Problem-Solving Methods in Combinatorics: An Approach to Olympiad Problems

178 Pages·2013·1.76 MB·English

by  Pablo Soberón (auth.)

Pablo Soberón
Problem-Solving
Methods in
Combinatorics
An Approach to Olympiad Problems
Problem-Solving Methods in Combinatorics
Pablo Soberón
Problem-Solving
Methods in
Combinatorics
An Approach to Olympiad Problems
PabloSoberón
DepartmentofMathematics
UniversityCollegeLondon
London,UK
ISBN978-3-0348-0596-4 ISBN978-3-0348-0597-1(eBook)
DOI10.1007/978-3-0348-0597-1
SpringerBaselHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013934547
MathematicsSubjectClassification: 05-01,97K20
©SpringerBasel2013
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Contents
1 FirstConcepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 SetsandFirstCountings . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 PathsinBoards . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 ACoupleofTricks. . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 ThePigeonholePrinciple. . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 ThePigeonholePrinciple . . . . . . . . . . . . . . . . . . . . . . 17
2.2 RamseyNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 TheErdo˝s-SzekeresTheorem . . . . . . . . . . . . . . . . . . . . 22
2.4 AnApplicationinNumberTheory . . . . . . . . . . . . . . . . . 23
2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3 Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1 DefinitionandFirstExamples . . . . . . . . . . . . . . . . . . . . 27
3.2 Colorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 ProblemsInvolvingGames . . . . . . . . . . . . . . . . . . . . . 33
3.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4 GraphTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.1 BasicConcepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 ConnectednessandTrees . . . . . . . . . . . . . . . . . . . . . . 47
4.3 BipartiteGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.4 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.1 FunctionsinCombinatorics . . . . . . . . . . . . . . . . . . . . . 59
5.2 Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3 CountingTwice . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4 TheErdo˝s-Ko-RadoTheorem . . . . . . . . . . . . . . . . . . . . 72
5.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6 GeneratingFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.1 BasicProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
v
vi Contents
6.2 FibonacciNumbers . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.3 CatalanNumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.4 TheDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.5 EvaluatingGeneratingFunctions . . . . . . . . . . . . . . . . . . 87
6.6 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.1 Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 StirlingNumbersoftheFirstKind . . . . . . . . . . . . . . . . . 94
7.3 StirlingNumbersoftheSecondKind . . . . . . . . . . . . . . . . 96
7.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8 HintsfortheProblems . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.1 HintsforChap.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
8.2 HintsforChap.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
8.3 HintsforChap.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.4 HintsforChap.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
8.5 HintsforChap.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.6 HintsforChap.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
8.7 HintsforChap.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
9 SolutionstotheProblems . . . . . . . . . . . . . . . . . . . . . . . . 113
9.1 SolutionsforChap.1 . . . . . . . . . . . . . . . . . . . . . . . . 113
9.2 SolutionsforChap.2 . . . . . . . . . . . . . . . . . . . . . . . . 120
9.3 SolutionsforChap.3 . . . . . . . . . . . . . . . . . . . . . . . . 128
9.4 SolutionsforChap.4 . . . . . . . . . . . . . . . . . . . . . . . . 138
9.5 SolutionsforChap.5 . . . . . . . . . . . . . . . . . . . . . . . . 146
9.6 SolutionsforChap.6 . . . . . . . . . . . . . . . . . . . . . . . . 156
9.7 SolutionsforChap.7 . . . . . . . . . . . . . . . . . . . . . . . . 164
Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Introduction
Everyyearthereisatleastonecombinatoricsproblemineachofthemajormath-
ematical olympiads of international level. These problems have the common trait
of needing a very high level of wit and creativity to find a solution. Even in the
most recent competitions there are difficult problems that can be solved almost
completelybyasinglebrilliantidea.However,tobeabletoattacktheseproblems
withcomfortitisnecessarytohavefacedproblemsofsimilardifficultypreviously
and to have a good knowledge of the techniques that are commonly used to solve
them.
I write this book with two purposes in mind. The first is to explain the tools
and tricks necessary to solve almost any combinatorics problems in international
olympiads, with clear examples of how they are used. The second way to offer to
the olympicstudents(and other interestedreaders) anamplelist of problemswith
hints and solutions. This book may be used for training purposes in mathematical
olympiadsoraspartofacourseincombinatorics.
Despitethefactthatthisbookisself-contained,previouscontactwithcombina-
toricsisadvisableinordertograsptheconceptswithease.Inthesection“Further
reading”wesuggestmaterialforthispurpose.Readingthiswouldbeespeciallyuse-
fulforfamiliarizingwiththenotationandbasicideas.Itisalsorequiredtohavea
basicunderstandingofcongruencesinnumbertheory.
The book is divided into 7 chapters of theory, a chapter of hints and a chapter
of solutions. The theory chapters provide examples and exercises along the text
andendwithaproblemssection.Intotalthereare 127 problemsinthebook.The
purposeoftheexercisesisforthereadertostartusingtheideasofthechapter.There
arenohintsnorsolutionsfortheexercises,astheyare(almostalways)easierthan
theproblems.Allexamplesandproblemshavesolutions.
Althoughthetextisintendedtobereadintheorderitispresented,itispossible
toreaditfollowingthediagrambelow.
vii
viii Introduction
Many of the problems and examples of this book have appeared in mathematical
contests,soIhavetriedtoprovidereferencestotheirfirstappearance.Iapologize
ifanyaremissingorjustplainwrong.Attheendofthebook,wherethenotationis
explained,thereisalistofabbreviationsthatwereusedtomakethesereferences.
IwouldliketothankRadmilaBulajichforbothteachingmethenecessaryLATEX
to write this book and to motivating me to do so. Without her support this project
wouldhaveneverbeenbroughttocompletion.Iwouldalsoliketothanktheexten-
sive corrections and comments by Leonardo Martínez Sandovalas well as the ob-
servationsbyAdriánGonzález-CasanovaSoberónandCarlaMárquezLuna.Their
helpshapedthisbooktoitspresentform.
PabloSoberón
FortheStudent
Eventhoughalargenumberofproblemsincombinatoricshaveaquickand/oreasy
solution,thatdoesnotmeantheproblemonehastosolveisnothard.Manytimes
thedifficultyofaproblemincombinatoricsliesinthefactthattheideathatworksis
verywell“hidden”.Duetothistheonlywaytoreallylearncombinatoricsissolving
manyproblems,ratherthanreadingalotoftheory.Thispracticeispreciselywhat
teachesyouhowtolookforthesecreativeorhiddenideas.
Asyoureadthetheoryyouwillfindexercisesandexamples.Itisimportanttodo
themastheycomealong,sincemanyofthemareanimportantpartofthetheoryand
willbeusedrepeatedlylateron.Whenyoufindanexample,trytosolveitbyyour-
selfbeforereadingthesolution.Thisistheonlywaytoseethecomplicationsthat
particularproblembrings.Bydoingthisyouwillhaveaneasiertimeunderstanding
whytheideasinthesolutionworkandwhytheyshouldbenatural.
Introduction ix
Attheendofeachchapterthereisasectionwithproblems.Theseproblemsare
meanttobeharderthantheexercisesandtheyhavehintsandsolutionsattheendof
the book. Many problems are of an international competition level, so you should
notbecomediscouragedifyoufindaparticularlydifficultone.
Someproblemshavemorethanonesolution.Thisdoesnotmeanthatthesesolu-
tionsaretheonlyones.Chaseyourownideasandyouwillprobablyfindsolutions
thatdifferfromtheonespresentedinthisbook(andperhapsareevenbetter).Thus,
when you reach a problems section, do not restrict yourself to the tools shown in
thatchapter.
Finally,mymainobjectivewhenwritingthisbookwasnotsimplytoteachcom-
binatorics, but for the book to be enjoyed by its readers. Take it easy. Remember
thatsolvingproblemsisadisciplinethatislearnedwithconstantpracticeandleads
toagreatsenseofsatisfaction.
1
First Concepts
1.1 SetsandFirstCountings
A set is defined to be a “collection of elements contained within a whole”. That
is, a set is defined only by its elements, which can also be sets. When we write
{a,b,c}=S,wemeanthat S is theset thathastheelements a, b, c.Twosets are
equalif(andonlyif)theyhavethesameelements.Toindicatethata isanelement
ofS,wewritea∈S.Inasetweneverrepeatelements,theycanjustbeinthesetor
notbethere.
Given a set A and a property ψ, we denote by {a ∈A|ψ(a)} the set of all
elementsofAthatsatisfypropertyψ.Forexample,{a∈R|a>2}isthesetofall
realnumbersthataregreaterthan2.Byconvention,thereisanemptyset.Thatis,
asetwithnoelements.Oneusuallydenotesthatsetwiththesymbol∅.
Whenwesaythatasethastobecontainedwithinawhole,wemeanthatitshould
notbe“toobig”.Thisisaverytechnicaldetail,buttherearecollections(suchasthe
collectionofallsets)thatbringdifficultiesifweconsiderthemassets.Throughout
thisbookwewillneverfacethiskindofdifficulties,evenwhenwetalkaboutinfinite
sets.However,itisimportanttoknowthattherearecollectionsthatarenotsets.1
GiventwosetsAandB,wesaythatAisasubsetofB,orthatAiscontained
in B, if every element of A is an element of B. We denote this by A⊂B. For
example{1,2}⊂{1,{1,3},2}since1and2areelementsofthesecondset.However,
{1,3}(cid:5)⊂{1,{1,3},2}since3isnotanelementinthesecondset.
GiventwosetsAandB,wecanusethemtogenerateothersets.
• A∩B,calledtheintersectionofAandB.Thisisthesetthatconsistsofallthe
elementsthatareinbothAandB.
• A∪B,calledtheunionofAandB.Thisisthesetthatconsistsalltheelements
thatareinatleastoneofAorB.
1Ifyouwantanexampleofthis,considerRussell’sparadox.LetS bethecollectionofallsets
thatdonotcontainthemselvesaselements,i.e.,allsetsAsuchthatA∈/A.IfSisconsideredasa
set,thenS∈SifandonlyifS∈/S,acontradiction!
P.Soberón,Problem-SolvingMethodsinCombinatorics, 1
DOI10.1007/978-3-0348-0597-1_1,©SpringerBasel2013