104 number theory problems: from the training of the USA IMO team

213 Pages·2007·0.685 MB·English

by  Titu Andreescu, Dorin Andrica, Zuming Feng

About the Authors
Titu Andreescu received his Ph.D. from the West University of Timisoara, Ro-
mania.Thetopicofhisdissertationwas“ResearchonDiophantineAnalysisand
Applications.”ProfessorAndreescucurrentlyteachesatTheUniversityofTexas
atDallas.HeispastchairmanoftheUSAMathematicalOlympiad,servedasdi-
rectoroftheMAAAmericanMathematicsCompetitions(1998–2003),coachof
theUSAInternationalMathematicalOlympiadTeam(IMO)for10years(1993–
2002), director of the Mathematical Olympiad Summer Program (1995–2002),
andleaderoftheUSAIMOTeam(1995–2002).In2002Tituwaselectedmember
oftheIMOAdvisoryBoard,thegoverningbodyoftheworld’smostprestigious
mathematics competition. Titu co-founded in 2006 and continues as director of
theAwesomeMathSummerProgram(AMSP).HereceivedtheEdythMaySliffe
Award for Distinguished High School Mathematics Teaching from the MAA in
1994anda“CertificateofAppreciation”fromthepresidentoftheMAAin1995
forhisoutstandingserviceascoachoftheMathematicalOlympiadSummerPro-
graminpreparingtheUSteamforitsperfectperformanceinHongKongatthe
1994 IMO. Titu’s contributions to numerous textbooks and problem books are
recognizedworldwide.
Dorin Andrica received his Ph.D in 1992 from “Babes¸-Bolyai” University in
Cluj-Napoca, Romania; his thesis treated critical points and applications to the
geometryofdifferentiablesubmanifolds.ProfessorAndricahasbeenchairmanof
the Department of Geometry at “Babes¸-Bolyai” since 1995. He has written and
contributedtonumerousmathematicstextbooks,problembooks,articlesandsci-
entificpapersatvariouslevels.Heisaninvitedlectureratuniversityconferences
around the world: Austria, Bulgaria, Czech Republic, Egypt, France, Germany,
Greece, Italy, the Netherlands, Portugal, Serbia, Turkey, and the USA. Dorin is
a member of the Romanian Committee for the Mathematics Olympiad and is a
memberontheeditorialboardsofseveralinternationaljournals.Also,heiswell
knownforhisconjectureaboutconsecutiveprimescalled“Andrica’sConjecture.”
HehasbeenaregularfacultymemberattheCanada–USAMathcampsbetween
2001–2005andattheAwesomeMathSummerProgram(AMSP)since2006.
ZumingFengreceivedhisPh.D.fromJohnsHopkinsUniversitywithemphasis
on Algebraic Number Theory and Elliptic Curves. He teaches at Phillips Exeter
Academy.ZumingalsoservedasacoachoftheUSAIMOteam(1997–2006),was
thedeputyleaderoftheUSAIMOTeam(2000–2002),andanassistantdirectorof
theUSAMathematicalOlympiadSummerProgram(1999–2002).Hehasbeena
memberoftheUSAMathematicalOlympiadCommitteesince1999,andhasbeen
the leader of the USA IMO team and the academic director of the USA Mathe-
matical Olympiad Summer Program since 2003. Zuming is also co-founder and
academic director of the AwesomeMath Summer Program (AMSP) since 2006.
HereceivedtheEdythMaySliffeAwardforDistinguishedHighSchoolMathe-
maticsTeachingfromtheMAAin1996and2002.
Titu Andreescu
Dorin Andrica
Zuming Feng
104 Number Theory
Problems
From the Training of the USA IMO Team
Birkha¨user
Boston • Basel • Berlin
TituAndreescu DorinAndrica
TheUniversityofTexasatDallas “Babes¸-Bolyai”University
DepartmentofScience/MathematicsEducation FacultyofMathematics
Richardson,TX75083 3400Cluj-Napoca
U.S.A. Romania
[email protected] [email protected]
ZumingFeng
PhillipsExeterAcademy
DepartmentofMathematics
Exeter,NH03833
U.S.A.
[email protected]
CoverdesignbyMaryBurgess.
MathematicsSubjectClassification(2000):00A05,00A07,11-00,11-XX,11Axx,11Bxx,11D04
LibraryofCongressControlNumber:2006935812
ISBN-10:0-8176-4527-6 e-ISBN-10:0-8176-4561-6
ISBN-13:978-0-8176-4527-4 e-ISBN-13:978-0-8176-4561-8
Printedonacid-freepaper.
(cid:1)c2007Birkha¨userBoston
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit-
tenpermissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233
SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor
scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic
adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde-
velopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey
arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare
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www.birkhauser.com (EB)
104 Number Theory Problems
TituAndreescu,DorinAndrica,ZumingFeng
October25,2006
Contents
Preface vii
Acknowledgments ix
AbbreviationsandNotation xi
1 FoundationsofNumberTheory 1
Divisibility 1
DivisionAlgorithm 4
Primes 5
TheFundamentalTheoremofArithmetic 7
G.C.D. 11
EuclideanAlgorithm 12
Be´zout’sIdentity 13
L.C.M. 16
TheNumberofDivisors 17
TheSumofDivisors 18
ModularArithmetics 19
ResidueClasses 24
Fermat’sLittleTheoremandEuler’sTheorem 27
Euler’sTotientFunction 33
MultiplicativeFunction 36
LinearDiophantineEquations 38
NumericalSystems 40
DivisibilityCriteriaintheDecimalSystem 46
FloorFunction 52
Legendre’sFunction 65
FermatNumbers 70
MersenneNumbers 71
PerfectNumbers 72
vi Contents
2 IntroductoryProblems 75
3 AdvancedProblems 83
4 SolutionstoIntroductoryProblems 91
5 SolutionstoAdvancedProblems 131
Glossary 189
FurtherReading 197
Index 203
Preface
This book contains 104 of the best problems used in the training and testing of
theU.S.InternationalMathematicalOlympiad(IMO)team. Itisnotacollection
of very difficult, and impenetrable questions. Rather, the book gradually builds
students’ number-theoretic skills and techniques. The first chapter provides a
comprehensive introduction to number theory and its mathematical structures.
This chapter can serve as a textbook for a short course in number theory. This
workaimstobroadenstudents’viewofmathematicsandbetterpreparethemfor
possibleparticipationinvariousmathematicalcompetitions. Itprovidesin-depth
enrichment in important areas of number theory by reorganizing and enhancing
students’problem-solvingtacticsandstrategies. Thebookfurtherstimulatesstu-
dents’interestforthefuturestudyofmathematics.
IntheUnitedStatesofAmerica,theselectionprocessleadingtoparticipation
intheInternationalMathematicalOlympiad(IMO)consistsofaseriesofnational
contests called the American Mathematics Contest 10 (AMC 10), the American
MathematicsContest12(AMC12), theAmericanInvitationalMathematicsEx-
amination (AIME), and the United States of America Mathematical Olympiad
(USAMO). Participation in the AIME and the USAMO is by invitation only,
basedonperformanceintheprecedingexamsofthesequence. TheMathematical
Olympiad Summer Program (MOSP) is a four-week intensive training program
forapproximatelyfiftyverypromisingstudentswhohaverisentothetopinthe
American Mathematics Competitions. The six students representing the United
States of America in the IMO are selected on the basis of their USAMO scores
andfurthertestingthattakesplaceduringMOSP.ThroughoutMOSP,fulldaysof
classesandextensiveproblemsetsgivestudentsthoroughpreparationinseveral
important areas of mathematics. These topics include combinatorial arguments
and identities, generating functions, graph theory, recursive relations, sums and
products,probability,numbertheory,polynomials,functionalequations,complex
numbersingeometry,algorithmicproofs,combinatorialandadvancedgeometry,
functionalequations,andclassicalinequalities.
Olympiad-styleexamsconsistofseveralchallengingessayproblems. Correct
solutionsoftenrequiredeepanalysisandcarefulargument. Olympiadquestions
viii Preface
canseemimpenetrabletothenovice,yetmostcanbesolvedwithelementaryhigh
schoolmathematicstechniques,whencleverlyapplied.
Hereissomeadviceforstudentswhoattempttheproblemsthatfollow.
• Takeyourtime! Veryfewcontestantscansolveallthegivenproblems.
• Try to make connections between problems. An important theme of this
workisthatallimportanttechniquesandideasfeaturedinthebookappear
morethanonce!
• Olympiad problems don’t “crack” immediately. Be patient. Try different
approaches. Experimentwithsimplecases. Insomecases,workingback-
wardfromthedesiredresultishelpful.
• Even if you can solve a problem, do read the solutions. They may con-
tainsomeideasthatdidnotoccurinyoursolutions,andtheymaydiscuss
strategicandtacticalapproachesthatcanbeusedelsewhere. Thesolutions
are also models of elegant presentation that you should emulate, but they
oftenobscurethetortuousprocessofinvestigation,falsestarts,inspiration,
andattentiontodetailthatledtothem. Whenyoureadthesolutions,tryto
reconstructthethinkingthatwentintothem. Askyourself,“Whatwerethe
keyideas? HowcanIapplytheseideasfurther?”
• Gobacktotheoriginalproblemlater, andseewhetheryoucansolveitin
adifferentway. Manyoftheproblemshavemultiplesolutions,butnotall
areoutlinedhere.
• Meaningful problem solving takes practice. Don’t get discouraged if you
havetroubleatfirst. Foradditionalpractice, usethebooksonthereading
list.
TituAndreescu
DorinAndrica
ZumingFeng
October2006
Acknowledgments
ThankstoSaraCampbell,Yingyu(Dan)Gao,SherryGong,KoeneHon,RyanKo,
Kevin Medzelewski, Garry Ri, and Kijun (Larry) Seo. They were the members
of Zuming’s number theory class at Phillips Exeter Academy. They worked on
thefirstdraftofthebook. Theyhelpedproofreadtheoriginalmanuscript,raised
criticalquestions,andprovidedacutemathematicalideas. Theircontributionim-
proved the flavor and the structure of this book. We thank Gabriel Dospinescu
(Dospi)formanyremarksandcorrectionstothefirstdraftofthebook. Somema-
terialsareadaptedfrom[11], [12], [13], and[14]. Wealsothankthosestudents
whohelpedTituandZumingeditthosebooks.
Manyproblemsareeitherinspiredbyoradaptedfrommathematicalcontests
indifferentcountriesandfromthefollowingjournals:
• TheAmericanMathematicalMonthly,UnitedStatesofAmerica
• Crux,Canada
• HighSchoolMathematics,China
• MathematicsMagazine,UnitedStatesofAmerica
• RevistaMatematicaˇ Timis¸oara,Romania
Wedidourbesttocitealltheoriginalsourcesoftheproblemsinthesolution
section. We express our deepest appreciation to the original proposers of the
problems.