Topics in Algebra and Analysis: Preparing for the Mathematical Olympiad

319 Pages·2015·3.4 MB·English

by  Radmila Bulajich Manfrino

Radmila Bulajich Manfrino
José Antonio Gómez Ortega
Rogelio Valdez Delgado
Topics in
Algebra
and Analysis
Preparing for the
Mathematical Olympiad

Radmila Bulajich Manfrino
José Antonio Gómez Ortega
Rogelio Valdez Delgado
Topics in Algebra and Analysis
Preparing for the Mathematical Olympiad
Radmila Bulajich Manfrino José Antonio Gómez Ortega
Facultad de Ciencias Facultad de Ciencias
Universidad Autónoma del Estado de Morelos Universidad Nacional Autónoma de México
Cuernavaca, Morelos, México Distrito Federal, México
Rogelio Valdez Delgado
Facultad de Ciencias
Universidad Autónoma del Estado de Morelos
Cuernavaca, Morelos, México
ISBN 978-3-319-11945-8 ISBN 978-3-319-11946-5 (eBook)
DOI 10.1007/978-3-319-11946-5

Library of Congress Control Number: 2015930195
Mathematics Subject Classification (2010): 00A07, 11B25, 11B65, 11C08, 39B22, 40A05, 97Fxx
S pringer Cham Heidelberg New York Dordrecht London
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Introduction
Topics in algebra and analysis have become fundamental for the mathematical
olympiad. Today, the problems in these topics that appear in the contests are
frequent, and the problems from other areas that use algebra and analysis in
their solutions are also frequent. In this book, we want to point out the principal
algebraandanalysistoolsthatastudentmustassimilateandlearntousegradually
in training for mathematical contests and olympiads. Some of the topics that we
studyinthebookarealsopartofthemathematicalsyllabusinhighschoolcourses,
butthereareothertopicsthatarepresentedatthecollegelevel.Thatis,thebook
can be used as a reference text for undergraduates in the first year of college who
will be facing algebra and analysis problems and will be interested in learning
techniques to solve them.
The book is dividedintenchapters.The firstfour correspondto topicsfrom
high school and they are basic for the students that are training for the math-
ematical olympiad contest, at a local and national level. The next four chapters
areusually studied in the first yearof college, but they have become fundamental
tools, for the students competing in an international level. The last two chapters
contain the problems and solutions of the theory studied in the book.
Thefirstchaptercoversthebasicalgebra,asarethenumericalsystems,abso-
lute value, notable products, andfactorization,amongothers.We expectthatthe
reader gain some skills for the manipulation of equations and algebraic formulae
to carry them in equivalent forms, which are easier to understand and work with
them.
In Chapter 2 the study of the finite sums of numbers is presented, for in-
stance,thesumofthesquaresofthefirstnnaturalnumbers.Thetelescopicsums,
arithmetic and geometricprogressionsareanalyzed, as well as some ofits proper-
ties.
Chapter 3 talks about the mathematical technique to prove mathematical
statements that involve natural numbers, knownas the principle of mathematical
induction.Itsuseisexemplifiedwithseveralproblems.Manyequivalentstatements
of the principle of mathematical induction are presented.
To complete the first part of the book, in Chapter 4 the quadratic and cu-
bic polynomials are studied, with emphasis in the study of the discriminant of a
quadratic polynomial and Vieta’s formulas for these two classes of polynomials.
v
vi Introduction
The secondpartof the text begins with Chapter 5, where the complex num-
bers are studied, as well as its properties and some applications are given. All
these with examples related to mathematical olympiad problems. In addition, a
proof of the fundamental theorem of algebra is included.
In Chapter 6, the principal propertiesof functions arestudied. Also, there is
an introduction to the functional equations theory, its properties and a series of
recommendations are given to solve the problems where appear functional equa-
tions.
Chapter 7 talks about the notion of sequence and series. Special sequences
arestudiedasbounded, periodic,monotone,recursive,amongothers.Inaddition,
the concept of convergence for sequences and series is introduced.
In Chapter 8, the study of polynomials from the first part of the book is
generalized. The theory of polynomials of arbitrary degree is presented, as well
as several techniques to analyze properties of the polynomials. At the end of the
chapter, the polynomials of several variables are studied. Most of the sections of
these firsteightchaptershaveatthe enda listofexercisesfor the reader,selected
and suitable to practice the topics in the correspondingsections. The difficulty of
the exercisesvaryfrombeing adirectapplicationofa resultseeninthe sectionto
being a contest problem that with the technique studied is possible to solve.
Chapter9isacollectionofproblems,eachoneofthemclosetooneormoreof
thetopicsseeninthebook.Theseproblemshaveadegreeofdifficultygreaterthan
the exercises.Mostofthe problems haveappearedinsome mathematical contests
around the world or olympiads. In the solution of each problem is implicit the
knowledge and skills that are need to manipulate algebraic expressions.
Finally, Chapter 10 contains the solutions to all exercises and problems pre-
sentedinthebook.Thereadercannoticethatattheendofsomesectionsthereis
a(cid:2) symbol, this means thatthe level ofthe sectionis harderthanotherssections.
In a first lecture, the reader can skip these sections; however, it is recommended
that the reader have them in mind for the techniques used in them.
WethankLeonardoIgnacioMart´ınezSandovalandRafaelMart´ınezEnr´ıquez
forhisalways-helpfulcommentsandsuggestions,whichcontributetotheimprove-
ment of the material presented in this book.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1 Preliminaries
1.1 Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Absolute value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Integer part and fractional part of a number . . . . . . . . . . . . 12
1.4 Notable products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Matrices and determinants . . . . . . . . . . . . . . . . . . . . . . 17
1.6 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.7 Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Progressionsand Finite Sums
2.1 Arithmetic progressions . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 Geometric progressions . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3 Other sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4 Telescopic sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Induction Principle
3.1 The principle of mathematical induction . . . . . . . . . . . . . . . 43
3.2 Binomial coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.3 Infinite descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Erroneous induction proofs . . . . . . . . . . . . . . . . . . . . . . 60
4 Quadratic and Cubic Polynomials
4.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.1 Vieta’s formulas . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Complex Numbers
5.1 Complex numbers and their properties . . . . . . . . . . . . . . . . 75
5.2 Quadratic polynomials with complex coefficients . . . . . . . . . . 79
5.3 The fundamental theorem of algebra . . . . . . . . . . . . . . . . . 81
5.4 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.5 Proof of the fundamental theorem of algebra (cid:2) . . . . . . . . . . . 85
vii
viii Contents
6 Functions and Functional Equations
6.1 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.2 Properties of functions . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.2.1 Injective, surjective and bijective functions . . . . . . . . . 94
6.2.2 Even and odd functions . . . . . . . . . . . . . . . . . . . . 96
6.2.3 Periodic functions . . . . . . . . . . . . . . . . . . . . . . . 97
6.2.4 Increasing and decreasing functions. . . . . . . . . . . . . . 98
6.2.5 Bounded functions . . . . . . . . . . . . . . . . . . . . . . . 99
6.2.6 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.3 Functional equations of Cauchy type . . . . . . . . . . . . . . . . . 102
6.3.1 The Cauchy equation f(x+y)=f(x)+f(y) . . . . . . . . 102
6.3.2 The other Cauchy functional equations (cid:2) . . . . . . . . . . 105
6.4 Recommendations to solve functional equations . . . . . . . . . . . 108
6.5 Difference equations. Iterations . . . . . . . . . . . . . . . . . . . . 111
7 Sequences and Series
7.1 Definition of sequence . . . . . . . . . . . . . . . . . . . . . . . . . 115
7.2 Properties of sequences . . . . . . . . . . . . . . . . . . . . . . . . 118
7.2.1 Bounded sequences . . . . . . . . . . . . . . . . . . . . . . . 118
7.2.2 Periodic sequences . . . . . . . . . . . . . . . . . . . . . . . 119
7.2.3 Recursive or recurrent sequences . . . . . . . . . . . . . . . 120
7.2.4 Monotone sequences . . . . . . . . . . . . . . . . . . . . . . 124
7.2.5 Totally complete sequences . . . . . . . . . . . . . . . . . . 125
7.2.6 Convergentsequences . . . . . . . . . . . . . . . . . . . . . 126
7.2.7 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . 127
7.3 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.3.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
7.3.2 Abel’s summation formula . . . . . . . . . . . . . . . . . . . 133
7.4 Convergence of sequences and series (cid:2) . . . . . . . . . . . . . . . . 135
8 Polynomials
8.1 Polynomials in one variable . . . . . . . . . . . . . . . . . . . . . . 139
8.2 The division algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 140
8.3 Roots of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3.1 Vieta’s formulas . . . . . . . . . . . . . . . . . . . . . . . . 145
8.3.2 Polynomials with integer coefficients . . . . . . . . . . . . . 145
8.3.3 Irreducible polynomials . . . . . . . . . . . . . . . . . . . . 146
8.4 The derivative and multiple roots (cid:2) . . . . . . . . . . . . . . . . . 150
8.5 The interpolation formula . . . . . . . . . . . . . . . . . . . . . . . 151
8.6 Other tools to find roots. . . . . . . . . . . . . . . . . . . . . . . . 153
8.6.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Contents ix
8.6.2 Conjugate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.6.3 Descartes’ rule of signs (cid:2). . . . . . . . . . . . . . . . . . . . 155
8.7 Polynomials that commute . . . . . . . . . . . . . . . . . . . . . . 157
8.8 Polynomials of several variables. . . . . . . . . . . . . . . . . . . . 161
9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10 Solutions to Exercises and Problems
10.1 Solutions to exercises of Chapter 1 . . . . . . . . . . . . . . . . . . 179
10.2 Solutions to exercises of Chapter 2 . . . . . . . . . . . . . . . . . . 190
10.3 Solutions to exercises of Chapter 3 . . . . . . . . . . . . . . . . . . 199
10.4 Solutions to exercises of Chapter 4 . . . . . . . . . . . . . . . . . . 214
10.5 Solutions to exercises of Chapter 5 . . . . . . . . . . . . . . . . . . 220
10.6 Solutions to exercises of Chapter 6 . . . . . . . . . . . . . . . . . . 229
10.7 Solutions to exercises of Chapter 7 . . . . . . . . . . . . . . . . . . 239
10.8 Solutions to exercises of Chapter 8 . . . . . . . . . . . . . . . . . . 248
10.9 Solutions to problems of Chapter 9 . . . . . . . . . . . . . . . . . . 261
Notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309