The Contest Problem Book VIII

229 Pages·2012·0.84 MB·English

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The Contest Problem
Book VIII
American Mathematics Competitions
(AMC 10)
2000–2007
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©2008 by
The Mathematical Association of America (Incorporated)
Library of Congress Catalog Card Number 2008920905
Print ISBN 978-0-88385-825-7
Electronic ISBN 978-0-88385-963-6
Printed in the United States of America
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The Contest Problem
Book VIII
American Mathematics Competitions
(AMC 10)
2000–2007
Compiled and Edited by
J. Douglas Faires & David Wells
®
Publishedanddistributedby
TheMathematicalAssociationofAmerica
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MAAPROBLEMBOOKSSERIES
ProblemBooksisaseriesoftheMathematicalAssociationofAmericaconsist-
ing of collections of problems and solutions from annual mathematical com-
petitions;compilationsofproblems(includingunsolvedproblems)specificto
particularbranchesofmathematics; booksontheartandpracticeofproblem
solving,etc.
CouncilonPublications
JamesDaniel,Chair
RichardA.GillmanEditor
RogerNelsen
MarkSaul
TatianaShubin
TheContestProblemBookVII:AmericanMathematicsCompetitions,1995–2000Con-
tests,compiledandaugmentedbyHaroldB.Reiter
TheContestProblemBookVIII:AmericanMathematicsCompetitions(AMC10),2000–
2007Contests,compiledandeditedbyJ.DouglasFaires&DavidWells
TheContestProblemBookIX:AmericanMathematicsCompetitions(AMC12),2000–
2007Contests,compiledandeditedbyDavidWells&J.DouglasFaires
AFriendlyMathematicsCompetition:35YearsofTeamworkinIndiana,editedbyRick
Gillman
TheInquisitiveProblemSolver,PaulVaderlind,RichardK.Guy,andLorenC.Larson
InternationalMathematicalOlympiads1986–1999,MarcinE.Kuczma
MathematicalOlympiads1998–1999:ProblemsandSolutionsFromAroundtheWorld,
editedbyTituAndreescuandZumingFeng
MathematicalOlympiads1999–2000:ProblemsandSolutionsFromAroundtheWorld,
editedbyTituAndreescuandZumingFeng
MathematicalOlympiads2000–2001:ProblemsandSolutionsFromAroundtheWorld,
editedbyTituAndreescu,ZumingFeng,andGeorgeLee,Jr.
TheWilliamLowellPutnamMathematicalCompetitionProblemsandSolutions:1938–
1964,A.M.Gleason,R.E.Greenwood,L.M.Kelly
TheWilliamLowellPutnamMathematicalCompetitionProblemsandSolutions:1965–
1984,GeraldL.Alexanderson,LeonardF.Klosinski,andLorenC.Larson
TheWilliamLowellPutnamMathematicalCompetition1985–2000: Problems, Solu-
tions,andCommentary,KiranS.Kedlaya,BjornPoonen,RaviVakil
USAandInternationalMathematicalOlympiads2000,editedbyTituAndreescuand
ZumingFeng
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USAandInternationalMathematicalOlympiads2001,editedbyTituAndreescuand
ZumingFeng
USAandInternationalMathematicalOlympiads2002,editedbyTituAndreescuand
ZumingFeng
USAandInternationalMathematicalOlympiads2003,editedbyTituAndreescuand
ZumingFeng
USA and International Mathematical Olympiads 2004, edited by Titu Andreescu,
ZumingFeng,andPo-ShenLoh
MAAServiceCenter
P.O.Box91112
Washington,DC20090-1112
1-800-331-1622 fax: [phone]
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Contents
Preface............................................................ ix
ABriefHistoryoftheContests . . . . . . . . . . . . . . . . . . . ix
ConstructionoftheContests . . . . . . . . . . . . . . . . . . . . . xi
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . xv
Problems.......................................................... 1
2000AMC10Problems . . . . . . . . . . . . . . . . . . . . . . . 1
2001AMC10Problems . . . . . . . . . . . . . . . . . . . . . . . 6
2002AMC10AProblems . . . . . . . . . . . . . . . . . . . . . . 12
2002AMC10BProblems . . . . . . . . . . . . . . . . . . . . . . 17
2003AMC10AProblems . . . . . . . . . . . . . . . . . . . . . . 21
2003AMC10BProblems . . . . . . . . . . . . . . . . . . . . . . 26
2004AMC10AProblems . . . . . . . . . . . . . . . . . . . . . . 31
2004AMC10BProblems . . . . . . . . . . . . . . . . . . . . . . 36
2005AMC10AProblems . . . . . . . . . . . . . . . . . . . . . . 41
2005AMC10BProblems . . . . . . . . . . . . . . . . . . . . . . 46
2006AMC10AProblems . . . . . . . . . . . . . . . . . . . . . . 50
2006AMC10BProblems . . . . . . . . . . . . . . . . . . . . . . 55
2007AMC10AProblems . . . . . . . . . . . . . . . . . . . . . . 60
2007AMC10BProblems . . . . . . . . . . . . . . . . . . . . . . 65
Solutions.......................................................... 71
2000AMC10Solutions . . . . . . . . . . . . . . . . . . . . . . . 72
2001AMC10Solutions . . . . . . . . . . . . . . . . . . . . . . . 80
2002AMC10ASolutions . . . . . . . . . . . . . . . . . . . . . . 89
2002AMC10BSolutions . . . . . . . . . . . . . . . . . . . . . . 99
2003AMC10ASolutions . . . . . . . . . . . . . . . . . . . . . . 107
2003AMC10BSolutions . . . . . . . . . . . . . . . . . . . . . . 116
2004AMC10ASolutions . . . . . . . . . . . . . . . . . . . . . . 124
2004AMC10BSolutions . . . . . . . . . . . . . . . . . . . . . . 132
2005AMC10ASolutions . . . . . . . . . . . . . . . . . . . . . . 142
vii
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viii TheContestProblemBookVIII
2005AMC10BSolutions . . . . . . . . . . . . . . . . . . . . . . 152
2006AMC10ASolutions . . . . . . . . . . . . . . . . . . . . . . 161
2006AMC10BSolutions . . . . . . . . . . . . . . . . . . . . . . 171
2007AMC10ASolutions . . . . . . . . . . . . . . . . . . . . . . 181
2007AMC10BSolutions . . . . . . . . . . . . . . . . . . . . . . 192
IndexofProblems.................................................200
AbouttheEditors ................................................. 211
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Preface
A Brief History of the Contests
Overview
Thecontestsfromwhichtheproblemsinthisbookweredrawnaredirectde-
scendantsofonecreatedin1950bytheMetropolitanNewYorkSectionofthe
Mathematical Association of America (MAA). Originally given only to stu-
dents in the New York City area, the contests were first offered nationally in
1957underthesponsorshipoftheMAAandtheSocietyofActuaries.Thepri-
maryobjectiveofthecontestswas,andstillis,topromotethestudyofmathe-
maticsbyprovidinghighschoolstudentswithapositiveexperienceincreative
problemsolving.Asecondaryobjectiveofidentifyingmathematicallytalented
students was introduced in1972, when about 100 topscorerswere invitedto
participateintheUSAMathematicalOlympiad(USAMO),anextremelychal-
lengingproof-orientedcontest.
The1950contestwasentitledsimply,“MathematicalContest.”Theword
“Annual” was added to the name in 1953. The solution booklet from 1959 is
titled, “Annual H.S. Mathematics Examination.” Between 1959 and 1982 the
titlesofthecontestandsolutionbookletsusedeither“Contest”or“Examina-
tion,”thelatterappearingmorefrequentlyinthelateryears.Thewords“High
School” or the initials H.S. were sometimes, but not always, included in the
title.In1983thecontestofficiallybecametheAmericanHighSchoolMathe-
maticsExamination(AHSME).Inthesameyearathirdcontest,theAmerican
InvitationalMathematicsExamination(AIME),wasintroducedasastepping
stone between the AHSME and the USAMO. Computational in nature, but
requiring more creativity than the AHSME, the AIME was deemed a more
reliablevehicleforidentifyingsuitableUSAMOparticipants.Twoyearslater
a contest for middle school students was created. The American Junior High
SchoolMathematicsExamination(AJHSME)wasmodeledaftertheAHSME.
Shortlythereafter,theentireprogramofcontestsbecameknownastheAmer-
icanMathematicsCompetitions(AMC).
ix
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