The math problems notebook PDF
Preview The math problems notebook
Valentin Boju Louis Funar The Math Problems Notebook Birkha¨user Boston • Basel • Berlin ValentinBoju LouisFunar MontrealTech InstitutFourierBP74 InstitutdeTechnologiedeMontreal URFMathematiques P.O.Box78575,StationWilderton Universite´deGrenobleI Montreal,Quebec,H3S2W9Canada 38402SaintMartind’Herescedex [email protected] France [email protected] CoverdesignbyAlexGerasev. MathematicsSubjectClassification(2000):00A07(Primary);05-01,11-01,26-01,51-01,52-01 LibraryofCongressControlNumber:2007929628 ISBN-13:978-0-8176-4546-5 e-ISBN-13:978-0-8176-4547-2 Printedonacid-freepaper. (cid:1)c2007Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233Spring Street,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarly analysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation, computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbid- den. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. 9 8 7 6 5 4 3 2 1 www.birkhauser.com (TXQ/SB) TheAuthors ValentinBojuwasprofessorofmathematicsattheUniversityofCraiova,Romania, untilhisretirementin2000.Hisresearchworkwasprimarilyinthefieldofgeome- try.Hefurtherpromotedbiostatisticsandbiomathematicsasadisciplinewithinthe DepartmentofMedicineoftheUniversityofCraiova.HeleftRomaniaforCanada, wherehehastaughtatMontrealTechsince2001.Hewasactivelyinvolvedincoach- ingcollegestudentsinproblem-solvingandintuitivemathematics.In2004,hewas honoredwiththetitleofOfficeroftheOrder“CulturalMerit,”Category“Scientific Research.” MontrealTech InstitutdeTechnologiedeMontréal P.O.Box78575StationWilderton Montréal,Quebec,H3S2W9 Canada e-mail:[email protected] Louis Funar has been a researcher at CNRS, at the Fourier Institute, University of Grenoble,France,since1994.Duringhiscollegeyears,heparticipatedthreetimes in the International Mathematics Olympiads with the Romanian team in the years 1983–1985,winningbronze,gold,andsilvermedals.Hismainresearchinterestsare ingeometrictopologyandmathematicalphysics. InstitutFourier,BP74 MathématiquesUMR5582 UniversitédeGrenobleI 38402Saint-Martin-d’Hèrescedex France e-mail:[email protected] Toallschoolteachers,particularlytomywonderfulparents, EvdochiaandNicolaeBoju,whogenuinelybelievedinthe educationoftheyoungergeneration. Tomyparents,MariaandIoanFunar,wholaidthefoundation forourcollaboration,asalaterewardfortheirhardworkand help,fortheirenthusiasm,determinedness,andstrength duringalltheseyears. Preface TheauthorsmetonaSundaymorningabout25yearsagoinRoom113.Oneofus wasacollegestudentandtheotherwasleadingtheSundayMathCircle.Thiscircle, withinthemathdepartmentoftheUniversityofCraiova,gatheredcollegestudents whopossessedacommonpassionformathematics.Mostofthemwereparticipating inthemathematicalcompetitionsinvogueatthattime,namelytheOlympiads.Others justwantedtohavegoodtime. Thereweresimilarmathcircleseverywhereinthecountry,inwhichhigh-school studentswerecommittedtoactivetrainingformathcompetitions.Thehigheststan- dard was achieved by the selective training camps organized at the national level, whichwereledbyprofessionalcoachesandmathematicianswhowereabletostimu- lateandelicithighperformance.Tomentionafewamongtheleaders,werecallDorin Andrica,TomaAlbu,TituAndreescu,MirceaBecheanu,IoanCuculescu,DorelMihet, andIoanTomescu. TheSundayMathCircleshiftedpartiallyfromitspurposeofOlympiadtraining byfollowingfreelytheleader’sideasandthusbecomingaplaceprimarilyintended for disseminating beautiful mathematics at an elementary level. The fundamental texts were the celebrated book of Richard Courant and Herbert Robbins with the mysterious title What is mathematics?, and the book of David Hilbert and Stefan Cohn–VossenentitledGeometryandtheImagination.Theparticipantssoonrealized thedifferencesinbothscaleandnoveltybetweenthecompetition-typemathproblems thattheyencounteredeverydayandthemathematicsbuiltupoverhundredsofyears andengagedinbyprofessionalmathematicians. Competition problems have to be solved in a short amount of time, and people competeagainsteachother.Thesemightbehighlynontrivial,unconventionalprob- lemsrequiringdeepinsightsandalotofimagination,butstilltheyhavetheadvantage thatoneknowsinadvancethattheyhaveasolution.Intherealmathematicalworld, problemsoftenhadtowaitnotmerelyyearsbutsometimescenturiestobesolved.A workingmathematiciancouldgoformonthsoryearstryingtosolveaparticularprob- lemthathadnotbeensolvedbeforeandtaketheriskofbitterlyfailing.Moreover,the sustainedeffortneededforaccomplishingsuchataskwouldhavetotakeintoaccount adelicatebalancebetweentheaccumulationofknowledge,methods,andtoolsand viii Preface thecreationfromscratchofapathleadingtoasolution.Thisintellectualadventure isfilledwithsuspenseandfrustration,sinceonedoesnotknowforsurewhatoneis tryingtoproveorwhetheritisindeedtrue. Real mathematics seemed to many of us to be too far away from competition problems.ThephilosophyoftheSundayMathCirclewasthat,incontrasttowhatwe mightthink,theborderbetweenthetwokindsofmathematicscouldbesovagueand flexiblethatattimesonecouldcrossitatanearlyage.Thehistoryofmathematics aboundsinexamplesinwhichafreshmindwasabletofindanunexpectedsolution thatspecialistshadbeenunabletofind.Oneneeds,however,therighttrainingandthe unconventional sense of finding the inspiring problem. Eventually, once a solution hasbeenfound,mathematiciansarethenwillingtotrytounderstanditevenbetter, and other solutions follow in time, each one simpler and clearer than the previous one.Insomesense,oncesolved,eventhehardestproblemsstartlosingslowlytheir auraofdifficultyandeventuallybecomejustproblems.Problemsthataretodaypart ofthecurriculumoftheaveragehigh-schoolstudentweredifficultresearchproblems threehundredyearsago,solvedonlybybrilliantmathematicians.Thisphenomenon demonstratestheevolutionthatlanguageandsciencehaveundergonesincethen. Theauthorsconceivedthepresentbookwiththenostalgiaofthe“goodoldtimes" oftheSundayMathCircles.Wewantedsomethingthatcarriesthemarkofthatphilos- ophy,namely,anumberofchallengingmathproblemsforOlympiadswithaglimpse ofrelatedproblemsofinterestforthemathematician.Thepresenttextisacollection ofproblemsthatwethinkwillbeusefulintrainingstudentsformathematicalcom- petitions. On the other hand, we hope that it might fulfill our second goal, namely, thatofawakeninginterestinadvancedmathematics.Thusitsaudiencemightrange fromcollegestudentsandteacherstoadvancedmathstudentsandmathematicians. Theproblemsineachsectionareinincreasingorderofdifficulty,sothatthereader givesomeoftheproblemsatry.Wewantedtohave25%easyproblemsconcerning basictoolsandmethodsandconsistingmainlyofinstructionalexercises.Thebeginner mightjumpdirectlytothesolutions,whereaconcentrateofthegeneraltheoryand basictrickscanbefound.Thelargestchunkcontainsabout50%problemsofmedium difficulty,whichcouldbeusefulintrainingformathematicalcompetitionsfromlocal to international levels. The remaining 25% might be considered difficult problems evenfortheexperiencedproblem-solver.Theseproblemsareoftenaccompaniedby commentsthatputtheresultsinabroaderperspectiveandmightincitethereaderto pursuetheresearchfurther. Theproblemsserveasanexcuseforintroducingallsortsofgeneralizationsand closelyrelatedopenproblems,whicharespreadamongthesolutions.Someofthese are truly outstanding problems that resisted the efforts of mathematicians over the centuries, such as the congruent numbers conjecture and the Riemann hypothesis, whichareamongsevenMillenniumPrizeProblemsthattheClayMathematicsInsti- tute recorded as some of the most difficult issues with which mathematicians were strugglingattheturnofthesecondmillenniumandofferedarewardofonemillion dollars for a solution to each one. In mathematics the frontier of our knowledge is still open, and it abounds in important unsolved problems, many of which can be Preface ix understoodattheundergraduatelevel.Thereadermightbesoondriventotheedge ofthatpartofmathematicswhereheorshecouldundertakeoriginalresearch. We drew inspiration from the spirit of the famous books by R. K. Guy and his collaborators.Nevertheless,ouraimwasnottobuildupacollectionofopenquestions inelementarymathematicsbutrathertoofferajourneyamongthebasicmethodsin problem-solving.Developingintuitionandstrengtheningthemostpopulartechniques inmathematicalcompetitionsareequallypartofthegoal.Inthemeantime,weofferthe enthusiasticreaderabriefglimpseofmathematicalresearch,aplacewhereproblems yetunsolvedlongfordeliverance. Thepresentcollectionofproblemsevolvedfromanotebookinwhichthesecond- named author collected the most interesting and unconventional problems that he encountered during his training for mathematical competitions in the 1980s. In the traditionoftheRomanianschoolofmathematicaltraining,heencounteredproblems inspiredbybothRussianOlympiadsandAmericanCompetitions.Some(ifnotmost) oftheproblemshavealreadyappearedelsewhere,especiallyinKvant,Matematika vShkole,AmericanMathematicalMonthly,ElementederMathematik,Matematikai Lapok, Gazeta Matematica, and so, in a sense, the collection gained a certain cos- mopolitanflavor.Wehavegiventhesourceintheproblemsection,whenknown,and moredetailedinformationinthecommentswithinthesolutionspart. The original set of problems is complemented by more basic exercises, which aimatintroducingmanyofthetricksandmethodsofwhichthecompetitorshouldbe aware.Yearslater,wefollowedthedestinyofsomeofthemostintriguingquestions fromthenotebook.Someofthemledtodevelopmentsthatarewellbeyondthescope ofthisbook,andwedecidedtooutlineafewofthem. We have supplied a short glossary containing some less usual definitions and identitiesinthegeometryoftrianglesandthesolutionofPell’sequation.Inorderto helpthereaderfindhisorherwaythroughthebook,wehaveprovidedbothanindex concerningallmathematicalresultsneededintheproofs,whichareusuallystatedat theplacewheretheyareusedfirst,andanindexofmathematicaltermsattheendof thebook. Theauthorshavebenefitedfromdiscussions,corrections,help,andfeedbackfrom several people, whom we want to thank warmly: DorinAndrica, Barbu Berceanu, Roland Bacher, Maxime Wolff, Mugurel Barcaˇu, Ioan Filip, and Simon György Szatmari. We also thank Ann Kostant, Editorial Director, Springer, and Avanti Paranjpye, Associate Editor, Birkhäuser Boston, for their productive suggestions. One of them led to the “Index of Topics and Methods,” which might be useful for a better reception of the book by readers.We are very grateful to Elizabeth Loew, our Production Editor, for her patience and dedication to accuracy and excellence. Finally,wearethankfultoDavidKramerforhisthoroughcopyeditingcorrections. ValentinBojuandLouisFunar Montreal,Canada and Saint-Martin-d’Hères,France July2006 Contents Preface ......................................................... vii PartI Problems 1 NumberTheory.............................................. 3 2 AlgebraandCombinatorics.................................... 11 2.1 Algebra................................................... 11 2.2 AlgebraicCombinatorics .................................... 13 2.3 GeometricCombinatorics.................................... 18 3 Geometry................................................... 21 3.1 SyntheticGeometry......................................... 21 3.2 CombinatorialGeometry .................................... 23 3.3 GeometricInequalities ...................................... 27 4 Analysis .................................................... 29 PartII SolutionsandCommentstotheProblems 5 NumberTheorySolutions ..................................... 37 6 AlgebraandCombinatoricsSolutions ........................... 85 6.1 Algebra................................................... 85 6.2 AlgebraicCombinatorics .................................... 95 6.3 GeometricCombinatorics.................................... 120 7 GeometrySolutions .......................................... 133 7.1 SyntheticGeometry......................................... 133 7.2 CombinatorialGeometry .................................... 142
The list of books you might like
