MAA HungProbVII Case:Layout 2 12/21/10 2:24 PM Page 1 HUNGARIAN PROBLEM BOOK IV M TRANSLATED AND EDITED BY ROBERT BARRINGTON LEIGH AND ANDY LIU A A H The Kürschák Mathematics Competition is the oldest high school U TRANSLATED AND EDITED BY mathematics competition in the world, dating back to 1894. This N ROBERT BARRINGTON LEIGH G book is a continuation of Hungarian Problem Book III and takes the AND ANDY LIU A contest through 1963. Forty-eight problems in all are presented in R I this volume. Problems are classified under combinatorics, graph A N theory, number theory, divisibility, sums and differences, algebra, P geometry, tangent lines and circles, geometric inequalities, R combinatorial geometry, trigonometry and solid geometry. O Multiple solutions to the problems are presented, along with back- B L ground material. There is a substantial chapter entitled “Looking E M Back,” which provides additional insights into the problems. B O O Hungarian Problem Book IV is intended for beginners, although K the experienced student will find much here. Beginners are I V encouraged to work the problems in each section, and then to compare their results against the solutions presented in the book. RT They will find ample material in each section to help them improve ORA BEN their problem-solving techniques. RTSLA BT AE RD RA INN GD TOED NIT LEED IGB HY A Theauthorsin2000. N D A ISBN978-0-88385-831-8 N D Y L IU 9 780883 858318 This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:33:59 UTC All use subject to http://about.jstor.org/terms i i “master” — 2010/11/17 — 9:17 — page i — #1 i i Hungarian Problem Book IV This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:33:59 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2012/12/13 — 9:32 — page ii — #2 i i (cid:13)c 2011by TheMathematicalAssociationofAmerica(Incorporated) LibraryofCongressCatalogCardNumber2010940176 PrinteditionISBN978-0-88385-831-8 ElectroniceditionISBN978-1-61444-405-3 PrintedintheUnitedStatesofAmerica CurrentPrinting(lastdigit): 10987654321 This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:33:59 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2010/11/17 — 9:17 — page iii — #3 i i Hungarian Problem Book IV Translated and Edited by Robert Barrington Leigh and Andy Liu University of Alberta PublishedandDistributedby TheMathematicalAssociationofAmerica This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:33:59 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2010/11/17 — 9:17 — page iv — #4 i i CommitteeonBooks FrankFarris,Chair RichardA.GillmanEditor ZumingFeng ElginH.Johnston RogerNelsen TatianaShubin RichardA.Stong PaulA.Zeitz This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:33:59 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2010/11/17 — 9:17 — page v — #5 i i MAAPROBLEMBOOKSSERIES ProblemBooksisaseriesoftheMathematicalAssociationofAmericacon- sistingof collections of problems and solutionsfrom annual mathematical competitions;compilationsofproblems(includingunsolvedproblems)spe- cifictoparticularbranchesofmathematics;booksontheartandpracticeof problemsolving,etc. Aha!Solutions,MartinErickson TheAlbertaHighSchoolMathCompetitions1957–2006:ACanadianProblemBook, compiledandeditedbyAndyLiu The Contest Problem BookVII: American MathematicsCompetitions, 1995–2000 Contests,compiledandaugmentedbyHaroldB.Reiter The Contest Problem Book VIII: AmericanMathematics Competitions (AMC 10), 2000–2007,compiledandeditedbyJ.DouglasFaires&DavidWells TheContestProblemBookIX:AmericanMathematicsCompetitions(AMC12),2000– 2007,compiledandeditedbyDavidWells&J.DouglasFaires FirstStepsforMathOlympians:UsingtheAmericanMathematicsCompetitions,by J.DouglasFaires AFriendlyMathematicsCompetition: 35YearsofTeamworkinIndiana,editedby RickGillman HungarianProblemBookIV,translatedandeditedbyRobertBarringtonLeighand AndyLiu 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. ProblemsfromMurrayKlamkin:TheCanadianCollection,editedbyAndyLiuand BruceShawyer The William Lowell Putnam Mathematical Competition Problems and Solutions: 1938–1964,A.M.Gleason,R.E.Greenwood,L.M.Kelly The William Lowell Putnam Mathematical Competition Problems and Solutions: 1965–1984,GeraldL.Alexanderson,LeonardF.Klosinski,andLorenC.Larson TheWilliamLowellPutnamMathematicalCompetition1985–2000:Problems,Solu- tions,andCommentary,KiranS.Kedlaya,BjornPoonen,RaviVakil USAandInternationalMathematicalOlympiads2000,editedbyTituAndreescuand ZumingFeng USAandInternationalMathematicalOlympiads2001,editedbyTituAndreescuand ZumingFeng This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:33:59 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2010/11/17 — 9:17 — page vi — #6 i i USAandInternationalMathematicalOlympiads2002,editedbyTituAndreescuand ZumingFeng USAandInternationalMathematicalOlympiads2003,editedbyTituAndreescuand ZumingFeng USAandInternationalMathematicalOlympiads2004, editedbyTitu Andreescu, ZumingFeng,andPo-ShenLoh MAAServiceCenter P.O.Box91112 Washington,DC20090-1112 1-800-331-1622 fax:1-301-206-9789 This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:33:59 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2010/11/17 — 9:17 — page vii — #7 i i Foreword George Berzsenyi Theappearanceofthepresentvolumeisatrulyimportanteventintheworld of mathematics. It is a huge step forward for itsHilbert Prize winningau- thor,sincehehadtoovercomethetragedyofthedeathofhisyoungprote´ge´e andco-author,andhadtofindthestrengthtocompletetheworkalone.Itis importanttothosewhoare involvedwiththeorganizationofmathematical competitions,sincetheynowhavemorecompleteaccesstotheproblemsof the famous Ku¨rscha´k Mathematical Competitionfor the years 1947–1963, andthesolutionsofthoseproblems.Moreover, itisimportanttothosewho are engaged inthe teachingand/orlearningofcreative mathematical prob- lem solving,since AndyLiu’sHungarianProblemBook IV isa wonderful vehicleformasteringtheprocessofproblemsolvinginthespiritofthelate George Po´lya, who was also a product of the Hungarian school of mathe- matics. Indeed,ProfessorLiu’srenditionoftheproblemsandsolutionsofthe51 problems covered in the second half of Ja´nos Sura´nyi’s original Part II of MatematikaiVersenyte´telek ismuchmorethanatranslation.Hegroupsthe problems according to subject matter, develops the background necessary forsolvingtheproblemsinthoseareas,andthensystematicallypresentsthe varioussolutions,addingseveralnewonesasheproceeds.Attheend,healso exemplifiesthefourthstepofPo´lya’sprocess,andreflectingonthesolutions ofsomeoftheproblems,arrivesatyetothersolutionsforsomeoftheother problems.Fortunately,theproblemsarevariedanddeepenoughtoallowfor suchatreatmentatthehandsofanexpert. InhisforewordtoDr.Liu’sHungarianProblemBookIII,Jo´zsefPelika´n, thedistinguishedleaderofHungary’steamstotheInternationalMathematics Olympiads,describedthecarefulprocessbywhichtheproblemsarechosen. In the present foreword, I want to reflect on the wisdom of the founding vii This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:16:07 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2010/11/17 — 9:17 — page viii — #8 i i viii fathers of the Ku¨rscha´k Mathematical Competitionin creating a most ad- mirable framework for that competition.Apart from minor adjustments, it haswithstoodthetestsoftimeforthelast115years. Byfixingthenumberofproblemsat3foreachcompetition,onecanpro- videenoughvariety,bothintopicareaandinmethodology,tohavethevery best competitors surface. Of course, all 3 problems must be of the right caliber. Hence the problems committee is always most carefully chosen. Since 1894, some of the best mathematicians of Hungary serve on it. The allowanceof4 hoursforthe solutionoftheproblemsturnedouttobe also ideal.Itisnottooshorttoundulyrushanyoneandlongenoughtocomplete one’swork,oftenwithgeneralizations,alternatesolutions,etc.Butthemost ingeniousideawastoallowthefree useofbooksandnotestakenalongby thecontestants.Thus, theproblemstestednotthefactualknowledgeofthe familiarmaterials,butthestudents’abilitytothinkcreatively,theiringenuity andknow-how.Hence,itisnotsurprisingthatsomanyofthewinnersofthe Ku¨rscha´kMathematicalCompetitionbecame outstandingscientistsintheir chosen fields. While in the beginning,onlythe first and second prize win- nerswererecognized,later,whenthenumberofcontestantsgrewtoseveral hundred, the originalformat allowed for the identificationof several more outstanding students. There are also some years when even the best con- testant doesn’tget first prize, while inotheryears several share the firstor secondprize—italldependsonthecaliberoftheworkturnedin. AsfarasIknow,noothercompetitionreflectsthestrengthofthefieldof contestantssorealistically.AtthispointIshouldalsoaddthattheorganizers ofthecompetitionmakeahugeefforttoprovideproperrecognitioninwell- publicizedceremonies notonlytothewinners, butto theirteachers too— yetanothertraitworthytobeemulated. With respect to the birth of the Ku¨rscha´k Mathematical Competition, it isinstructivetoreflectfirstonthespecialcircumstances whichmadeitrea- sonableforsuchaprogramtobeconceivedinthemindsofthescientistsof Hungaryback inthe1890s.Tothisend, Iwillbrieflyreviewthehistoryof Hungary. Originatingsomewhere in the Northern parts of present-day China, the HungariansarrivedattheCarpathianBasinaround895A.D.,andestablished a Christiankingdomthere inthe Year 1000. Later they withstoodthe dev- astationoftheMongolians(13thCentury),werepartiallyconqueredbythe Turks (16th–17thCentury), and became more and more subjugatedby the Austrians as part of the Habsburg Empire. Followingseveral unsuccessful revolutions, they finally managed to arrive at a compromise with Austria in 1867,which restored most of the territoryand some of the earlier inde- This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:16:07 UTC All use subject to http://about.jstor.org/terms i i i i i i “master” — 2010/11/17 — 9:17 — page ix — #9 i i ix pendenceoftheKingdomofHungary.Atthattime,Hungarywasnearly31 2 timesitspresentsize,withapopulationofover20million,andrichinminer- als,forests,andlandforagriculturaluse.Thefuturewaspromising,andthe countrywas feverishlypreparingfor its upcomingMillennium,the 1000th anniversaryofTheConquest.Hence,itwasnotunreasonabletothinkbigon thathistoricoccasion,forHungarywasalargecountryatthattime. Inahugespurtofeffort,hundredsofpublicbuildingswereerectedacross thecountry,includingschools,hospitals,banks,museums,etc.InBudapest in1895alone,595newapartmenthouseswerebuilt,containingnearly13,000 rooms.Itisalsonotablethat400newschoolswerebuiltthroughoutthecoun- tryforthemillennialyear. The world-famousParliamentwas completedat thattimetoo,aswellasthesubway,whichwasthefirstinContinentalEu- rope,andthreemorebridgesacrosstheRiverDanube.Thisperiodalsosaw the construction of several public buildings of later fame, like the Opera House,theFishermen’sBastion,theHeroes’Square,theSouthernandWest- ern RailwayStations,theMuseum ofFine Arts,theMain CustomsHouse, andtheArtGallery.Infact,theentirecitywasredesigned,withthreerings ofboulevardsleadingtothebridgesandanavenueleadingfromthecitycen- ter(ofPest)totheCityPark,whichwasthemainvenueforthecelebratory events. A year-long World Exhibitionwas organized there, where a proud nationdisplayeditspursuitsinthenameofprogressandpeace. Thus, it was not surprising that some of the intellectual leaders of the countrywere thinkingon a larger scale too. The groundworkfor the latter was laid by the physicist Baron Lora´nd Eo¨tvo¨s, who was the founder and firstpresidentoftheHungarianPhysicalandMathematicalSocietyin1891. Hesaidtohiscolleagues:“Wehave toraisetheflagofscience sohighthat itshouldbevisiblebeyondourborders”.Inthatspirit,uponhisappointment asMinisterofCultureandEducationin1894,membersofthatSocietyiniti- ateda“StudentCompetition”.ItwasnamedafterEo¨tvo¨sfollowinghisdeath in 1919, and renamed after Jo´zsef Ku¨rscha´k in 1947 when the physicists wantedtohavetheircompetitionbearthenameofEo¨tvo¨s. Ku¨rscha´kwasaprofessorofmathematics andastrongproponentofthis competition;itwas a revised editionof hiscompilationof the competition materials covering the years 1894–1928 that was translated as Hungarian ProblemBooksI&IIin1960.FollowingAndyLiu’swonderfulHungarian Problem Books III & IV, one might ask: What about the problems of the Ku¨rscha´k Mathematical Competitionsince 1963? It turns out that the late Professor Ja´nos Sura´nyi published Parts III and IV of the compilation in Hungarian; they cover the years 1964–1997. Reports on the last 12 years canbefoundinKo¨MaL,whichistheacceptedabbreviationforKo¨ze´piskolai This content downloaded from 130.64.11.153 on Thu, 25 Aug 2016 04:16:07 UTC All use subject to http://about.jstor.org/terms i i i i