Official Competition Rules 2008

From MathClub

These are the official rules governing the Princeton University Mathematics Competition

Students are expected to obey the rules and respect the property of Princeton University, the Undergraduate Mathematics Club, and their respective teams. Failure to comply may lead to removal from participation in the competition, and, if deemed serious enough by Princeton University Faculty or Staff, and/or Mathematics Club Officers, removal from the premises by a parent or guardian.

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 The Math Bowl 5.6 The Individual Finals

Eligibility

1. A full team will consist of 8 members.
2. A team member must not have reached his or her 20th birthday before January 1, 2008.
3. A team member must not have been enrolled full-time in a post-secondary institution before January 1, 2008.
4. 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. There will be no exceptions.
5. See Registration Guidelines for registration deadlines. Teams that register after the late registration deadline, or are not among the first 320 students to register, 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.
6. No teams can register for the competition after the late registration/payment deadline (Friday, December 12th, 2008). 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.
7. No team can be added to the competition after Friday, January 23rd, 2009.
8. 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.
9. Teams may register for the A or the B Division. The A Division is for very strong and/or experienced teams. Though the B Division exists partly to help students gain experience and skill, it is also competitive, and prizes will be awarded. Divisions will be given different tests on the day of the competition. The Power Test will be identical for both divisions, but teams in different divisions will be held to different grading standards. Awards for the two divisions will be separately calculated.
10. Teams that do not come from a single high school must compete in the A Division.
11. 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.

1. 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.
2. 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.
3. 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.
4. 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…
5. The letter i will be used for complex numbers (i² = − 1).
6. 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.
7. Prime numbers refer to positive primes only.
8. 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).
9. Diagrams are not necessarily drawn to scale.
10. 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}.
11. 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

1. The Power Test will present an interesting definition or advanced concept. It will be sent to the teams in advance of the competition, and should be opened no sooner than 2:00 PM on Friday, January 23rd, 2009.
2. The entire team may collaborate on this test. There is no time limit.
3. Any resources may be used, including calculators, books, and web sites. However, students may not confer with anyone except their team mates. If the Princeton University Mathematics Competition staff have reason to believe that someone outside of the team has aided in the Power Test in any way, that team will be subject to disqualification from that round.
4. Students will be required to submit written solutions with justification. Solutions are to be submitted at registration on the day of the competition.
5. The Power Test will have several parts. Results from previous parts may be assumed true in later parts, even if the team fails to prove these previous results. Students should cite these results when they use them. For example, “From Part II, Question 1, …”
6. Solutions should be written on one side of the paper only.

The Day of the Competition

1. The on-site portion of the competition will begin at 9:00 AM, and consist of the Team Test, the Individual Test, and the Math Bowl. The Power Test is to be submitted at registration, no later than 9:00 AM.
2. 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.
3. If a team arrives at the competition with fewer than 8 members, they can still participate as a team.
4. 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 during the next official break in the competition (before the Individual Test or the Math Bowl.)
5. The use of “borrowed” students is prohibited. No student who is registered with one team can participate with another team.
6. As a courtesy to fellow competitors, cell phones should be turned off during competition events and the awards ceremony.

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

1. No calculators may be used for any test on the day of the competition.
2. Rulers (straight edges), protractors, and compasses will not be allowed for any test on the day of the competition.
3. 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.
4. 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 Team Test 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.
5. No books, websites, notes, or other references may be used for any test on the day of the competition.

The Team Test

1. The Team Test will consist of 10 short-answer questions, and will last 60 minutes. The entire team may collaborate on this test.
2. Each correct answer to the first 5 problems will be worth 4 points, and each correct answer to the last 5 problems will be worth 6 points. Under no circumstances will partial credit be given for an answer.
3. Proctors will give 5-minute and 1-minute warnings, and no other information.
4. At the end of the 60 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.
5. 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.
6. Cell phones will not be allowed in the room during the Team Test. If a cell phone is discovered in the room then that team will be disqualified from that round. Proctors will collect cell phones before the test begins, and will return them after the answer sheet has been submitted.

The Individual Tests

1. Four different Individual Tests will be offered in the following topics: Algebra, Geometry, Combinatorics, and Number Theory.
2. Each student will take two different Individual Tests.
3. A team may have as many students take each test as they choose, but only the top four scores in each test will be considered for the team’s total score. To prevent conflicts of interest on the day of the competition for half teams that have been paired together, only the top two scores in each test will be considered from each half team. Of course, the optimal strategy for a well-rounded team would be to have exactly four students take each Individual Test. However, the Undergraduate Math Club recognizes that certain teams may not have any experience in, say, Combinatorics. The purpose of this rule is to allow all students from such teams to take tests that they are comfortable with, while keeping the competition fair for other teams.
4. The Individual Tests that each student is taking must be declared by the late registration/payment received deadline (Friday, December 12th, 2008), and may be changed up until 9:00 AM on the day of the competition. All changes must be reported to the Registration Table prior to the beginning of the Team Test on the day of the competition.
5. If a student takes an Individual Test that he or she is not signed up for, his or her results will be ignored.
6. All Individual Tests will consist of 10 short-answer questions, and will last 60 minutes.
7. Each correct answer will be worth 1 point, so that a full team of ten students can gain a maximum of 160 points from the Individual Tests. Partial teams with x<8 students can gain a maximum of 20x points from the Individual Tests.
8. Proctors will give 5-minute and 1-minute warnings, and no other information.
9. During an Individual Test, there will be absolutely no communication between the students from the time the question papers are handed out to the time the answer sheets are collected.
10. At the end of 60 minutes, students must submit completed official answer sheets to the Proctor. Copying of answers from other sheets of paper to the official answer sheet after the time limit will not be allowed.
11. Students will be placed in the same room as those taking the same pair of tests. After the first 60-minute session, answer sheets will be collected, there will be a 5-minute break, and the second Individual Test will begin.

Challenging Errata for Short Answer Tests

1. After each of the Team Test and the Individual Tests, team coaches may pick up answers from the Registration Table. Coaches may confer with students during lunch, and challenge any answers that appear to be incorrect.
2. All challenges must be made by the end of lunch. Tests will be re-graded if necessary.

The Math Bowl

1. Each team will send a sub-team of up to 4 members to participate in the Math Bowl, an elimination-style tournament with buzzers. A full list of Math Bowl rules will be sent to teams in a packet after registration.
2. The math bowl is optional, and independent of the rest of the competition.

The Individual Finals

1. Top-scoring students in each Individual Test topic will be invited to participate in the Individual Finals during lunch. Coaches should check the Registration Table at the start of lunch to see if any of their students were invited to participate in the Individual Finals.
2. The Individual Finals will be a very challenging 60-minute test in which students will be required to submit fully-written solutions with justification.
3. Proctors will give 5-minute and 1-minute warnings, and no other information.
4. During the Individual Finals, there will be absolutely no communication between the students from the time the question papers are handed out to the time the solution sheets are collected.
5. At the end of 60 minutes, competitors must submit their solutions written on official solution paper, in order, with numbered pages within each solution (if that solution has more than one page). Collecting, copying, and/or re-ordering the solutions after the time limit will not be allowed. Solutions written on pages other than the provided official solution paper will not be accepted.
6. The Individual Finals will determine individual awards only, and will have no bearing on team scores.