Topics in Algebra and Analysis

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

4 Radmila Bulajich Manfrino Facultad de Ciencias Universidad Autónoma del Estado de Morelos Cuernavaca, Morelos, México José Antonio Gómez Ortega Facultad de Ciencias Universidad Nacional Autónoma de México Distrito Federal, México Rogelio Valdez Delgado Facultad de Ciencias Universidad Autónoma del Estado de Morelos Cuernavaca, Morelos, México ISBN ISBN (ebook) DOI / Library of Congress Control Number: Mathematics Subject Classification (2010): 00A07, 11B25, 11B65, 11C08, 39B22, 40A05, 97Fxx Springer Cham Heidelberg New York Dordrecht London Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (

5 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 algebra and analysis tools that a student must assimilate and learn to use gradually in training for mathematical contests and olympiads. Some of the topics that we study in the book are also part of the mathematical syllabus in high school courses, but there are other topics that are presented at the college level. That is, the book 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 divided in ten chapters. The first four correspond to topics from high school and they are basic for the students that are training for the mathematical olympiad contest, at a local and national level. The next four chapters are usually studied in the first year of 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. The first chapter covers the basic algebra, as are the numerical systems, absolute value, notable products, and factorization, among others. We expect that the 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 instance, the sum of the squares of the first n natural numbers. The telescopic sums, arithmetic and geometric progressions are analyzed, as well as some of its properties. Chapter 3 talks about the mathematical technique to prove mathematical statements that involve natural numbers, known as the principle of mathematical induction. Its use is exemplified with several problems. Many equivalent statements of the principle of mathematical induction are presented. To complete the first part of the book, in Chapter 4 the quadratic and cubic 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

6 vi Introduction The second part of the text begins with Chapter 5, where the complex numbers 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 properties of functions are studied. 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 equations. Chapter 7 talks about the notion of sequence and series. Special sequences are studied as bounded, periodic, monotone, recursive, among others. In addition, 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 first eight chapters have at the end a list of exercises for the reader, selected and suitable to practice the topics in the corresponding sections. The difficulty of the exercises vary from being a direct application of a result seen in the section to being a contest problem that with the technique studied is possible to solve. Chapter 9 is a collection of problems, each one of them close to one or more of the topics seen in the book. These problems have a degree of difficulty greater than the exercises. Most of the problems have appeared in some 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 presented in the book. The reader can notice that at the end of some sections there is a symbol, this means that the level of the section is harder than others sections. 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. We thank LeonardoIgnacio Martínez Sandoval and Rafael Martínez Enríquez for his always-helpful comments and suggestions, which contribute to the improvement of the material presented in this book.

7 Contents Introduction v 1 Preliminaries 1.1 Numbers Absolute value Integer part and fractional part of a number Notable products Matrices and determinants Inequalities Factorization Progressions and Finite Sums 2.1 Arithmetic progressions Geometric progressions Other sums Telescopic sums Induction Principle 3.1 The principle of mathematical induction Binomial coefficients Infinite descent Erroneous induction proofs Quadratic and Cubic Polynomials 4.1 Definition and properties Vieta s formulas Roots Complex Numbers 5.1 Complex numbers and their properties Quadratic polynomials with complex coefficients The fundamental theorem of algebra Roots of unity Proof of the fundamental theorem of algebra vii

8 viii Contents 6 Functions and Functional Equations 6.1 Functions Properties of functions Injective, surjective and bijective functions Even and odd functions Periodic functions Increasing and decreasing functions Bounded functions Continuity Functional equations of Cauchy type The Cauchy equation f(x + y) =f(x)+f(y) The other Cauchy functional equations Recommendations to solve functional equations Difference equations. Iterations Sequences and Series 7.1 Definition of sequence Properties of sequences Bounded sequences Periodic sequences Recursive or recurrent sequences Monotone sequences Totally complete sequences Convergent sequences Subsequences Series Power series Abel s summation formula Convergence of sequences and series Polynomials 8.1 Polynomials in one variable The division algorithm Roots of a polynomial Vieta s formulas Polynomials with integer coefficients Irreducible polynomials The derivative and multiple roots The interpolation formula Other tools to find roots Parameters

9 Contents ix Conjugate Descartes rule of signs Polynomials that commute Polynomials of several variables Problems Solutions to Exercises and Problems 10.1 Solutions to exercises of Chapter Solutions to exercises of Chapter Solutions to exercises of Chapter Solutions to exercises of Chapter Solutions to exercises of Chapter Solutions to exercises of Chapter Solutions to exercises of Chapter Solutions to exercises of Chapter Solutions to problems of Chapter Notation Bibliography Index