Table Of ContentMAA HungProbVII Case:Layout 2 12/21/10 2:24 PM Page 1
HUNGARIAN PROBLEM BOOK IV
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TRANSLATED AND EDITED BY ROBERT BARRINGTON LEIGH AND ANDY LIU
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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
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book is a continuation of Hungarian Problem Book III and takes the AND ANDY LIU
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contest through 1963. Forty-eight problems in all are presented in R
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this volume. Problems are classified under combinatorics, graph A
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theory, number theory, divisibility, sums and differences, algebra,
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geometry, tangent lines and circles, geometric inequalities,
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combinatorial geometry, trigonometry and solid geometry. O
Multiple solutions to the problems are presented, along with back- B
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ground material. There is a substantial chapter entitled “Looking E
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Back,” which provides additional insights into the problems.
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Hungarian Problem Book IV is intended for beginners, although
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the experienced student will find much here. Beginners are
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encouraged to work the problems in each section, and then to
compare their results against the solutions presented in the book.
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They will find ample material in each section to help them improve ORA
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their problem-solving techniques. RTSLA
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Theauthorsin2000.
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ISBN978-0-88385-831-8 N
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9 780883 858318
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Hungarian Problem
Book IV
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(cid:13)c 2011by
TheMathematicalAssociationofAmerica(Incorporated)
LibraryofCongressCatalogCardNumber2010940176
PrinteditionISBN978-0-88385-831-8
ElectroniceditionISBN978-1-61444-405-3
PrintedintheUnitedStatesofAmerica
CurrentPrinting(lastdigit):
10987654321
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Hungarian Problem
Book IV
Translated and Edited by
Robert Barrington Leigh
and
Andy Liu
University of Alberta
PublishedandDistributedby
TheMathematicalAssociationofAmerica
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CommitteeonBooks
FrankFarris,Chair
RichardA.GillmanEditor
ZumingFeng
ElginH.Johnston
RogerNelsen
TatianaShubin
RichardA.Stong
PaulA.Zeitz
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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
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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
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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
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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-
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pendenceoftheKingdomofHungary.Atthattime,Hungarywasnearly31
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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
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