Titu Andreescu Razvan˘ Gelca Mathematical Olympiad Challenges SECOND EDITION Foreword by Mark Saul Birkhäuser Boston • Basel • Berlin Contents Foreword xi Preface to the Second Edition xiii Preface to the First Edition xv IProblems 1 1 Geometry and Trigonometry 3 1.1APropertyofEquilateralTriangles................... 4 1.2CyclicQuadrilaterals.......................... 6 1.3PowerofaPoint............................. 10 1.4 Dissections of Polygonal Surfaces . ................ 15 1.5 Regular Polygons . ........................... 20 1.6GeometricConstructionsandTransformations............. 25 1.7ProblemswithPhysicalFlavor..................... 27 1.8TetrahedraInscribedinParallelepipeds................. 29 1.9 Telescopic Sums and Products in Trigonometry ............ 31 1.10 Trigonometric Substitutions . .................... 34 2 Algebra and Analysis 39 2.1NoSquareIsNegative......................... 40 2.2 Look at the Endpoints . .................... 42 2.3 Telescopic Sums and Products in Algebra . ............ 44 2.4 On an Algebraic Identity . .................... 48 2.5SystemsofEquations.......................... 50 2.6Periodicity................................ 55 2.7TheAbelSummationFormula..................... 58 2.8 x + 1/x ................................. 62 2.9Matrices................................. 64 2.10TheMeanValueTheorem....................... 66 vi Contents 3 Number Theory and Combinatorics 69 3.1ArrangeinOrder............................ 70 3.2SquaresandCubes........................... 71 3.3Repunits ................................ 74 3.4DigitsofNumbers............................ 76 3.5Residues................................. 79 3.6 Diophantine Equations with the Unknowns as Exponents . ..... 83 3.7NumericalFunctions.......................... 86 3.8Invariants................................ 90 3.9PellEquations.............................. 94 3.10PrimeNumbersandBinomialCoefficients............... 99 II Solutions 103 1 Geometry and Trigonometry 105 1.1APropertyofEquilateralTriangles................... 106 1.2CyclicQuadrilaterals.......................... 110 1.3PowerofaPoint............................. 118 1.4 Dissections of Polygonal Surfaces . ................ 125 1.5 Regular Polygons . ........................... 134 1.6GeometricConstructionsandTransformations............. 145 1.7ProblemswithPhysicalFlavor..................... 151 1.8TetrahedraInscribedinParallelepipeds................. 156 1.9 Telescopic Sums and Products in Trigonometry ............ 160 1.10 Trigonometric Substitutions . .................... 165 2 Algebra and Analysis 171 2.1NoSquareisNegative......................... 172 2.2 Look at the Endpoints . .................... 176 2.3 Telescopic Sums and Products in Algebra . ............ 183 2.4 On an Algebraic Identity . .................... 188 2.5SystemsofEquations.......................... 190 2.6Periodicity................................ 197 2.7TheAbelSummationFormula..................... 202 2.8 x + 1/x .................................. 209 2.9Matrices................................. 214 2.10TheMeanValueTheorem....................... 217 3 Number Theory and Combinatorics 223 3.1ArrangeinOrder............................ 224 3.2SquaresandCubes........................... 227 3.3Repunits................................. 232 3.4DigitsofNumbers............................ 235 3.5Residues................................. 242 3.6 Diophantine Equations with the Unknowns as Exponents . ..... 246 Contents vii 3.7NumericalFunctions.......................... 252 3.8Invariants................................ 260 3.9PellEquations.............................. 264 3.10PrimeNumbersandBinomialCoefficients............... 270 Appendix A: Definitions and Notation 277 A.1GlossaryofTerms............................ 278 A.2GlossaryofNotation.......................... 282 Matematica,˘ matematica,˘ matematica,˘ atataˆ matematica?˘ Nu, mai multa.˘ 1 Grigore Moisil 1Mathematics, 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 appe- tizer, 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 xii Foreword practice material that will leave them stronger and more ready to take on the next challenge, from whatever corner of mathematics it may originate. Newcomers to Olympiads will find an organized introduction to the experience. There is also something here for the more general reader who is interested in mathe- matics. Simply perusing the problems, letting their beauty catch the eye, and working through the authors’ solutions will add to the reader’s understanding. The multiple solutions link together areas of mathematics that are not apparently related. They often illustrate how a simple mathematical tool—a geometric transformation, or an algebraic identity—can be used in a novel way, stretched or reshaped to provide an unexpected solution to a daunting problem. These problems are daunting on any level. True to its title, the book is a challenging one. There are no elementary problems—although there are elementary solutions. The content of the book begins just at the edge of the usual high school curriculum. The calculus is sometimes referred to, but rarely leaned on, either for solution or for moti- vation. Properties of vectors and matrices, standard in European curricula, are drawn upon freely. Any reader should be prepared to be stymied, then stretched. Much is demanded of the reader by way of effort and patience, but the reader’s investment is greatly repaid. In this, it is not unlike mathematics as a whole. Mark Saul Bronxville School Preface to the Second Edition The second edition is a significantly revised and expanded version. The introductions to many sections were rewritten, adopting a more user-friendly style with more accessible and inviting examples. The material has been updated with more than 70 recent problems and examples. Figures were added in some of the solutions to geometry problems. Reader suggestions have been incorporated. We would like to thank Dorin Andrica and Iurie Boreico for their suggestions and contributions. Also, we would like to express our deep gratitude to Richard Stong for reading the entire manuscript and considerably improving its content. Titu Andreescu University of Texas at Dallas Razvan˘ Gelca Texas Tech University April 2008 Preface to the First Edition At the beginning of the twenty-first century, elementary mathematics is undergoing two major changes. The first is in teaching, where one moves away from routine exercises and memorized algorithms toward creative solutions to unconventional problems. The second consists in spreading problem-solving culture throughout the world. Mathe- matical Olympiad Challenges reflects both trends. It gathers essay-type, nonroutine, open-ended problems in undergraduate mathematics from around the world. As Paul Halmos said, “problems are the heart of mathematics,” so we should “emphasize them more and more in the classroom,