Table Of Content50th IMO – 50 Years of International Mathematical
Olympiads
.
Hans-Dietrich Gronau (cid:129) Hanns-Heinrich Langmann
Dierk Schleicher
Editors
50th IMO – 50 Years
of International Mathematical
Olympiads
123
Editors
Hans-DietrichGronau Hanns-HeinrichLangmann
UniversitätRostock Bildung&BegabunggemeinnützigeGmbH
InstitutfürMathematik BundeswettbewerbMathematik
Ulmenstr.69 KortrijkerStr.1
18051Rostock 53177Bonn
Germany Germany
[email protected] [email protected]
DierkSchleicher
JacobsUniversity
ResearchI
CampusRing1
28759Bremen
Germany
[email protected]
ISBN978-3-642-14564-3 e-ISBN978-3-642-14565-0
DOI10.1007/978-3-642-14565-0
SpringerHeidelbergDordrechtLondonNewYork
MathematicsSubjectClassification(2010):00A09
(cid:2)c Springer-VerlagBerlinHeidelberg2011
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Preface
From 10 to 22 July 2009 Germany hosted the 50th International Mathematical
Olympiad(IMO).ThisjubileeanniversaryeventtookplaceonthecampusofJacobs
UniversityBremen.Forthefirsttimethenumberofparticipatingcountriesexceeded
100, with 104 countriesfromall continentssendinga delegationconsisting of six
students,aleaderandadeputyleader.
Celebrating the 50th anniversary of the IMO provides an ideal opportunity to
look back over the past five decades and to review its development from modest
beginningstobecomeaworldwideevent.ThisbookisaimedatstudentsintheIMO
agegroupandseekstodemonstratethatmathematicsisanactiveandlivelysubject,
thousandsofyearsoldandyetyoungandfreshasever.WehaveinvitedseveralIMO
pioneerstotellusaboutthedevelopmentofIMOsinceitfirstbegan:theyrevealthat
initiallynobodythoughtitwoulddevelopintoanannualevent,letalonegrowasbig
asitistoday.
We include three accounts from IMO pioneers: Mircea Becheanu from Roma-
nia, speaking on behalf of the country that created the IMO and hosted the initial
Olympiadevent;IstvánReimanfromHungarywhopersonallyexperiencedtheearly
years;andWolfgangEngelfrom(thenEast)Germanywhowitnessedtheimpactthe
Olympiads have had from the outset on mathematical activities for students, par-
ticularly in Germany.These reports give a vivid and personaldescription of what
graduallybecameaninternationalsuccessstory.
Alsoincludedisdataaboutthe50annualOlympiadswhichillustratesthedevel-
opmentofthecompetitionfromjustsevencountriesin1959to104countries,with
565participants,in2009.Welistthemostsuccessfulcontestants,theresultsofthe
50Olympiadsandthenamesofall112countriesthathaveevertakenpart.
Inmathematics,talentisoftendevelopedanddemonstratedatanearlyage.Thus
it is impressive to see that many of the world’s leading research mathematicians
wereamongthemostsuccessfulIMOparticipantsintheir youth.Toillustrate this
weincludealistshowingwinnersofprestigiousresearchawardswhoparticipated
intheIMOwhentheywereyoung.
InadditiontothemathematicscompetitionanimportantaspectoftheOlympiads
isthatstudentsfromallovertheworldmeetandexchangeideas.Itisbothapersonal
v
vi Preface
highlightoftheirlivesthattheytreasureforalongtimeandaplacewhereinterna-
tionalfriendshipsdevelop.Someofthepersonaloramusinganecdotesofprevious
IMOsareincludedinthelastchapterofthebook.
The outstandingeventof the 2009 Olympiadwas the celebrationof the IMO’s
50thanniversary.Sixoftheworld’sleadingmathematicians,allofthemhighlysuc-
cessfulformerIMOparticipants,wereinvitedasguestsofhonourtogivepresenta-
tions:BélaBollobás,TimothyGowers,LászlóLovász,StanislavSmirnov,Terence
TaoandJean-ChristopheYoccoz.Theircontributionsarealsoincludedinthisbook.
We would like to express our sincere gratitude to all those who supported the
organisationofthe50th IMO,makingitamemorableanduniqueeventforallpar-
ticipants.
Finally, we wish to thank the SpringerVerlag for publishingthis book. We are
indebtedtoClemensHeineforhissupportandcollaboration.
Rostock,Bonn,Bremen Hans-DietrichGronau
May2010 Hanns-HeinrichLangmann
DierkSchleicher
Welcome Message
DearIMOparticipants,
This year’s IMO has a special significance: we celebrate the 50th anniversary
of the IMO. In the beginningsit started as a one-time eventfor studentsfrom the
countrieswhichatthetimeweresaidtoformthe“socialistblock”or“Sovietblock”,
towit:Bulgaria,Czechoslovakia,GermanDemocraticRepublic,Hungary,Poland,
RomaniaandtheSovietUnion.Itisremarkablethatofthose7countries,3donot
existanymore.
ButtheIMO,which,encouragedbythesuccessofthefirstcompetitionbecame
a yearlyevent,flourishes.Somekeydates:1967—thefirstWest-Europeancoun-
tries(UnitedKingdom,France,Italy,Sweden)takepartinthe9thIMOinCetinje,
Yugoslavia.(Finland’sone-timeparticipationin 1965hadno immediatecontinua-
tion.)The Netherlandsfollowedsuit in 1969,Austria in 1970.Two non-European
socialistcountriesparticipatedinthe60’sandearly70’s:Mongoliasince1964and
Cubasince1971.The16thIMOin1974inErfurt(EastGermany)sawatruebreak-
through:theteamsoftheUSAandofVietnamparticipatedforthefirsttime.
Lienz(Austria)wasthevenueofthefirst“Western”IMOin1976,Washington,
DC of the first non-EuropeanIMO in 1981.Australia organizedthe 1988 IMO in
Canberra,thefirstAsianIMOtookplacein1990inBeijing,China.Argentinawas
the venueof the1997IMO;we arelookingforwardtoan IMO onAfricansoilin
thenot-too-far-awayfuture.
MysincerehopeisthatmoreandmorecountrieswilljointheIMOcommunity;
old participatingcountrieswill see more and more benefitfromthe IMO for their
ownmathematicaleducationsystem andmathematicaltalentsearch;andtheIMO
willcontinuetobeasuccessstoryinthenext50years.
Bremen,July2009 JózsefPelikán
ChairmanoftheIMOAdvisoryBoard
vii
viii WelcomeMessage
JózsefPelikán
Hungary
József Pelikán took part in 4 IMOs in his student
days: In 1963 he won a silver medal, and in 1964-
65-66threegoldmedals.Hewasalsoawardedtwice
with a special prize for an outstanding solution of a
problem.
Since1971hehasbeenteachingatEo˝tvo˝sLoránd
University,Budapest,Hungary.Heparticipatedinthe
organizationofthe1970IMOheldinKeszthely,Hun-
gary,andwasoneofthechieforganizersofthe1982
IMOinBudapest,Hungary.
HehasbeentheleaderoftheHungarianIMOteam
since 1988. He was an elected member of the IMO
AdvisoryBoardbetween1992and2002,andhasbeen
theChairmanoftheIMOAdvisoryBoardsince2002.
Contents
PartI 50thInternationalMathematicalOlympiad–Germany2009
1 Committees ................................................ 5
1.1 IMOAdvisoryBoard ...................................... 5
1.2 SteeringCommittee ....................................... 5
1.3 Organizers............................................... 6
1.4 ProblemSelectionCommittee............................... 6
1.5 Jury..................................................... 7
1.6 Coordinators ............................................. 8
1.7 Invigilators............................................... 9
1.8 Guides .................................................. 10
1.9 50thIMOAnniversaryCommittee ........................... 12
1.10 Volunteers ............................................... 12
2 Regulations ................................................ 13
3 CarlFriedrichGauss,Curvature,andtheCoverArtofthe50thIMO 21
4 Programme ................................................ 27
4.1 IMOProgrammeOverview................................. 27
4.2 ProgrammeDetails........................................ 28
4.3 DailyNews .............................................. 32
5 Participants ................................................ 47
5.1 NationalTeams........................................... 48
5.2 ObserverCountries........................................100
ix
x Contents
6 MathematicalCompetition ................................... 101
6.1 Problems ................................................103
6.2 Solutions ................................................113
6.2.1 Problem1–NumberTheory–Australia ..............113
6.2.2 Problem2–Geometry–Russia .....................116
6.2.3 Problem3–Algebra–UnitedStates .................118
6.2.4 Problem4–Geometry–Belgium....................120
6.2.5 Problem5–Algebra–France.......................122
6.2.6 Problem6–Combinatorics–Russia .................123
6.3 Awards..................................................126
6.3.1 GoldMedals-TopScores ..........................126
6.3.2 GoldMedals......................................126
6.3.3 SilverMedals.....................................127
6.3.4 BronzeMedals....................................129
6.3.5 HonourableMentions ..............................131
6.4 IndividualScores .........................................134
7 MathematicalGuestsofHonour ............................... 155
7.1 BélaBollobás ............................................156
TheLionandtheChristian,andOtherPursuitandEvasionGames ......157
7.2 TimothyGowers ..........................................169
HowdoIMOProblemsComparewithResearchProblems? ............171
7.3 LászlóLovász ............................................185
GraphTheoryOver45Years......................................187
7.4 StanislavSmirnov.........................................197
HowdoResearchProblemsComparewithIMOProblems? ............199
7.5 TerenceTao..............................................210
StructureandRandomnessinthePrimeNumbers ....................211
7.6 Jean-ChristopheYoccoz....................................217
SmallDivisors:NumberTheoryinDynamicalSystems ...............219
PartII History:50YearsInternationalMathematicalOlympiads
8 BriefSurvey................................................ 229
9 AllIMOs .................................................. 233
10 AllCountries ............................................... 235
