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Problem Books in Mathematics Edited by K. Bencsath P.R. Halmos Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo Problem Books in Mathematics Series Editors: K. Bencsdth and P.R. Halmos Polynomials by Edward J. Barbeau Problems in Geometry by Marcel Berger, Pierre Pansu, Jean-Pic Berry, and Xavier Saint-Raymond Problem Book for First Year Calculus by George W. Bluman Exercises in Probability by T. CacouUos An Introduction to HUbert Space and Quantum Logic by David W. Cohen Unsolved Problems in Geometry by Mallard T. Croft, Kenneth J. Falconer, and Richard K. Guy Problem-Solving Strategies by Arthur Engel Problems in Analysis by Bernard R. Gelbaum Problems in Real and Complex Analysis by Bernard R. Gelbaum Theorems and Counterexamples in Mathematics by Bernard R. Gelbaum and John M.H. Olmsted Exercises in Integration by Claude George Algebraic Logic by S.G. Gindikin Unsolved Problems in Number Theory (2nd ed.) by Richard K. Guy (continued after index) Arthur Engel Problem-Solving Strategies With223Figures 1 3 AngelEngel Institutfu¨rDidaktikderMathematik JohannWolfgangGoethe–Universita¨tFrankfurtamMain Senckenberganlage9–11 60054FrankfurtamMain11 Germany SeriesEditor: PaulR.Halmos DepartmentofMathematics SantaClaraUniversity SantaClara,CA95053 USA MathematicsSubjectClassification(1991):00A07 LibraryofCongressCataloging-in-PublicationData Engel,Arthur. Problem-solvingstrategies/ArthurEngel. p. cm.—(Problembooksinmathematics) Includesindex. ISBN0-387-98219-1(softcover:alk.paper) 1.Problemsolving. I.Title. II.Series. QA63.E54 1997 (cid:1) 510.76—dc21 97-10090 ©1998Springer-VerlagNewYork,Inc. Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Springer-VerlagNewYork,Inc.,175FifthAvenue,NewYork,NY10010, USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnection withanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orby similarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseofgeneraldescriptivenames,tradenames,trademarks,etc.,inthispublication,evenifthe formerarenotespeciallyidentified,isnottobetakenasasignthatsuchnames,asunderstoodbythe TradeMarksandMerchandiseMarksAct,mayaccordinlybeusedfreelybyanyone. ISBN0–387–98219–1 Springer-Verlag NewYork Berlin Heidelburg SPIN10557554 Preface ThisbookisanoutgrowthofthetrainingoftheGermanIMOteamfromatime whenwehadonlyashorttrainingtimeof14days,including6half-daytests.This hasforceduponusatrainingofenormouscompactness.“GreatIdeas”werethe leading principles. A huge number of problems were selected to illustrate these principles.Notonlytopicsbutalsoideaswereefficientmeansofclassification. Forwhomisthisbookwritten? • Fortrainersandparticipantsofcontestsofallkindsuptothehighestlevelof internationalcompetitions,includingtheIMOandthePutnamCompetition. • Fortheregularhighschoolteacher,whoisconductingamathematicsclub andislookingforideasandproblemsforhis/herclub.Here,he/shewillfind problemsofanylevelfromverysimpleonestothemostdifficultproblems everproposedatanycompetition. • Forhighschoolteacherswhowanttoposetheproblemoftheweek,problem ofthemonth,andresearchproblemsoftheyear.Thisisnotsoeasy.Manyfail, butsomepersevere,andafterawhiletheysucceedandgenerateacreative atmospherewithcontinuousdiscussionsofmathematicalproblems. • Fortheregularhighschoolteacher,whoisjustlookingforideastoenrich his/herteachingbysomeinterestingnonroutineproblems. • Forallthosewhoareinterestedinsolvingtoughandinterestingproblems. Thebookisorganizedintochapters.Eachchapterstartswithtypicalexamples illustrating the main ideas followed by many problems and their solutions. The vi Preface solutionsaresometimesjusthints,givingawaythemainidealeadingtothesolu- tion.Inthisway,itwaspossibletoincreasethenumberofexamplesandproblems toover1300.Thereadercanincreasetheeffectivenessofthebookevenmoreby tryingtosolvetheexamples. The problems are almost exclusively competition problems from all over the world.MostofthemarefromtheformerUSSR,somefromHungary,andsome fromWesterncountries,especiallyfromtheGermanNationalCompetition.The competitionproblemsareusuallyvariationsofproblemsfromjournalswithprob- lemsections.Soitisnotalwayseasytogivecredittotheoriginatorsoftheproblem. If you see a beautiful problem, you first wonder at the creativity of the problem proposer. Later you discover the result in an earlier source. For this reason, the referencestocompetitionsaresomewhatsporadic.UsuallynosourceisgivenifI haveknowntheproblemformorethan25years.Anyway,mostoftheproblems areresultsthatareknowntoexpertsintherespectivefields. Thereisahugeliteratureofmathematicalproblems.But,asatrainer,Iknow thattherecanneverbeenoughproblems.Youarealwaysindesperateneedofnew problemsoroldproblemswithnewsolutions.Anynewproblembookhassome newproblems,andabigbook,asthisone,usuallyhasquiteafewproblemsthat arenewtothereader. Theproblemsarearrangedinnoparticularorder,andespeciallynotinincreasing orderofdifficulty.Wedonotknowhowtorateaproblem’sdifficulty.EventheIMO jury, now consisting of 75 highly skilled problem solvers, commits grave errors in rating the difficulty of the problems it selects. The over 400 IMO contestants are also an unreliable guide. Too much depends on the previous training by an ever-changingsetofhundredsoftrainers.Aproblemchangesfromimpossibleto trivialifarelatedproblemwassolvedintraining. I would like to thank Dr. Manfred Grathwohl for his help in implementing a variousLT XversionsontheworkstationattheinstituteandonmyPCathome. E Whendifficultiesarose,hewasacompetentandfriendlyadvisor. There will be some errors in the proofs, for which I take full responsibility, since none of my colleagues has read the manuscript before. Readers will miss importantstrategies.SodoI,butIhavesetmyselfalimittothesizeofthebook. Especially,advancedmethodsaremissing.Still,itisprobablythemostcomplete training book on the market. The gravest gap is the absence of new topics like probability and algorithmics to counter the conservative mood of the IMO jury. OneexceptionisChapter13ongames,atopicalmostnonexistentintheIMO,but verypopularinRussia. FrankfurtamMain,Germany ArthurEngel Contents Preface............................................................ v AbbreviationsandNotations......................................... ix 1 TheInvariancePrinciple.......................................... 1 2 ColoringProofs................................................. 25 3 TheExtremalPrinciple........................................... 39 4 TheBoxPrinciple............................................... 59 5 EnumerativeCombinatorics...................................... 85 6 NumberTheory................................................. 117 7 Inequalities..................................................... 161 8 TheInductionPrinciple.......................................... 205 9 Sequences...................................................... 221 10 Polynomials.................................................... 245 11 FunctionalEquations ............................................ 271 viii Contents 12 Geometry....................................................... 289 13 Games.......................................................... 361 14 FurtherStrategies................................................ 373 References......................................................... 397 Index.............................................................. 401 Abbreviations and Notations Abbreviations ARO AllrussianMathematicalOlympiad ATMO AustrianMathematicalOlympiad AuMO AustralianMathematicalOlympiad AUO AllunionMathematicalOlympiad BrMO BritishMathematicalOlympiad BWM GermanNationalOlympiad BMO BalkanMathematicalOlympiad ChNO ChineseNationalOlympiad HMO HungarianMathematicalOlympiad(Ku˝rschakCompetition) IIM InternationalIntellectualMarathon(Mathematics/PhysicsCompetition) IMO InternationalMathematicalOlympiad LMO LeningradMathematicalOlympiad MMO MoskovMathematicalOlympiad PAMO Polish-AustrianMathematicalOlympiad