To Alina and to Our Mothers Titu Andreescu ˘ Razvan Gelca Mathematical Olympiad Challenges SECOND EDITION Foreword by Mark Saul Birkhäuser Boston • Basel • Berlin Titu Andreescu University of Texas at Dallas School of Natural Sciences and Mathematics Richardson, TX 75080 USA titu.andreescu@utdallas.edu ISBN: 978-0-8176-4528-1 DOI: 10.1007/978-0-8176-4611-0 Răzvan Gelca Texas Tech University Department of Mathematics and Statistics Lubbock, TX 79409 USA rgelca@gmail.com e-ISBN: 978-0-8176-4611-0 Mathematics Subject Classification (2000): 00A05, 00A07, 05-XX, 11-XX, 51XX © Birkhäuser Boston, a part of Springer Science+Business Media, LLC, Second Edition 2009 © Birkhäuser Boston, First Edition 2000 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper springer.com Contents Foreword xi Preface to the Second Edition xiii Preface to the First Edition xv I Problems 1 Geometry and Trigonometry 1.1 A Property of Equilateral Triangles 1.2 Cyclic Quadrilaterals 1.3 Power of a Point 1.4 Dissections of Polygonal Surfaces 1.5 Regular Polygons 1.6 Geometric Constructions and Transformations 1.7 Problems with Physical Flavor 1.8 Tetrahedra Inscribed in Parallelepipeds 1.9 Telescopic Sums and Products in Trigonometry 1.10 Trigonometric Substitutions 10 15 20 25 27 29 31 34 Algebra and Analysis 2.1 No Square Is Negative 2.2 Look at the Endpoints 2.3 Telescopic Sums and Products in Algebra 2.4 On an Algebraic Identity 2.5 Systems of Equations 2.6 Periodicity 2.7 The Abel Summation Formula 2.8 x + 1/x 2.9 Matrices 2.10 The Mean Value Theorem 39 40 42 44 48 50 55 58 62 64 66 Contents vi Number Theory and Combinatorics 3.1 Arrange in Order 3.2 Squares and Cubes 3.3 Repunits 3.4 Digits of Numbers 3.5 Residues 3.6 Diophantine Equations with the Unknowns as Exponents 3.7 Numerical Functions 3.8 Invariants 3.9 Pell Equations 3.10 Prime Numbers and Binomial Coefficients II Solutions 69 70 71 74 76 79 83 86 90 94 99 103 Geometry and Trigonometry 1.1 A Property of Equilateral Triangles 1.2 Cyclic Quadrilaterals 1.3 Power of a Point 1.4 Dissections of Polygonal Surfaces 1.5 Regular Polygons 1.6 Geometric Constructions and Transformations 1.7 Problems with Physical Flavor 1.8 Tetrahedra Inscribed in Parallelepipeds 1.9 Telescopic Sums and Products in Trigonometry 1.10 Trigonometric Substitutions 105 106 110 118 125 134 145 151 156 160 165 Algebra and Analysis 2.1 No Square is Negative 2.2 Look at the Endpoints 2.3 Telescopic Sums and Products in Algebra 2.4 On an Algebraic Identity 2.5 Systems of Equations 2.6 Periodicity 2.7 The Abel Summation Formula 2.8 x + 1/x 2.9 Matrices 2.10 The Mean Value Theorem 171 172 176 183 188 190 197 202 209 214 217 Number Theory and Combinatorics 3.1 Arrange in Order 3.2 Squares and Cubes 3.3 Repunits 3.4 Digits of Numbers 3.5 Residues 3.6 Diophantine Equations with the Unknowns as Exponents 223 224 227 232 235 242 246 Contents 3.7 3.8 3.9 3.10 vii Numerical Functions Invariants Pell Equations Prime Numbers and Binomial Coefficients 252 260 264 270 Appendix A: Definitions and Notation 277 A.1 Glossary of Terms 278 A.2 Glossary of Notation 282 Matematic˘a, matematic˘a, matematic˘a, atˆata matematic˘a? Nu, mai mult˘a.1 Grigore Moisil Mathematics, mathematics, mathematics, that much mathematics? No, even more Foreword Why Olympiads? Working mathematicians often tell us that results in the field are achieved after long experience and a deep familiarity with mathematical objects, that progress is made slowly and collectively, and that flashes of inspiration are mere punctuation in periods of sustained effort The Olympiad environment, in contrast, demands a relatively brief period of intense concentration, asks for quick insights on specific occasions, and requires a concentrated but isolated effort Yet we have found that participants in mathematics Olympiads have often gone on to become first-class mathematicians or scientists and have attached great significance to their early Olympiad experiences For many of these people, the Olympiad problem is an introduction, a glimpse into the world of mathematics not afforded by the usual classroom situation A good Olympiad problem will capture in miniature the process of creating mathematics It’s all there: the period of immersion in the situation, the quiet examination of possible approaches, the pursuit of various paths to solution There is the fruitless dead end, as well as the path that ends abruptly but offers new perspectives, leading eventually to the discovery of a better route Perhaps most obviously, grappling with a good problem provides practice in dealing with the frustration of working at material that refuses to yield If the solver is lucky, there will be the moment of insight that heralds the start of a successful solution Like a well-crafted work of fiction, a good Olympiad problem tells a story of mathematical creativity that captures a good part of the real experience and leaves the participant wanting still more And this book gives us more It weaves together Olympiad problems with a common theme, so that insights become techniques, tricks become methods, and methods build to mastery Although each individual problem may be a mere appetizer, the table is set here for more satisfying fare, which will take the reader deeper into mathematics than might any single problem or contest The book is organized for learning Each section treats a particular technique or topic Introductory results or problems are provided with solutions, then related problems are presented, with solutions in another section The craft of a skilled Olympiad coach or teacher consists largely in recognizing similarities among problems Indeed, this is the single most important skill that the coach can impart to the student In this book, two master Olympiad coaches have offered the results of their experience to a wider audience Teachers will find examples and topics for advanced students or for their own exercise Olympiad stars will find ... international competitions such as the IMO, Balkan Mathematical Olympiad, Ibero-American Mathematical Olympiad, Asian-Pacific Mathematical Olympiad, Austrian-Polish Mathematical Competition, Tournament of... (Mathematics in School), American Mathematical Monthly, and Matematika Sofia More than 60 problems were created by the authors and have yet to be circulated Mathematical Olympiad Challenges is written as... problems The second consists in spreading problem-solving culture throughout the world Mathematical Olympiad Challenges reflects both trends It gathers essay-type, nonroutine, open-ended problems