AMS / MAA PROBLEM BOOKS VOL 31 VII The Contest Problem Book American Mathematics Competitions 1995–2000 Contests Compiled and augmented by Harold B. Reiter The Contest Problem Book VII American Mathematics Competitions 1995–2000 Contests Originally published by The Mathematical Association of America, 2006. ISBN: 978-1-4704-4970-4 LCCN: 2005937659 Copyright © 2006, held by the American Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2019 The American Mathematical Society retains all rights except those granted to the United States Government. ⃝1 The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 24 23 22 21 20 19 10.1090/prb/031 AMS/MAA PROBLEM BOOKS VOL 31 The Contest Problem Book VII American Mathematics Competitions 1995–2000 Contests Compiled and augmented by Harold B. Reiter MAA PROBLEM BOOKS SERIES Problem Books is a series of the Mathematical Association of America consisting of collections of problems and solutions from annual mathemat- ical competitions; compilations of problems (including unsolved problems) specific to particular branches of mathematics; books on the art and practice of problem solving, etc. Council on Publications Roger Nelsen, Chair Roger Nelsen Editor Irl C. Bivens Richard A. Gibbs Richard A. Gillman Gerald Heuer Elgin Johnston Kiran Kedlaya Loren C. Larson Margaret M. Robinson Mark Saul Tatiana Shubin The Contest Problem Book VII: American Mathematics Competitions, 1995{2000 Contests, compiled and augmented by Harold B. Reiter The Contest Problem Book VIII: American Mathematics Competitions (AMC 10), 2000{2007 Contests, compiled and edited by J. Douglas Faires & David Wells The Contest Problem Book IX: American Mathematics Competitions (AMC 12), 2001{2007 Contests, compiled and edited by J. Douglas Faires & David Wells A Friendly Mathematics Competition: 35 Years of Teamwork in Indiana, edited by Rick Gillman The Inquisitive Problem Solver, Paul Vaderlind, Richard K. Guy, and Loren C. Larson International Mathematical Olympiads 1986{1999, Marcin E. Kuczma Mathematical Olympiads 1998{1999: Problems and Solutions From Around the World, edited by Titu Andreescu and Zuming Feng Mathematical Olympiads 1999{2000: Problems and Solutions From Around the World, edited by Titu Andreescu and Zuming Feng Mathematical Olympiads 2000{2001: Problems and Solutions From Around the World, edited by Titu Andreescu, Zuming Feng, and George Lee, Jr. The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938{1964, A. M. Gleason, R. E. Greenwood, L. M. Kelly The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965{1984, Gerald L. Alexanderson, Leonard F. Klosinski, and Loren C. Larson The William Lowell Putnam Mathematical Competition 1985{2000: Problems, Solutions, and Commentary, Kiran S. Kedlaya, Bjorn Poonen, Ravi Vakil USA and International Mathematical Olympiads 2000, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2001, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2002, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2003, edited by Titu Andreescu and Zuming Feng USA and International Mathematical Olympiads 2004, edited by Titu Andreescu, Zuming Feng, and Po-Shen Loh Contents Preface . ix 46th AHSME, 1995 . 1 47th AHSME, 1996 . 9 48th AHSME, 1997 . 17 49th AHSME, 1998 . 25 50th AHSME, 1999 . 33 Sample AMC 10, 1999 . 39 51st AMC 12, 2000 . 45 1st AMC 10, 2000 . 53 50th Anniversary AHSME . 59 46th AHSME solutions, 1995 . 71 47th AHSME solutions, 1996 . 83 48th AHSME solutions, 1997 . 95 49th AHSME solutions, 1998 . 107 50th AHSME solutions, 1999 . 119 Sample AMC 10 solutions, 1999 . 129 51st AMC 12 solutions, 2000 . 135 1st AMC 10 solutions, 2000 . 145 Additional Problems . .153 Solutions to Additional Problems . 159 Classification . 175 About the Editor . 183 vii Preface History Name and sponsors The exam now known as the AMC 12 began in 1950 as the Annual High School Contest under the sponsorship of the Metropolitan (New York) Section of the Mathematical Association of America (MAA). It was offered only in New York state until 1952 when it became a national contest under the sponsorship of the MAA and the Society of Actuaries. By 1982, sponsorship had grown to include the national high school and two-year college honorary mathematics society Mu Alpha Theta, the National Council of Teachers of Mathematics (NCTM), and the Casualty Actuary Society. Today there are twelve sponsoring organizations, which, in addition to the above, include the American Statistical Association, the American Mathematical Association of Two-Year Colleges, the American Mathematical Society, the American Society of Pension Actuaries, the Consortium for Mathematics and its Applications, the national collegiate honorary mathematics society Pi Mu Epsilon, and the National Association of Mathematicians. During the years 1973{1982 the exam was called the Annual High School Mathematics Examination. The name American High School Mathematics Examination and the better known acronym AHSME, were introduced in 1983. At this time, the organizational unit became the American Mathematics Competitions (AMC), a subcommittee of the Mathematical Association of America. Also in 1983, a new exam, the American Invitational Math Exam (AIME), was introduced. Two years later, the AMC introduced the American Junior High School Mathematics Examination (AJHSME). In February 2000, the AMC introduced the AMC ix x The Contest Problem Book VII 10 for students in grade ten and below. At the same time, the AMC changed the name AJHSME to AMC 8 and AHSME to AMC 12. The two high school exams became 25 question, 75 minute exams. Participation Before 1992, the scoring of the exam was done locally, in some states by the teacher-managers themselves, and in other states by the volunteer state director. Beginning in 1992, all the scoring was done at the AMC office in Lincoln, Nebraska. Beginning in 1994, students were asked to indicate their sex on the answer form. The following table shows the degree of participation and average scores among females versus that for males. Year Females Mean Males Mean Unspecified Mean 1994 104,471 68.8 120,058 76.0 6,530 70.6 1995 115,567 72.3 133,523 78.5 6,877 73.7 1996 124,491 65.8 142,750 71.2 6,659 67.8 1997 120,649 63.8 140,359 69.8 7,944 65.5 1998 108,386 66.5 128,172 71.9 7,438 67.8 1999 105,705 66.1 126,992 71.1 8,200 67.5 2000(12) 71,272 61.0 89,965 67.9 5671 64.3 2000(10) 49,288 60.8 52,836 67.5 4870 63.6 Related Exams Until the introduction of the AIME in 1983, the AHSME was used for several purposes. It was introduced in order to promote interest in problem solving and mathematics among high school students. It was also used to select participants in the United States of America Mathematical Olympiad (USAMO), the six question, nine hour exam given each May to honor and reward the top high school problem solvers in America. The USAMO was used to pick the six-student United States Mathematical Olympiad team for the International Mathematical Olympiad competition held each July. With the introduction of the AIME, which was given the primary role of selecting USAMO participants, the AHSME question writing committee began to focus on the primary objective: providing students with an enjoyable problem-solving adventure. The AHSME became accessible to a much larger body of students. Some 7th and 8th graders, encouraged by their successes on the AJHSME, began participating. Preface xi Calculators In 1994, calculators were allowed for the first time. At that time, the AMC established the policy that every problem had to have a solution without a calculator that was no harder than a calculator solution. In 1996, this rule was modified to read ”every problem can be solved without the aid of a calculator'. Of course the availability of the graphing calculator, and now calculators with computer algebra systems (CAS) capabilities has changed the types of questions that can be asked. Allowing the calculator has had the effect of limiting the use of certain computational problems. Referring to the Special Fiftieth Anniversary AHSME, problems [1954{ 38], [1961{5], [1969{29], [1974{20], [1976{30], [1980{18], [1981{24], and [1992{14] would all have to be eliminated, either because of the graphing calculator's —solve and graphing" capabilities or because of the symbolic algebra capabilities of some recent calculators. But the AMC has felt, like NCTM, that students must learn when not to use the calculator as well as when and how to use it. Thus questions which become more difficult when the calculator is used indiscriminately are becoming increasingly popular with the committee. For example, consider [1999{21] below: how many solutions does cos.log x/ 0 have on the interval D .0; 1/? Students whose first inclination is to construct the graph of the function will be led to the answer 2 since in each viewing window, the function appears to have just two intercepts. However, the composite function has infinitely many x-intercepts. Scoring The number of problems and the scoring system has changed over the history of the exam. In the first years of the AHSME, there were a total of 50 questions with point values of 1, 2, or 3. In 1960, the number of questions was reduced from 50 to 40 and, in 1967, was further reduced from 40 to 35. The exam was reduced to 30 questions in 1974. In 1978, the scoring system was changed to the formula 30 4R W, where R is C the number of correct answers and W is the number of wrong answers.