The Contest Problem Book VII - American Mathematics Competitions 1995-2000

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The Contest Problem Book VII - American Mathematics Competitions 1995-2000Compiled and augmented by Harold B. Reiter

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2006 byThe Mathematical Association of America (Incorporated)Library of Congress Catalog Card Number 2005937659ISBN 0-88385-821-5Printed in the United States of AmericaCurrent Printing (last digit):10987654321The Contest ro lemBook v1:1America athematics Compet-tio s995-2000 Co testsComp"led a d Augmented byHarold B. ReiterPublished anddistributed byThe Mathematical A sociation of AmericaMAA PROBLEM BOOKS SERIESProblem Books is a series of the Mathematical Association of Americaconsisting 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 andpractice of problem solving, etc.Council on PublicationsRoger Nelsen, ChairRoger Nelsen EditorIrl C. Bivens Richard A. GibbsRichard A. Gillman Gerald HeuerElgin Johnston Kiran KedlayaLoren C. Larson Margaret M. RobinsonMark Saul Tatiana ShubinThe Contest Problem Book VII: American Mathematics Competitions,1995-2000 Contests, compiled and augmented by Harold B. ReiterA Friendly Mathematics Competition: 35 Years of Teamwork in Indiana,edited by Rick GillmanThe Inquisitive Problem Solver, Paul Vaderlind, Richard K. Guy, and LorenC. LarsonInternational Mathematical Olympiads 1986-1999, Marcin E. KuczmaMathematical Olympiads 1998-1999: Problems and Solutions FromAround the World, edited by Titu Andreescu and Zuming FengMathematical Olympiads 1999-2000: Problems and Solutions FromAround the World, edited by Titu Andreescu and Zuming FengMathematical Olympiads 2000-2001: Problems and Solutions FromAround the World, edited by Titu Andreescu, Zuming Feng, and GeorgeLee, Jr.The William Lowell Putnam Mathematical Competition Problems andSolutions: 1938-1964, A. M. Gleason, R. E. Greenwood, L. M. KellyThe William Lowell Putnam Mathematical Competition Problems andSolutions: 1965-1984, Gerald L. Alexanderson, Leonard F. Klosinski,and Loren C. LarsonThe William Lowell Putnam Mathematical Competition 1985-2000: Prob-lems, Solutions, and Commentary, Kiran S. Kedlaya, Bjorn Poonen,Ravi VakilUSA and International Mathematical Olympiads 2000, edited by TituAndreescu and Zuming FengUSA and International Mathematical Olympiads 2001, edited by TituAndreescu and Zuming FengUSA and International Mathematical Olympiads 2002, edited by TituAndreescu and Zuming FengUSA .and International Mathematical Olympiads 2003, edited by TituAndreescu and Zuming FengUSA and International Mathematical Olympiads 2004, edited by TituAndreescu, Zuming Feng, and Po-Shen LohMAA Service CenterP. O. Box 91112Washington, DC 20090-11121-800-331-1622 fax: 1-301-206-9789ContentsPreface IX46th AHSME, 1995 147th AHSME, 1996 948th AHSME, 1997 1749th AHSME, 1998 2550th AHSME, 1999 33Sample AMC 10, 1999 3951st AMC 12, 2000 451st AMC 10, 2000 5350th Anniversary AHSME 5946th AHSME solutions, 1995 7147th AHSME solutions, 1996 8348th AHSME solutions, 1997 9549th AHSME solutions, 1998 10750th AHSME solutions, 1999 119Sample AMC 10 solutions, 1999 12951st AMC 12 solutions, 2000 1351st AMC 10 solutions, 2000 145Additional Problems 153Solutions to Additional Problems 159Classification 175About the Editor 183viiPrefaceHistoryName and sponsorsThe exam now known as the AMC 12 began in 1950 as the AnnualHigh School Contest under the sponsorship of the Metropolitan (NewYork) Section of the Mathematical Association of America (MAA). Itwas offered only in New York state until 1952 when it became a nationalcontest under the sponsorship of the MAA and the Society of Actuaries.By 1982, sponsorship had grown to include the national high schooland two-year college honorary mathematics society Mu Alpha Theta, theNational Council of Teachers of Mathematics (NCTM), and the CasualtyActuary Society. Today there are twelve sponsoring organizations, which,in addition to the above, include the American Statistical Association, theAmerican Mathematical Association of Two-Year Colleges, the AmericanMathematical Society, the American Society of Pension Actuaries, theConsortium for Mathematics and its Applications, the national collegiatehonorary mathematics society Pi Mu Epsilon, and the National Associationof Mathematicians. During the years 1973-1982 the exam was called theAnnual High School Mathematics Examination. The name American HighSchool Mathematics Examination and the better known acronym AHSME,were introduced in 1983. At this time, the organizational unit becamethe American Mathematics Competitions (AMC), a subcommittee of theMathematical Association of America. Also in 1983, a new exam, theAmerican Invitational Math Exam (AIME), was introduced. Two yearslater, the AMC introduced the American Junior High School MathematicsExamination (AJHSME). In February 2000, the AMC introduced the AMCixx The Conlest Problem Book VII10 for students in grade ten and below. At the same time, the AMC changedthe name AJHSME to AMC 8 and AHSME to AMC 12. The two highschool exams became 25 question, 75 minute exams.ParticipationBefore 1992, the scoring of the exam was done locally, in some states bythe teacher.managers themselves, and in other states by the volunteer statedirector. Beginning in 1992, all the scoring was done at the AMC officein Lincoln, Nebraska. Beginning in 1994, students were asked to indicatetheir sex on the answer form. The following table shows the degree ofparticipation and average scores among females versus that for males.Year Females Mean Males Mean Unspecified Mean1994 104,471 68.8 120,058 76.0 6,530 70.61995 115,567 72.3 133,523 78.5 6,877 73.71996 124,491 65.8 142,750 71.2 6,659 67.81997 120,649 63.8 140,359 69.8 7,944 65.51998 108,386 66.5 128,172 71.9 7,438 67.81999 105,705 66.1 126,992 71.1 8,200 67.52000(12) 71,272 61.0 89,965 67.9 5671 64.32000(10) 49,288 60.8 52,836 67.5 4870 63.6Related ExamsUntil the introduction of the AIME in 1983, the AHSME was used forseveral purposes. It was introduced in order to promote interest in problemsolving and mathematics among high school students. It was also used toselect participants in the United States of America Mathematical Olympiad(USAMO), the six question, nine hour exam given each May to honor andreward the top high school problem solvers in America. The USAMO wasused to pick the six-student United States Mathematical Olympiad team forthe International Mathematical Olympiad competition held each July. Withthe introduction ofthe AIME, which was given the primary role of selectingUSAMO participants, the AHSME question writing committee began tofocus on the primary objective: providing students with an enjoyableproblem-solving adventure. The AHSME became accessible to a muchlarger body of students. Some 7th and 8th graders, encouraged by theirsuccesses on the AJHSME, began participating.PrefaceCalculatorsxiIn 1994, calculators were allowed for the first time. At that time, the AMCestablished the policy that every problem had to have a solution withouta calculator that was no harder than a calculator solution. In 1996, thisrule was modified to read 'every problem can be solved without the aidof a calculator'. Of course the availability of the graphing calculator,and now calculators with computer algebra systems (CAS) capabilities haschanged the types of questions that can be asked. Allowing the calculatorhas 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 thegraphing calculator's "solve and graphing" capabilities or because of thesymbolic algebra capabilities of some recent calculators. But the AMC hasfelt, like NCTM, that students must learn when not to use the calculatoras well as when and how to use it. Thus questions which becomemore difficult when the calculator is used indiscriminately are becomingincreasingly popular with the committee. For example, consider [1999-21]below: how many solutions does cos(logx) = 0 have on the interval(0, I)? Students whose first inclination is to construct the graph of thefunction will be led to the answer 2 since in each viewing window, thefunction appears to have just two intercepts. However, the compositefunction has infinitely many x-intercepts.ScoringThe number of problems and the scoring system has changed over thehistory of the exam. In the first years of the AHSME, there were a totalof 50 questions with point values of 1, 2, or 3. In 1960, the number ofquestions was reduced from 50 to 40 and, in 1967, was further reducedfrom 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 i