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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 Birkhaüser 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)c2007BirkhaüserBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit- tenpermissionofthepublisher(BirkhaüserBoston,c/oSpringerScience+BusinessMediaLLC,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. 9 8 7 6 5 4 3 2 1 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.