The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009 Second Edition

825 Pages·2011·4.31 MB·English

by  Dušan Djukić

Problem Books in Mathematics
Series Editor
Peter Winkler
Department of Mathematics
Dartmouth College
Hanover, NH 03755
USA
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Dusˇan Djukic´ • Vladimir Jankovic´
Ivan Matic´ • Nikola Petrovic´
The IMO Compendium
A Collection of Problems Suggested
for The International Mathematical
Olympiads: 1959-2009
Second Edition
Dušan Djukić Vladimir Janković
Department of Mathematics Department of Mathematics
University of Toronto University of Belgrade
Toronto Ontario, M5S3G3 Studentski Trg 16
Canada 11000 Belgrade
[email protected] Serbia
[email protected]

Ivan Matić Nikola Petrović
Department of Mathematics Science Department
Duke University Texas A&M University
Durham, North Carolina 27708 PO Box 23874
USA Doha
[email protected] Qatar
[email protected]

ISSN 0941-3502
ISBN 978-1-4419-9853-8 e-ISBN 978-1-4419-9854-5
DOI 10.1007/978-1-4419-9854-5
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011926996
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Preface
TheInternationalMathematicalOlympiad(IMO)existsformorethan50yearsand
hasalreadycreatedaveryrichlegacyandfirmlyestablisheditselfasthemostpresti-
giousmathematicalcompetitioninwhichahigh-schoolstudentcouldaspiretopar-
ticipate.Apartfromtheopportunitytotackleinterestingandverychallengingmath-
ematicalproblems,theIMOrepresentsagreatopportunityforhigh-schoolstudents
toseehowtheymeasureupagainststudentsfromtherestoftheworld.Perhapseven
more importantly, it is an opportunity to make friends and socialize with students
whohavesimilarinterests,possiblyeventobecomeacquaintedwiththeirfuturecol-
leaguesonthisfirstlegoftheirjourneyintotheworldofprofessionalandscientific
mathematics.Aboveall, howeverpleasingor disappointingthe finalscore maybe,
preparingforanIMOandparticipatinginoneisanadventurethatwillundoubtedly
lingerinone’smemoryfortherestofone’slife.Itistothehigh-school-agedaspiring
mathematicianandIMOparticipantthatwedevotethisentirebook.
ThegoalofthisbookistoincludeallproblemsevershortlistedfortheIMOsin
asinglevolume.Uptothispoint,onlyscatteredmanuscriptstradedamongdifferent
teamshavebeenavailable,andanumberofmanuscriptswerelostformanyyearsor
unavailabletomany.
In this book, all manuscriptshave been collected into a single compendiumof
mathematicsproblemsof the kindthat usually appearon the IMOs. Therefore,we
believethatthisbookwillbethedefinitiveandauthoritativesourceforhigh-school
studentspreparingfortheIMO,andwesuspectthatitwillbeofparticularbenefitin
countrieslackingadequatepreparationliterature.Ahigh-schoolstudentcouldspend
anenjoyableyeargoingthroughthenumerousproblemsandnovelideaspresented
in the solutionsand emergeready to tackle even the most difficultproblemson an
IMO.Inaddition,theskillacquiredintheprocessofsuccessfullyattackingdifficult
mathematicsproblemswillprovetobeinvaluableinaseriousandprosperouscareer
inmathematics.
However,wemustcautionouraspiringIMOparticipantontheuseofthisbook.
Any book of problems, no matter how large, quickly depletes itself if the reader
merelyglancesataproblemandthenfiveminuteslater,havingdeterminedthatthe
problemseemsunsolvable,glancesatthesolution.
VI Preface
Theauthorsthereforeproposethefollowingplanforworkingthroughthebook.
Each problem is to be attempted at least half an hour before the reader looks at
thesolution.Thereaderisstronglyencouragedtokeeptryingtosolvetheproblem
without looking at the solution as long as he or she is coming up with fresh ideas
and possibilities for solving the problem.Only after all venues seem to have been
exhausted is the reader to look at the solution, and then only in order to study it
in close detail, carefully noting any previously unseen ideas or methods used. To
condense the subject matter of this already very large book, most solutions have
been streamlined, omitting obviousderivationsand algebraic manipulations.Thus,
readingthesolutionsrequiresacertainmathematicalmaturity,andinanycase,the
solutions, especially in geometry, are intended to be followed through with pencil
and paper, the reader filling in all the omitted details. We highly recommend that
thereadermarksuchunsolvedproblemsandreturntotheminafewmonthstosee
whethertheycanbesolvedthistimewithoutlookingatthesolutions.Webelievethis
tobethemostefficientandsystematicway(aswithanybookofproblems)toraise
one’slevelofskillandmathematicalmaturity.
We now leave our reader with final words of encouragement to persist in this
journey even when the difficulties seem insurmountable and a sincere wish to the
readerforallmathematicalsuccessonecanhopetoaspireto.
Belgrade, DušanDjukic´
November2010 VladimirJankovic´
IvanMatic´
NikolaPetrovic´
Over the previous years we have created the website: www.imomath.com.
Thereyoucanfind themostcurrentinformationregardingthebook,thelist of de-
tectederrorswithcorrections,andtheresultsfromthepreviousolympiads.Thissite
alsocontainsproblemsfromothercompetitionsandolympiads,anda collectionof
trainingmaterialsforstudentspreparingforcompetitions.
Weareawarethatthisbookmaystillcontainerrors.Ifyoufindany,pleasenotify
us at [email protected]. If you have any questions, comments, or sugges-
tionsregardingbothourbookandourwebsite,pleasedonothesitatetowritetous
attheaboveemailaddress.Wewouldbemorethanhappytohearfromyou.
Preface VII
Acknowledgements
Themakingofthisbookwouldhaveneverbeenpossiblewithoutthehelpofnumer-
ousindividuals,whomwewishtothank.
Firstandforemost,obtainingmanuscriptscontainingsuggestionsforIMOswas
vitalinorderforustoprovidethemostcompletelistingofproblemspossible.Weob-
tainedmanuscriptsformanyoftheyearsfromtheformerandcurrentIMOteamlead-
ersofYugoslavia/Serbia,whocarefullypreservedthesevaluablepapersthroughout
the years. Special thanks are due to Prof. Vladimir Mic´ic´, for some of the oldest
manuscripts, and to Prof. Zoran Kadelburg. We also thank Prof. Djordje Dugošija
and Prof. Pavle Mladenovic´. In collecting shortlisted and longlisted problems we
werealsoassistedbyProf.IoanTomescufromRomania,Hà Duy HưngfromViet-
nam,andZhaolifromChina.
Alotofworkwasinvestedincleaningupourgiantmanuscriptoferrors.Special
thanks in this respect go to David Kramer, our copy-editor, and to Prof. Titu An-
dreescu and his group for checking, in great detail, the validity of the solutions in
thismanuscript,andfortheirproposedcorrectionsandalternativesolutionstosev-
eral problems. We also thank Prof. Abderrahim Ouardini from France for sending
us the list of countries of origin for the shortlisted problems of 1998, Prof. Dorin
Andrica for helpingus compile the list of booksfor reference,and Prof. Ljubomir
Cˇukic´forproofreadingpartofthemanuscriptandhelpinguscorrectseveralerrors.
We wouldalso liketo expressourthanksto allanonymousauthorsoftheIMO
problems. Without them, the IMO would obviously not be what it is today. It is a
pitythatauthors’namesarenotregisteredtogetherwiththeirproposedproblems.In
anattempttochangethis,wehavetriedtotracedowntheauthorsoftheproblems,
with partial success. We are thankfulto all people who were so kind to help us in
ourinvestigation.ThenameswehavefoundsofararelistedinAppendixC.Inmany
cases,theoriginalsolutionsoftheauthorswereused,andwedulyacknowledgethis
immensecontributionto ourbook,thoughonceagain,we regretthatwe cannotdo
thisindividually.Inthesamevein,wealsothankallthestudentsparticipatinginthe
IMOs,sincewehavealsoincludedsomeoftheiroriginalsolutionsinthisbook.
Wethankthefollowingindividualswhodiscussedproblemswithusandhelped
uswithcorrectingthemistakesfromthepreviouseditionofthebook:XiaominChen,
OrlandoDöhring,MarijaJelic´,RudolfsKreicbergs,StefanMehner,YasserAhmady
Phoulady,DominicShauChin,JuanIgnacioRestrepo,ArkadiiSlinko,HarunŠiljak,
JosefTkadlec,IlanVardi,GerhardWoeginger,andYufeiZhao.
TheillustrationsofgeometryproblemsweredoneinWinGCLC,aprogramcre-
atedbyProf.PredragJanicˇic´.Thisprogramisspecificallydesignedforcreatinggeo-
metricpicturesofunparalleledcomplexityquicklyandefficiently.Eventhoughitis
stillinitstestingphase,itscapabilitiesandutilityarealreadyremarkableandworthy
ofhighestcompliment.
Finally,wewouldliketothankourfamiliesforalltheirloveandsupportduring
themakingofthisbook.

Contents
1 Introduction................................................... 1
1.1 TheInternationalMathematicalOlympiad...................... 1
1.2 TheIMOCompendium...................................... 2
2 BasicConceptsandFacts ....................................... 5
2.1 Algebra................................................... 5
2.1.1 Polynomials........................................ 5
2.1.2 RecurrenceRelations ................................ 6
2.1.3 Inequalities ......................................... 7
2.1.4 GroupsandFields.................................... 9
2.2 Analysis .................................................. 10
2.3 Geometry ................................................. 12
2.3.1 TriangleGeometry ................................... 12
2.3.2 VectorsinGeometry.................................. 13
2.3.3 Barycenters......................................... 14
2.3.4 Quadrilaterals ....................................... 14
2.3.5 CircleGeometry..................................... 15
2.3.6 Inversion ........................................... 16
2.3.7 GeometricInequalities................................ 17
2.3.8 Trigonometry ....................................... 17
2.3.9 FormulasinGeometry................................ 18
2.4 NumberTheory............................................ 19
2.4.1 DivisibilityandCongruences .......................... 19
2.4.2 ExponentialCongruences ............................. 20
2.4.3 QuadraticDiophantineEquations....................... 21
2.4.4 FareySequences..................................... 22
2.5 Combinatorics ............................................. 22
2.5.1 CountingofObjects.................................. 22
2.5.2 GraphTheory ....................................... 23