two Individual Round tests will contribute 25% of a participant’s overall score, and Individual Finals will contribute 50%.** - In the case of a tie in the individual rankings, the advantage will be given to the participant solving the more difficult questions on the Individual Finals round. If a tie persists, the participant with the more elegant proof will be given the advantage.
- In the awards ceremony, ties for subject test rankings will be broken based on the results of the individual finals similarly to how ties in individual rankings will be broken. This will not affect the overall team scores.