Advanced book on Mathematics Olympiad

807 Pages·2007·5.34 MB·English

Putnam and Beyond
Ra˘zvan Gelca
Titu Andreescu
Putnam and Beyond
Ra˘zvanGelca TituAndreescu
TexasTechUniversity UniversityofTexasatDallas
DepartmentofMathematicsandStatistics SchoolofNaturalSciencesandMathematics
MA229 2601NorthFloydRoad
Lubbock,TX79409 Richardson,TX75080
USA USA
[email protected] [email protected]
CoverdesignbyMaryBurgess.
LibraryofCongressControlNumber:2007923582
ISBN-13:978-0-387-25765-5 e-ISBN-13:978-0-387-68445-1
Printedonacid-freepaper.
(cid:1)c2007SpringerScience+BusinessMedia,LLC
All rights reserved. This work may not be translated or copied in whole or in part without the written permission
ofthepublisher(SpringerScience+BusinessMediaLLC,233SpringStreet,NewYork,NY10013,USA)andthe
author,exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformof
informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyarenotidentified
assuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrights.
9 8 7 6 5 4 3 2 1
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Lifeisgoodforonlytwothings,discovering
mathematicsandteachingmathematics.
SiméonPoisson
Contents
Preface ............................................................ xi
AStudyGuide ...................................................... xv
1 MethodsofProof ................................................ 1
1.1 ArgumentbyContradiction ...................................... 1
1.2 MathematicalInduction ......................................... 3
1.3 ThePigeonholePrinciple........................................ 11
1.4 OrderedSetsandExtremalElements .............................. 14
1.5 InvariantsandSemi-Invariants ................................... 19
2 Algebra ........................................................ 25
2.1 IdentitiesandInequalities ....................................... 25
2.1.1 AlgebraicIdentities....................................... 25
2.1.2 x2 ≥ 0.................................................. 28
2.1.3 TheCauchy–SchwarzInequality............................ 32
2.1.4 TheTriangleInequality ................................... 36
2.1.5 TheArithmeticMean–GeometricMeanInequality............. 39
2.1.6 Sturm’sPrinciple......................................... 42
2.1.7 OtherInequalities ........................................ 45
2.2 Polynomials................................................... 45
2.2.1 AWarmup .............................................. 45
2.2.2 Viète’sRelations ......................................... 47
2.2.3 TheDerivativeofaPolynomial............................. 52
2.2.4 TheLocationoftheZerosofaPolynomial ................... 54
2.2.5 IrreduciblePolynomials ................................... 56
2.2.6 ChebyshevPolynomials ................................... 58
viii Contents
2.3 LinearAlgebra................................................. 61
2.3.1 OperationswithMatrices .................................. 61
2.3.2 Determinants ............................................ 63
2.3.3 TheInverseofaMatrix ................................... 69
2.3.4 SystemsofLinearEquations ............................... 73
2.3.5 VectorSpaces,LinearCombinationsofVectors,Bases ......... 77
2.3.6 LinearTransformations,Eigenvalues,Eigenvectors ............ 79
2.3.7 TheCayley–HamiltonandPerron–FrobeniusTheorems ........ 83
2.4 AbstractAlgebra ............................................... 87
2.4.1 BinaryOperations........................................ 87
2.4.2 Groups ................................................. 90
2.4.3 Rings .................................................. 95
3 RealAnalysis ................................................... 97
3.1 SequencesandSeries ........................................... 98
3.1.1 SearchforaPattern....................................... 98
3.1.2 LinearRecursiveSequences ............................... 100
3.1.3 LimitsofSequences ...................................... 104
3.1.4 MoreAboutLimitsofSequences ........................... 111
3.1.5 Series .................................................. 117
3.1.6 TelescopicSeriesandProducts ............................. 120
3.2 Continuity,Derivatives,andIntegrals ............................. 125
3.2.1 LimitsofFunctions....................................... 125
3.2.2 ContinuousFunctions..................................... 128
3.2.3 TheIntermediateValueProperty............................ 131
3.2.4 DerivativesandTheirApplications.......................... 134
3.2.5 TheMeanValueTheorem ................................. 138
3.2.6 ConvexFunctions ........................................ 142
3.2.7 IndefiniteIntegrals ....................................... 147
3.2.8 DefiniteIntegrals......................................... 150
3.2.9 RiemannSums .......................................... 153
3.2.10 InequalitiesforIntegrals................................... 156
3.2.11 TaylorandFourierSeries .................................. 159
3.3 MultivariableDifferentialandIntegralCalculus..................... 167
3.3.1 PartialDerivativesandTheirApplications.................... 167
3.3.2 MultivariableIntegrals .................................... 174
3.3.3 TheManyVersionsofStokes’Theorem...................... 179
3.4 EquationswithFunctionsasUnknowns............................ 185
3.4.1 FunctionalEquations ..................................... 185
3.4.2 OrdinaryDifferentialEquationsoftheFirstOrder ............. 191
Contents ix
3.4.3 OrdinaryDifferentialEquationsofHigherOrder .............. 195
3.4.4 ProblemsSolvedwithTechniquesofDifferentialEquations ..... 198
4 GeometryandTrigonometry ...................................... 201
4.1 Geometry..................................................... 201
4.1.1 Vectors ................................................. 201
4.1.2 TheCoordinateGeometryofLinesandCircles................ 206
4.1.3 ConicsandOtherCurvesinthePlane........................ 212
4.1.4 CoordinateGeometryinThreeandMoreDimensions .......... 219
4.1.5 IntegralsinGeometry..................................... 225
4.1.6 OtherGeometryProblems ................................. 228
4.2 Trigonometry.................................................. 231
4.2.1 TrigonometricIdentities ................................... 231
4.2.2 Euler’sFormula.......................................... 235
4.2.3 TrigonometricSubstitutions................................ 238
4.2.4 TelescopicSumsandProductsinTrigonometry ............... 242
5 NumberTheory ................................................. 245
5.1 Integer-ValuedSequencesandFunctions........................... 245
5.1.1 SomeGeneralProblems................................... 245
5.1.2 Fermat’sInfiniteDescentPrinciple.......................... 248
5.1.3 TheGreatestIntegerFunction .............................. 250
5.2 Arithmetic .................................................... 253
5.2.1 FactorizationandDivisibility .............................. 253
5.2.2 PrimeNumbers .......................................... 254
5.2.3 ModularArithmetic....................................... 258
5.2.4 Fermat’sLittleTheorem................................... 260
5.2.5 Wilson’sTheorem........................................ 264
5.2.6 Euler’sTotientFunction................................... 265
5.2.7 TheChineseRemainderTheorem........................... 268
5.3 DiophantineEquations.......................................... 270
5.3.1 LinearDiophantineEquations .............................. 270
5.3.2 TheEquationofPythagoras................................ 274
5.3.3 Pell’sEquation .......................................... 276
5.3.4 OtherDiophantineEquations............................... 279
6 CombinatoricsandProbability .................................... 281
6.1 CombinatorialArgumentsinSetTheoryandGeometry............... 281
6.1.1 SetTheoryandCombinatoricsofSets ....................... 281
6.1.2 Permutations ............................................ 283
6.1.3 CombinatorialGeometry .................................. 286
x Contents
6.1.4 Euler’sFormulaforPlanarGraphs .......................... 289
6.1.5 RamseyTheory .......................................... 291
6.2 BinomialCoefficientsandCountingMethods....................... 294
6.2.1 CombinatorialIdentities................................... 294
6.2.2 GeneratingFunctions ..................................... 298
6.2.3 CountingStrategies....................................... 302
6.2.4 TheInclusion–ExclusionPrinciple .......................... 308
6.3 Probability .................................................... 310
6.3.1 EquallyLikelyCases ..................................... 310
6.3.2 EstablishingRelationsAmongProbabilities .................. 314
6.3.3 GeometricProbabilities ................................... 318
Solutions
MethodsofProof.................................................... 323
Algebra............................................................ 359
RealAnalysis ....................................................... 459
GeometryandTrigonometry.......................................... 603
NumberTheory..................................................... 673
CombinatoricsandProbability ........................................ 727
IndexofNotation ................................................... 791
Index.............................................................. 795
Preface
Aproblembookatthecollegelevel. AstudyguideforthePutnamcompetition. Abridge
betweenhighschoolproblemsolvingandmathematicalresearch. Afriendlyintroduction
tofundamentalconceptsandresults. Allthesedesiresgavelifetothepagesthatfollow.
TheWilliamLowellPutnamMathematicalCompetitionisthemostprestigiousmath-
ematics competition at the undergraduate level in the world. Historically, this annual
event began in 1938, following a suggestion of William Lowell Putnam, who realized
themeritsofanintellectualintercollegiatecompetition. Nowadays,over2500students
frommorethan300collegesanduniversitiesintheUnitedStatesandCanadatakepart
in it. The name Putnam has become synonymous with excellence in undergraduate
mathematics.
UsingthePutnamcompetitionasasymbol, welaythefoundationsofhighermath-
ematics from a unitary, problem-based perspective. As such, Putnam and Beyond is a
journey through the world of college mathematics, providing a link between the stim-
ulating problems of the high school years and the demanding problems of scientific
investigation. Itgivesmotivatedstudentsachancetolearnconceptsandacquirestrate-
gies,honetheirskillsandtesttheirknowledge,seekconnections,anddiscoverrealworld
applications. Itsultimategoalistobuildtheappropriatebackgroundforgraduatestudies,
whetherinmathematicsorappliedsciences.
Ourpointofviewisthatinmathematicsitismoreimportanttounderstandwhythan
toknowhow. Becauseofthisweinsistonproofsandreasoning. Afterall,mathematics
means, as the Romanian mathematician Grigore Moisil once said, “correct reasoning.’’
Thewaysofmathematicalthinkingareuniversalintoday’sscience.
Putnam and Beyond targets primarily Putnam training sessions, problem-solving
seminars,andmathclubsatthecollegelevel,fillingagapintheundergraduatecurriculum.
But it does more than that. Written in the structured manner of a textbook, but with
strongemphasisonproblemsandindividualwork,itcoverswhatwethinkarethemost
importanttopicsandtechniquesinundergraduatemathematics,broughttogetherwithin
the confines of a single book in order to strengthen one’s belief in the unitary nature of