Official Main Competition Rules 2009

From MathClub

Update: The new PUMaC website is up. It can be found here . All of the information on this website, the wiki, will no longer be updated. For the most recent news and information regarding PUMaC, please go to the new website.

Welcome to the rules page for the 2009-10 Princeton University Mathematics Competition. This page will continually be updated until all of the rules are written up. Until then, the following rules are subject to change without notice. We promise to have the rules completed well before registration begins. Until then, feel free to contact us with any questions or concerns you may have.

Unless otherwise stated, all of the rules apply to both division A and division B.

Contents 1 Eligibility 2 Formatting of Test Questions and Answers 3 The Power Test 4 The Day of the Competition 5 The On-Site Tests 5.1 On Using Aids for On-Site Tests 5.2 The Team Test 5.3 The Individual Tests 5.4 Challenging Errata for Short Answer Tests 5.5 Mini Events 5.6 The Individual Finals 6 Preparation 7 Other Competition Pages

Eligibility

• A full team will consist of 8 members.
• A team member must not have reached his or her 20th birthday before January 1, 2010.
• A team member must not have been enrolled full-time in a post-secondary institution before January 1, 2010.
• 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 Registration Guidelines 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.
• No teams can register for the competition after the late registration/payment deadline (Sunday, November 1, 2009). If a registered team’s payment is not postmarked by this date, the team will automatically be placed at the end of the waiting list, and a team from the waiting list will be invited to participate instead. Though a place will be held for teams who are invited from the waiting list, teams will not be officially added to the competition until payment is received.
• No team can be added to the competition after Sunday, November 8, 2009.
• 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 (Divison 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 Undergraduate Mathematics Club reserves the right to ask high-scoring teams in the B Division to compete in the A Division for the following three years.

Formatting of Test Questions and Answers

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 words “compute,” “find,” or “evaluate” will always call for an answer in simplest form. Final answers like 9/6, 4+3, 33, and 4sin 30°, for example, would not be satisfactory. In cases where there is ambiguity as to what is the “most simplified” answer, alternate answers will be accepted. For example, 3/2, 1½, and 1.5 are all acceptable.
• When an answer calls for an “ordered pair (a, b),” it must be given precisely in that form, including the parentheses and the comma. The same applies for other choices of letters and for ordered n-tuples.
• The naming of polygons will involve letters 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.) When referring to polygons, it is assumed that they are non-degenerate.
• Logarithms are base 10 unless otherwise indicated. The use of log(x) also implies that x is positive. In general, when bases are not indicated, the numbers referred to are in base 10. If another base is being used, that base will be written as a subscript. For example, log₃(81) (which equals 4). ln(x) will refer to log_e(x), where e=2.71828…
• The letter i will be used for complex numbers (i² = − 1).
• Divisors (or factors) of an integer refer to positive integer divisors only. Proper divisors of an integer are divisors that are less than the integer itself.
• Prime numbers refer to positive primes only.
• Some problems refer to the digits of a number. In these cases the digits are usually underlined. Examples: “Let x=3 3 3 … 3 3, where the digit 3 occurs 1000 times,” or “Find the missing digits A and B if k= A 2 5 B and k is a multiple of 72” (the number k is not to be interpreted as 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}. The least upper bound of a set is the smallest number which is greater than or equal to all elements of the set. Thus 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 Power Test

The power test is a way for PUMaC to really expose a team to what real mathematics is about - questions that take more than an hour or two and require creative thinking for solutions. This year, two different versions of the power round will be released; BOTH of these versions are EQUIVALENT. One of the versions is more like what a mathematician would prefer because it uses the high level mathematics language an style. The other version is a bit more drawn out, and is intended for students with less exposure to proof-based mathematics. We recommend teams to consult both versions when taking the exam.

General Overview:

• The Power Test will present an interesting definition or advanced concept. It will be made available online for teams to upload about 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.
• 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.
• 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.

The Day of the Competition

• The on-site portion of the competition will begin at 9:00 AM, and consist of the individual tests and the team test. The Power Test is to be submitted at registration, no later than 9:00 AM.
• No substitutions can be made after 9:00 AM. All substitutions must be reported to the Registration Table prior to the beginning of the Team Test.
• If a team arrives at the competition with fewer than 8 members, they can still participate as a team.
• Should a pre-registered student arrive late for any reason, and no substitution was made, he or she may join the team on which they are registered once his or her team begins another round of the competition. If this should happen, please contact a volunteer as soon as possible. Any missed tests will result in a score of 0 for those rounds.
• The use of “borrowed” students is prohibited. No student who is registered with one team can participate with another team.
• As a courtesy to fellow competitors, cell phones should be turned off during competition events and the awards ceremony. Proctors may require students to hand in their cell phones during the competition. The Proctor will be in possession of the cell phone at all times, and in case of emergency, he/she will be available to answer any phone calls.

The On-Site Tests

Any infraction of general test rules or rules for specific tests may result in disqualification of those involved, removal of said students from the competition, and deletion of those students’ scores.

On Using Aids for On-Site Tests

• No calculators may be used for any test on the day of the competition.
• Rulers (straight edges), protractors, and compasses will not be allowed for any test on the day of the competition.
• Use of a dictionary in book form only (no computerized versions or versions printed on loose pages) will be allowed for participants whose first language is not English.
• Use of a visual aid such as a magnifying glass is allowed for participants with a visual handicap. These must be inspected prior to the exams by the Proctor. If any question arises as to the legitimacy of the device, the current Director of the Princeton University Mathematics Competition will have the final word.
• No books, websites, notes, or other references may be used for any test for this competition.

The Team Test

This year's team test will differ from team tests in the past. We hope the format chosen for this year's team test will encourage more teamwork than traditional team exams.

General Overview:

• The Team Test will consist of 10 short-answer questions, and will last 30 minutes. Team members may collaborate during this round of the competition.
• Proctors will give 15-minute, 5-minute, and 1-minute warnings, and no other information. If you would like your proctor to give more or fewer time announcements, you may request so. However, no other changes could be made to the administration of the test.
• At the end of the 30 minutes, teams must submit the completed official answer sheet to their Proctor. Copying of answers from the blackboard or other sheets of paper to the official answer sheet after the time limit will not be allowed.
• During the Team Test, any team member found communicating with anyone other than his or her teammates or Proctor will result in disqualification of the entire team from the Team Test.
• It is recommended that at the 1-minute mark, teams begin to write answers down onto the official PUMaC answer sheet if they have not done so yet. Points will not be taken off for guessing, so it is also recommended to have an answer for each question.

The Individual Tests

The individual tests make up the largest percentage of the final team's score, and a lot of time has been spent creating each exam. Individuals should choose exams based on what they are most comfortable in, but teams should keep in mind that only the top five scores in each category will count. No team should have more than 5 people taking each subject exam if they wish to attain the highest score possible.

General Overview:

• Each contestant will choose two individual tests to take during the competition.
• This year, we offer individual tests in the following categories:
  • Algebra
  • Combinatorics
  • Geometry
  • Number Theory
• Each test will have a one hour time limit.
• Teams cannot collaborate during this part of the competition.
• Individual tests will be comprised of 10 questions, with different point values for different questions, for a total of 42 points.

Challenging Errata for Short Answer Tests

Due to the efforts put into the test making this year, we hope to ensure that there will be no errors on the test. As such, we will not have formal arrangements for contestants who feel certain problems were ambiguous. As with any problems with the competition, however, feel free to contact any volunteer about any concerns you may have.

Mini Events

Minievents are a way for competitors to finally take a break from the morning of test-taking. Competitors are encouraged to participate in some of the scheduled Minievents and enjoy campus while test scores are (efficiently) compiled.

General Overview:

• After lunch, teams are invited to attend any of the campus Mini Events.
• Many Mini Events may offer small prizes, but none of the results from any of the mini events will affect the team's final results in the main competition.
• The following is a list of Minievents that will be available for students to participate in:
  • Math Bowl
  • Rubik's Cube
  • Useless Math Olympiad
  • Chess

The Individual Finals

The individual finals will be the ultimate deciding factor in the final individual rankings in both divisions. The top individuals in each subject (approximately 10 from each test for each division) will be announced in many places throughout the competition area, and they will be asked to find their way to an individual finals testing area so they can be considered in the final individual competition rankings. The exam will be comprised of questions requiring answers with proof, similar to the Power Round but requiring much less time to finish. This year, the test will actually be available to whoever wants to take it, and will be available to take at three different times after the individual and team rounds. During earlier times, official individual finalists may not have been announced yet, but those that feel they did well are encouraged to take the individual finals early so that there is time later to enjoy the minievents.

General Overview:

• The exam will be comprised of approximately two or three questions requiring proofs for answers. There will be a 60-minute time limit on the exam.
• Grading priority will be given to those that are official individual finalists.
• Scores on the individual finals exam will NOT be considered in the final team score. Individual finals scores are used solely for the final individual rankings.
• Teams are NOT allowed to collaborate during this round.
• Tests will be graded based on how correct, succinct, and simple the solutions given by competitors are.
• Although correctness is the most important aspect of a person's proof, ties will be broken by considering which competitor solved the more difficult problems. Beyond that, ties will be broken by considering which competitor has the highest quality proof.
• Volunteers will be aware of each testing site, and will be able to escort students to the correct testing sites if necessary.

Preparation

First and foremost, we recommend a good night's sleep the night before the competition.

The Art of Problem Solving offers online math courses and is home to a very active high school contest mathematics forum. Also, the website sells many texts that cover material that could be seen on PUMaC exams.

Other Competition Pages

PUMaC 2009 -- Main Page || Rules

PUMaC 2008 -- Main Page || Rules || Tests || Results

PUMaC 2007 -- Main Page

PUMaC 2006 -- Main Page