Mathematical Olympiad Challenges

291 Pages·2009·2.87 MB·English

by  Titu Andreescu

To Alina and to Our Mothers
Titu Andreescu
R˘azvan Gelca
Mathematical
Olympiad
Challenges
SECOND EDITION
Foreword by Mark Saul
Birkhäuser
Boston • Basel • Berlin
Titu Andreescu Răzvan Gelca
University of Texas at Dallas Texas Tech University
School of Natural Sciences Department of Mathematics
and Mathematics and Statistics
Richardson, TX 75080 Lubbock, TX 79409
USA USA
[email protected] [email protected]
ISBN:978-0-8176-4528-1 e-ISBN:978-0-8176-4611-0
DOI:10.1007/978-0-8176-4611-0
MathematicsSubjectClassification(2000):00A05, 00A07, 05-XX, 11-XX, 51XX
© Birkhäuser Boston, a part of Springer Science+Business Media, LLC, Second Edition 2009
© Birkhäuser Boston, First Edition 2000
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Contents
Foreword xi
PrefacetotheSecondEdition xiii
PrefacetotheFirstEdition xv
I Problems 1
1 GeometryandTrigonometry 3
1.1 APropertyofEquilateralTriangles. . . . . . . . . . . . . . . . . . . 4
1.2 CyclicQuadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 PowerofaPoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 DissectionsofPolygonalSurfaces . . . . . . . . . . . . . . . . . . . 15
1.5 RegularPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 GeometricConstructionsandTransformations . . . . . . . . . . . . . 25
1.7 ProblemswithPhysicalFlavor . . . . . . . . . . . . . . . . . . . . . 27
1.8 TetrahedraInscribedinParallelepipeds . . . . . . . . . . . . . . . . . 29
1.9 TelescopicSumsandProductsinTrigonometry . . . . . . . . . . . . 31
1.10 TrigonometricSubstitutions . . . . . . . . . . . . . . . . . . . . . . 34
2 AlgebraandAnalysis 39
2.1 NoSquareIsNegative . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.2 LookattheEndpoints . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 TelescopicSumsandProductsinAlgebra . . . . . . . . . . . . . . . 44
2.4 OnanAlgebraicIdentity . . . . . . . . . . . . . . . . . . . . . . . . 48
2.5 SystemsofEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.6 Periodicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.7 TheAbelSummationFormula . . . . . . . . . . . . . . . . . . . . . 58
2.8 x+1/x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.9 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.10 TheMeanValueTheorem . . . . . . . . . . . . . . . . . . . . . . . 66
vi Contents
3 NumberTheoryandCombinatorics 69
3.1 ArrangeinOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.2 SquaresandCubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.3 Repunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.4 DigitsofNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 DiophantineEquationswiththeUnknownsasExponents . . . . . . . 83
3.7 NumericalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.8 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.9 PellEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.10 PrimeNumbersandBinomialCoefficients . . . . . . . . . . . . . . . 99
II Solutions 103
1 GeometryandTrigonometry 105
1.1 APropertyofEquilateralTriangles. . . . . . . . . . . . . . . . . . . 106
1.2 CyclicQuadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . 110
1.3 PowerofaPoint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
1.4 DissectionsofPolygonalSurfaces . . . . . . . . . . . . . . . . . . . 125
1.5 RegularPolygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
1.6 GeometricConstructionsandTransformations . . . . . . . . . . . . . 145
1.7 ProblemswithPhysicalFlavor . . . . . . . . . . . . . . . . . . . . . 151
1.8 TetrahedraInscribedinParallelepipeds . . . . . . . . . . . . . . . . . 156
1.9 TelescopicSumsandProductsinTrigonometry . . . . . . . . . . . . 160
1.10 TrigonometricSubstitutions . . . . . . . . . . . . . . . . . . . . . . 165
2 AlgebraandAnalysis 171
2.1 NoSquareisNegative . . . . . . . . . . . . . . . . . . . . . . . . . 172
2.2 LookattheEndpoints . . . . . . . . . . . . . . . . . . . . . . . . . . 176
2.3 TelescopicSumsandProductsinAlgebra . . . . . . . . . . . . . . . 183
2.4 OnanAlgebraicIdentity . . . . . . . . . . . . . . . . . . . . . . . . 188
2.5 SystemsofEquations . . . . . . . . . . . . . . . . . . . . . . . . . . 190
2.6 Periodicity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
2.7 TheAbelSummationFormula . . . . . . . . . . . . . . . . . . . . . 202
2.8 x+1/x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
2.9 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
2.10 TheMeanValueTheorem . . . . . . . . . . . . . . . . . . . . . . . 217
3 NumberTheoryandCombinatorics 223
3.1 ArrangeinOrder . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
3.2 SquaresandCubes . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
3.3 Repunits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
3.4 DigitsofNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
3.5 Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
3.6 DiophantineEquationswiththeUnknownsasExponents . . . . . . . 246
Contents vii
3.7 NumericalFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 252
3.8 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
3.9 PellEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
3.10 PrimeNumbersandBinomialCoefficients . . . . . . . . . . . . . . . 270
AppendixA:DefinitionsandNotation 277
A.1 GlossaryofTerms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
A.2 GlossaryofNotation . . . . . . . . . . . . . . . . . . . . . . . . . . 282
Matematica˘,matematica˘,matematica˘,ataˆtamatematica˘?
Nu,maimulta˘.1
GrigoreMoisil
1Mathematics,mathematics,mathematics,thatmuchmathematics?No,evenmore.
Foreword
WhyOlympiads?
Workingmathematiciansoftentellusthatresultsinthefieldareachievedafterlong
experience and a deep familiarity with mathematical objects, that progress is made
slowlyandcollectively,andthatflashesofinspirationaremerepunctuationinperiods
ofsustainedeffort.
TheOlympiadenvironment,incontrast,demandsarelativelybriefperiodofintense
concentration,asksforquickinsightsonspecificoccasions,andrequiresaconcentrated
butisolatedeffort.YetwehavefoundthatparticipantsinmathematicsOlympiadshave
oftengoneontobecomefirst-classmathematiciansorscientistsandhaveattachedgreat
significancetotheirearlyOlympiadexperiences.
For many of these people, the Olympiad problem is an introduction, a glimpse
intotheworldofmathematicsnotaffordedbytheusualclassroomsituation. A good
Olympiadproblemwillcaptureinminiaturetheprocessofcreatingmathematics. It’s
all there: the period of immersion in the situation, the quiet examination of possible
approaches,thepursuitofvariouspathstosolution. Thereisthefruitlessdeadend,as
well as the path that endsabruptlybut offersnew perspectives, leading eventuallyto
thediscoveryofabetterroute.Perhapsmostobviously,grapplingwithagoodproblem
providespracticeindealingwiththefrustrationofworkingatmaterialthatrefusesto
yield.Ifthesolverislucky,therewillbethemomentofinsightthatheraldsthestartof
a successfulsolution. Like a well-craftedworkof fiction, a goodOlympiadproblem
tellsastoryofmathematicalcreativitythatcapturesagoodpartoftherealexperience
andleavestheparticipantwantingstillmore.
And this book gives us more. It weaves together Olympiad problems with a
common theme, so that insights become techniques, tricks become methods, and
methods build to mastery. Although each individual problem may be a mere appe-
tizer, the table is set here for more satisfying fare, which will take the reader deeper
intomathematicsthanmightanysingleproblemorcontest.
The book is organized for learning. Each section treats a particular technique
or topic. Introductory results or problems are provided with solutions, then related
problemsarepresented,withsolutionsinanothersection.
The craft of a skilled Olympiad coach or teacher consists largely in recognizing
similarities among problems. Indeed, this is the single most important skill that the
coach can impart to the student. In this book, two master Olympiad coaches have
offeredtheresultsoftheirexperiencetoawideraudience.Teacherswillfindexamples
and topics for advanced students or for their own exercise. Olympiad stars will find