Table Of ContentTitu Andreescu • Cristinel Mortici (cid:129) Marian Tetiva
Mathematical Bridges
TituAndreescu CristinelMortici
UniversityofTexasatDallasNatural ValahiaUniversityofTargoviste
SciencesandMathematics Targoviste,Romania
Richardson,Texas,USA
MarianTetiva
GheorgheRoscaCodreanuNationalCollege
Barlad,Romania
ISBN978-0-8176-4394-2 ISBN978-0-8176-4629-5 (eBook)
DOI10.1007/978-0-8176-4629-5
LibraryofCongressControlNumber:2017932379
MathematicsSubjectClassification(2010):00A07,08A50,15-XX,26-XX,97H60,97I20,97I30
©SpringerScience+BusinessMediaLLC2017
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
Printedonacid-freepaper
ThisbookispublishedunderthetradenameBirkhäuser,www.birkhauser-science.com
TheregisteredcompanyisSpringerScience+BusinessMediaLLC
Theregisteredcompanyaddressis:233SpringStreet,NewYork,NY10013,U.S.A.
Preface
Manybreakthroughsinresearchand,moregenerally,solutionstoproblemscomeas
theresultofsomeonemakingconnections.Theseconnectionsaresometimesquite
subtle, and at first blush, they may not appear to be plausible candidates for part
of the solution to a difficult problem. In this book, we think of these connections
as bridges. A bridge enables the possibility of a solution to a problem that may
haveaveryelementarystatementbutwhosesolutionmayinvolvemorecomplicated
realmsthatmaynotbedirectlyindicatedbytheproblemstatement.Bridgesextend
andbuildonexistingideasandprovidenewknowledgeandstrategiesforthesolver.
The ideal audience for this book consists of ambitious students who are seeking
usefultoolsandstrategiesforsolvingdifficultproblems(manyofolympiadcaliber),
primarilyintheareasofrealanalysisandlinearalgebra.
Theopeningchapter(aptlycalled“Chapter1”)exploresthemetaphorofbridges
by presenting a myriad of problems that span a diverse set of mathematical
fields. In subsequent chapters, it is left to the reader to decide what constitutes
a bridge. Indeed, different people may well have different opinions of whether
somethingisa(useful)bridgeornot.Eachchapteriscomposedofthreeparts:the
theoreticaldiscussion,proposedproblems,andsolutionstotheproposedproblems.
In each chapter, the theoretical discussion sets the stage for at least one bridge
by introducing and motivating the themes of that chapter—often with a review of
some definitions and proofs of classical results. The remainder of the theoretical
part of each chapter (and indeed the majority) is devoted to examining illustrative
examples—that is, several problems are presented, each followed by at least one
solution. It is assumed that the reader is intimately familiar with real analysis and
linear algebra, including their theoretical developments. There is also a chapter
that assumes a detailed knowledge of abstract algebra, specifically, group theory.
However,forthenotsofamiliarwithhighermathematicsreader,werecommenda
fewbooksinthebibliographythatwillsurelyhelp,like [5–9,11,12].
Bridgescanbefoundeverywhere—andnotjustinmathematics.Onesuchfinal
bridge is from us to our friends who carefully read the manuscript and made
extremely valuable comments that helped us a lot throughout the making of the
book.Itis,ofcourse,abridgeofacknowledgmentsandthanks;so,lastbutnotleast,
v
vi Preface
wemustsaythatwearedeeplygratefultoGabrielDospinescuandChrisJewellfor
alltheirhelpalongthewaytothefinalformofourwork.
Inclosing,asyoureadthisbook,weinviteyoutodiscoversomeofthesebridges
andembracetheirpowerinsolvingchallengingproblems.
Richardson,TX,USA TituAndreescu
Targoviste,Romania CristinelMortici
Barlad,Romania MarianTetiva
Contents
Preface............................................................................. v
1 Mathematical(andOther)Bridges....................................... 1
ProposedProblems .......................................................... 8
Solutions ..................................................................... 9
2 Cardinality .................................................................. 25
ProposedProblems .......................................................... 29
Solutions ..................................................................... 31
3 PolynomialFunctionsInvolvingDeterminants.......................... 43
ProposedProblems .......................................................... 48
Solutions ..................................................................... 50
4 SomeApplicationsoftheHamilton-CayleyTheorem.................. 57
ProposedProblems .......................................................... 64
Solutions ..................................................................... 66
5 ADecompositionTheoremRelatedtotheRankofaMatrix.......... 75
ProposedProblems .......................................................... 82
Solutions ..................................................................... 83
6 EquivalenceRelationsonGroupsandFactorGroups ................. 93
ProposedProblems .......................................................... 99
Solutions ..................................................................... 100
7 Density....................................................................... 109
ProposedProblems .......................................................... 114
Solutions ..................................................................... 116
8 TheNestedIntervalsTheorem............................................ 131
ProposedProblems .......................................................... 138
Solutions ..................................................................... 139
vii
viii Contents
9 TheSplittingMethodandDoubleSequences ........................... 149
ProposedProblems .......................................................... 160
Solutions ..................................................................... 162
10 TheNumbere............................................................... 175
ProposedProblems .......................................................... 181
Solutions ..................................................................... 182
11 TheIntermediateValueTheorem......................................... 189
ProposedProblems .......................................................... 192
Solutions ..................................................................... 193
12 TheExtremeValueTheorem.............................................. 201
ProposedProblems .......................................................... 205
Solutions ..................................................................... 207
13 UniformContinuity ........................................................ 213
ProposedProblems .......................................................... 219
Solutions ..................................................................... 220
14 DerivativesandFunctions’Variation .................................... 227
ProposedProblems .......................................................... 234
Solutions ..................................................................... 236
15 RiemannandDarbouxSums.............................................. 253
ProposedProblems .......................................................... 264
Solutions ..................................................................... 267
16 Antiderivatives.............................................................. 289
ProposedProblems .......................................................... 294
Solutions ..................................................................... 297
Bibliography...................................................................... 309
Chapter 1
Mathematical (and Other) Bridges
Manypeoplewhoreadthisbookwillprobablybefamiliarwiththefollowingresult
(veryfolkloric,ifwemaysayso).
Problem1. The midpoints of the bases of a trapezoid, the point at which its
lateral sides meet, and the point of intersection of its diagonals are four collinear
points.
E
N
D C
F
A M(cid:2)=M B
Solution. Indeed, let ABCD be a trapezoid with AB k CD; let M and N be the
midpoints of the linesegments AB andCD,respectively; and letfEg D AD\BC
and fFg D AC \BD. We intend to prove that M, N, E, and F are four collinear
points.
DenotebyM0theintersectionofEFwithAB.ByCeva’stheorem,wehave
AM0 BC ED
(cid:2) (cid:2) D1:
M0B CE DA
©SpringerScience+BusinessMediaLLC2017 1
T.Andreescuetal.,MathematicalBridges,DOI10.1007/978-0-8176-4629-5_1
2 1 Mathematical(andOther)Bridges
Also,Thales’interceptiontheoremsaysthat
DE CE CB DE
D , (cid:2) D1;
DA CB CE DA
andbyputtingtogethertheabovetwoequations,wegetM0A D M0B,thatis,M0 is
actually the midpoint of AB; therefore, M D M0 belongs to EF, which is (part of)
whatweintendedtoprove.(cid:2)
The reader will definitely find a similar way (or will be able to use the already
provedfactaboutthecollinearityofM,E,andF)toshowthatN,E,andFarealso
collinear.
Onecanalsoprovethataconverseofthistheoremisvalid,thatis,forinstance,if
M,E,andFarecollinear,thenABandCDareparallel(justproceedanalogously,but
goingintheoppositedirection).Ortrytoprovethatifthemidpointsoftwoopposite
sides of a trapezoid and the intersection point of its diagonals are three collinear
points,thenthequadrilateralisactuallyatrapezoid(thesideswhosemidpointswe
aretalkingaboutaretheparallelsides);thiscouldbemorechallengingtoprove.
As we said, this is a well-known theorem in elementary Euclidean geometry,
so why bother to mention it here? Well, this is because we find in it a very good
exampleofaproblemthatneedsa(mathematical)bridge.Namely,younoticedthat
theproblemstatementisveryeasytounderstandevenforapersonwhoonlyhasa
veryhumblebackgroundingeometry—butthatpersonwouldn’tbeabletosolvethe
problem. You could be familiar with basic notions as collinearity and parallelism,
youcouldknowsuchthingsaspropertiesofanglesdeterminedbytwoparallellines
andatransversal,butanyattempttosolvetheproblemwithsuchtoolswillfail.One
needsmuchmoreinordertoachievesuchagoal,namely,oneneedsanewtheory—
we are talking about the theory of similarity. In other words, if you want to solve
thisproblem,youhavetoraiseyourknowledgetonewfactsthatarenotmentioned
initsstatement.Youneedtothrowabridgefromthenarrowrealmwhereyouare
stucktoalargerextent.
Thefollowingproblemillustratesthesamesituation.
Problem2. Determineallmonotonefunctionsf WN(cid:2) !Rsuchthat
f.xy/Df.x/Cf.y/forallx;y2N(cid:2)
(N(cid:2)denotesthesetofpositiveintegers,whileRdenotesthereals).
Solution. Allthatonecangetfromthegivenrelationsatisfiedbyf isf.1/ D 0
and the obvious generalization f.x1(cid:2)(cid:2)(cid:2)xk/ D f.x1/ C (cid:2)(cid:2)(cid:2) C f.xk/ for all positive
integers x1;:::;xk (an easy and canonical induction leads to this formula) with its
corollary f.xk/ D kf.x/ for all positive integers x and k. But nothing else can be
done if you don’t step into a higher domain (mathematical analysis, in this case)
andifyoudon’tcomeupwithanidea.Theideaispossibleinthatsuperiordomain,
beingsomehownaturalifyouwanttouselimits.
1 Mathematical(andOther)Bridges 3
Let n be a positive integer (arbitrary, but fixed for the moment), and let us
consider, for any positive integer k, the unique nonnegative integer n such that
k
2nk (cid:3)nk <2nkC1.Rewritingtheseinequalitiesintheform
lnn 1 n lnn
(cid:4) < k (cid:3)
ln2 k k ln2
oneseesimmediatelythat lim n =kDlnn=ln2.
k
k!1
Because if f is increasing, (cid:4)f is decreasing and satisfies the same functional
equation (and conversely), we can assume, without loss of generality, that f is
increasing.Thenfromtheinequalitiessatisfiedbythenumbersn andbyapplying
k
thenoticedpropertyoff,weobtain
(cid:2) (cid:3)
n n 1
f.2nk/(cid:3)f.nk/(cid:3)f.2nkC1/, kf.2/(cid:3)f.n/(cid:3) k C f.2/:
k k k
Nowwecanletkgotoinfinity,yielding
lnn
f.n/Df.2/
ln2
for any positive integer n. So, all solutions are given by a formula of type f.n/ D
alnn, for a fixed real constant a. If f is strictly increasing (or strictly decreasing),
we get f.2/ > f.1/ D 0 (respectively, f.2/ < f.1/ D 0); thus f.2/ ¤ 0, and,
with b D 21=f.2/, the formula becomes f.n/ D log n (with greater, respectively
b
lesserthan1basebofthelogarithmaccordingtowhetherf isstrictlyincreasing,or
strictlydecreasing).Thenullfunction(f.n/D0foralln)canbeconsideredamong
thesolutions,ifwedonotaskonlyforstrictlymonotonicfunctions.Bytheway,if
we drop the monotonicity condition, we can find numerous examples of functions
thatonlysatisfythefirstcondition.Forinstance,definef.n/Da1Ca2C(cid:2)(cid:2)(cid:2)Cakfor
n D pa11(cid:2)(cid:2)(cid:2)pakk,withp1;:::;pk distinctprimesanda1;:::;ak positiveintegersand,
of course, f.1/ D 0, and we have a function with property f.mn/ D f.m/Cf.n/
for all positive integers m and n. The interested reader can verify for himself this
condition and the fact that f is not of the form f.n/ D log n, for some positive
b
b¤1(or,equivalently,thatthisfunctionisnotmonotone).(cid:2)
Again,oneseesthatinordertosolvesuchaproblem,oneneedstobuildabridge
betweentheveryelementarystatementoftheproblemandthemuchmoreinvolved
realmofmathematicalanalysis,wheretheproblemcanbesolved.
However,thereismoreaboutthisproblemfortheauthorsofthisbook.Namely,
it also demonstrates another kind of bridge—a bridge over the troubled water of
time, a bridge connecting moments of our lives. As youngsters are preoccupied
by mathematics, we had (behind the Iron Curtain, during the Cold War) very
few sources of information and very few periodicals to work with. There were,
say, in the 80s of the former century, Gazeta Matematica˘ and Revista matematica˘
din Timis¸oara—only two mathematical magazines. The first one was a monthly
