putnam and beyond

Putnam and Beyond R ˘ azvan Gelca Titu Andreescu Putnam and Beyond R ˘ azvan Gelca Texas Tech University Department of Mathematics and Statistics MA 229 Lubbock, TX 79409 USA rgelca@gmail.com Titu Andreescu University of Texas at Dallas School of Natural Sciences and Mathematics 2601 North Floyd Road Richardson, TX 75080 USA titu.andreescu@utdallas.edu Cover design by Mary Burgess. Library of Congress Control Number: 2007923582 ISBN-13: 978-0-387-25765-5 e-ISBN-13: 978-0-387-68445-1 Printed on acid-free paper. c  2007 Springer Science+Business Media, LLC All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. 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Siméon Poisson Contents Preface xi A Study Guide xv 1 Methods of Proof 1 1.1 Argument by Contradiction 1 1.2 Mathematical Induction 3 1.3 The Pigeonhole Principle 11 1.4 Ordered Sets and Extremal Elements 14 1.5 Invariants and Semi-Invariants 19 2 Algebra 25 2.1 Identities and Inequalities 25 2.1.1 Algebraic Identities 25 2.1.2 x 2 ≥ 0 28 2.1.3 The Cauchy–Schwarz Inequality 32 2.1.4 The Triangle Inequality 36 2.1.5 The Arithmetic Mean–Geometric Mean Inequality 39 2.1.6 Sturm’s Principle 42 2.1.7 Other Inequalities 45 2.2 Polynomials 45 2.2.1 A Warmup 45 2.2.2 Viète’s Relations 47 2.2.3 The Derivative of a Polynomial 52 2.2.4 The Location of the Zeros of a Polynomial 54 2.2.5 Irreducible Polynomials 56 2.2.6 Chebyshev Polynomials 58 viii Contents 2.3 Linear Algebra 61 2.3.1 Operations with Matrices 61 2.3.2 Determinants 63 2.3.3 The Inverse of a Matrix 69 2.3.4 Systems of Linear Equations 73 2.3.5 Vector Spaces, Linear Combinations of Vectors, Bases 77 2.3.6 Linear Transformations, Eigenvalues, Eigenvectors 79 2.3.7 The Cayley–Hamilton and Perron–Frobenius Theorems 83 2.4 Abstract Algebra 87 2.4.1 Binary Operations 87 2.4.2 Groups 90 2.4.3 Rings 95 3 Real Analysis 97 3.1 Sequences and Series 98 3.1.1 Search for a Pattern 98 3.1.2 Linear Recursive Sequences 100 3.1.3 Limits of Sequences 104 3.1.4 More About Limits of Sequences 111 3.1.5 Series 117 3.1.6 Telescopic Series and Products 120 3.2 Continuity, Derivatives, and Integrals 125 3.2.1 Limits of Functions 125 3.2.2 Continuous Functions 128 3.2.3 The Intermediate Value Property 131 3.2.4 Derivatives and Their Applications 134 3.2.5 The Mean Value Theorem 138 3.2.6 Convex Functions 142 3.2.7 Indefinite Integrals 147 3.2.8 Definite Integrals 150 3.2.9 Riemann Sums 153 3.2.10 Inequalities for Integrals 156 3.2.11 Taylor and Fourier Series 159 3.3 Multivariable Differential and Integral Calculus 167 3.3.1 Partial Derivatives and Their Applications 167 3.3.2 Multivariable Integrals 174 3.3.3 The Many Versions of Stokes’ Theorem 179 3.4 Equations with Functions as Unknowns 185 3.4.1 Functional Equations 185 3.4.2 Ordinary Differential Equations of the First Order 191 Contents ix 3.4.3 Ordinary Differential Equations of Higher Order 195 3.4.4 Problems Solved with Techniques of Differential Equations 198 4 Geometry and Trigonometry 201 4.1 Geometry 201 4.1.1 Vectors 201 4.1.2 The Coordinate Geometry of Lines and Circles 206 4.1.3 Conics and Other Curves in the Plane 212 4.1.4 Coordinate Geometry in Three and More Dimensions 219 4.1.5 Integrals in Geometry 225 4.1.6 Other Geometry Problems 228 4.2 Trigonometry 231 4.2.1 Trigonometric Identities 231 4.2.2 Euler’s Formula 235 4.2.3 Trigonometric Substitutions 238 4.2.4 Telescopic Sums and Products in Trigonometry 242 5 Number Theory 245 5.1 Integer-Valued Sequences and Functions 245 5.1.1 Some General Problems 245 5.1.2 Fermat’s Infinite Descent Principle 248 5.1.3 The Greatest Integer Function 250 5.2 Arithmetic 253 5.2.1 Factorization and Divisibility 253 5.2.2 Prime Numbers 254 5.2.3 Modular Arithmetic 258 5.2.4 Fermat’s Little Theorem 260 5.2.5 Wilson’s Theorem 264 5.2.6 Euler’s Totient Function 265 5.2.7 The Chinese Remainder Theorem 268 5.3 Diophantine Equations 270 5.3.1 Linear Diophantine Equations 270 5.3.2 The Equation of Pythagoras 274 5.3.3 Pell’s Equation 276 5.3.4 Other Diophantine Equations 279 6 Combinatorics and Probability 281 6.1 Combinatorial Arguments in Set Theory and Geometry 281 6.1.1 Set Theory and Combinatorics of Sets 281 6.1.2 Permutations 283 6.1.3 Combinatorial Geometry 286 x Contents 6.1.4 Euler’s Formula for Planar Graphs 289 6.1.5 Ramsey Theory 291 6.2 Binomial Coefficients and Counting Methods 294 6.2.1 Combinatorial Identities 294 6.2.2 Generating Functions 298 6.2.3 Counting Strategies 302 6.2.4 The Inclusion–Exclusion Principle 308 6.3 Probability 310 6.3.1 Equally Likely Cases 310 6.3.2 Establishing Relations Among Probabilities 314 6.3.3 Geometric Probabilities 318 Solutions Methods of Proof 323 Algebra 359 Real Analysis 459 Geometry and Trigonometry 603 Number Theory 673 Combinatorics and Probability 727 Index of Notation 791 Index 795 Preface A problem book at the college level. A study guide for the Putnam competition. A bridge between high school problem solving and mathematical research. A friendly introduction to fundamental concepts and results. All these desires gave life to the pages that follow. The William Lowell Putnam Mathematical Competition is the most prestigious math- 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 the merits of an intellectual intercollegiate competition. Nowadays, over 2500 students from more than 300 colleges and universities in the United States and Canada take part in it. The name Putnam has become synonymous with excellence in undergraduate mathematics. Using the Putnam competition as a symbol, we lay the foundations of higher math- 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. It gives motivated students a chance to learn concepts and acquire strate- gies, hone their skills and test their knowledge, seek connections, and discover real world applications. Its ultimate goal is to build the appropriate background for graduate studies, whether in mathematics or applied sciences. Our point of view is that in mathematics it is more important to understand why than to know how. Because of this we insist on proofs and reasoning. After all, mathematics means, as the Romanian mathematician Grigore Moisil once said, “correct reasoning.’’ The ways of mathematical thinking are universal in today’s science. Putnam and Beyond targets primarily Putnam training sessions, problem-solving seminars, and math clubs at thecollege level, filling a gap inthe undergraduate curriculum. But it does more than that. Written in the structured manner of a textbook, but with strong emphasis on problems and individual work, it covers what we think are the most important topics and techniques in undergraduate mathematics, brought together within the confines of a single book in order to strengthen one’s belief in the unitary nature of xii Preface mathematics. It is assumed that the reader possesses a moderate background, familiarity with the subject, and a certain level of sophistication, for what we cover reaches beyond the usual textbook, both in difficulty and in depth. When organizing the material, we were inspired by Georgia O’Keeffe’s words: “Details are confusing. It is only by selection, by elimination, by emphasis that we get at the real meaning of things.’’ The book can be used to enhance the teaching of any undergraduate mathematics course, since it broadens the database of problems for courses in real analysis, linear algebra, trigonometry, analytical geometry, differential equations, number theory, com- binatorics, and probability. Moreover, it can be used by graduate students and educators alike to expand their mathematical horizons, for many concepts of more advanced math- ematics can be found here disguised in elementary language, such as the Gauss–Bonnet theorem, the linear propagation of errors in quantum mechanics, knot invariants, or the Heisenberg group. The way of thinking nurtured in this book opens the door for true scientific investigation. As for the problems, they are in the spirit of mathematics competitions. Recall that the Putnam competition has two parts, each consisting of six problems, numbered A1 through A6, and B1 through B6. It is customary to list the problems in increasing order of difficulty, with A1 and B1 the easiest, and A6 and B6 the hardest. We keep the same ascending pattern but span a range from A0 to B7. This means that we start with some inviting problems below the difficulty of the test, then move forward into the depths of mathematics. As sources of problems and ideas we used the Putnam exam itself, the Interna- tional Competition in Mathematics for University Students, the International Mathemat- ical Olympiad, national contests from the United States of America, Romania, Rus- sia, China, India, Bulgaria, mathematics journals such as the American Mathemati- cal Monthly, Mathematics Magazine, Revista Matematic ˘ a din Timi¸soara (Timi¸soara Mathematics Gazette), Gazeta Matematic ˘ a (Mathematics Gazette, Bucharest), Kvant (Quantum), K ˝ ozépiskolai Matematikai Lapok (Mathematical Magazine for High Schools (Budapest)), and a very rich collection of Romanian publications. Many problems are original contributions of the authors. Whenever possible, we give the historical back- ground and indicate the source and author of the problem. Some of our sources are hard to find; this is why we offer you their most beautiful problems. Other sources are widely circulated, and by selecting some of their most representative problems we bring them to your attention. Here is a brief description of the contents of the book. The first chapter is introductory, giving an overview of methods widely used in proofs. The other five chapters reflect areas of mathematics: algebra, real analysis, geometry and trigonometry, number theory, combinatorics and probability. The emphasis is placed on the first two of these chapters, since they occupy the largest part of the undergraduate curriculum. Within each chapter, problems are clustered by topic. We always offer a brief theoret- ical background illustrated by one or more detailed examples. Several problems are left [...]... xiii for the reader to solve And since our problems are true brainteasers, complete solutions are given in the second part of the book Considerable care has been taken in selecting the most elegant solutions and writing them so as to stir imagination and stimulate research We always “judged mathematical proofs,’’ as Andrew Wiles once said, “by their beauty.’’ Putnam and Beyond is the fruit of work of... them at the first encounter and return to them as you become more experienced If you are a Putnam competitor, then as you go on with the study of the book try your hand at the true Putnam problems (which have been published in three excellent volumes) Identify your weaknesses and insist on those chapters of Putnam and Beyond Every once in a while, for a problem that you solved, write down the solution in... Proof, Algebra, Real Analysis, Geometry and Trigonometry, Number Theory, Combinatorics and Probability, divided into subchapters such as LinearAlgebra, Sequences and Series, Geometry, andArithmetic All subchapters are self-contained and independent of each other and can be studied in any order In most cases they reflect standard undergraduate courses or fields of mathematics The sections within each subchapter... your solutions be correct, structured, convincing, and easy to follow An instructor can add some of the problems from the book to a regular course in order to stimulate and challenge the better students Some of the theoretical subjects can also be incorporated in the course to give better insight and a new perspective Putnam xvi A Study Guide and Beyond can be used as a textbook for problem-solving... Induction 5 The third example is a problem from the 5th W.L Putnam Mathematical Competition, and it was selected because its solution combines several proofs by induction If you find it too demanding, think of Vincent van Gogh’s words: “The way to succeed is to keep your courage and patience, and to work energetically.’’ Example For m a positive integer and n an integer greater than 2, define f1 (n) = n, f2... regular textbook and make sure that you understand it very well Then choose the appropriate chapter or subchapter of this book and proceed section by section Read first the theoretical background and the examples from the introductory part; then do the problems These are listed in increasing order of difficulty, but even the very first can be tricky Don’t get discouraged; put effort and imagination into... Lowell Putnam Mathematical Competition In conclusion, we would like to thank Elgin Johnston, Dorin Andrica, Chris Jeuell, Ioan Cucurezeanu, Marian Deaconescu, Gabriel Dospinescu, Ravi Vakil, Vinod Grover, V.V Acharya, B.J Venkatachala, C.R Pranesachar, Bryant Heath, and the students of the International Mathematical Olympiad training programs of the United States and India for their suggestions and contributions... Kramer, and Paul Stanford for carefully reading the manuscript and considerably improving its quality We would be delighted to receive further suggestions and corrections; these can be sent to rgelca@gmail.com May 2007 Razvan Gelca ˘ Texas Tech University Titu Andreescu University of Texas at Dallas A Study Guide The book has six chapters: Methods of Proof, Algebra, Real Analysis, Geometry and Trigonometry,... set, and the principle of invariance The basic nature of these methods and their universal use throughout mathematics makes this separate treatment necessary In each case we have selected what we think are the most appropriate examples, solving some of them in detail and asking you to train your skills on the others And since these are fundamental methods in mathematics, you should try to understand... Michigan and Texas Tech University Putnam teams and of the International Mathematical Olympiad teams of the United States and India, as well as the product of the vast experience of the second author as head coach of the United States International Mathematical Olympiad team, coach of the Romanian International Mathematical Olympiad team, director of the American Mathematics Competitions, and member . Putnam and Beyond R ˘ azvan Gelca Titu Andreescu Putnam and Beyond R ˘ azvan Gelca Texas Tech University Department of Mathematics and Statistics MA 229 Lubbock,. 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. mathematical thinking are universal in today’s science. Putnam and Beyond targets primarily Putnam training sessions, problem-solving seminars, and math clubs at thecollege level, filling a gap inthe