To Alina and to our Mothers - Home - Springer978-1-4612-2138-8/1.pdf · Titu Andreescu American Mathematics Competitions University of Nebraska Lincoln, NE 68588-0658 U.S.A. Ri\zvan

To Alina and to our Mothers

Titu Andreescu Razvan Gelca

MATHEMATICAL OLYMPIAD

CHALLENGES

Foreword by Mark Saul

Birkhauser

Boston • Basel • Berlin

Titu Andreescu American Mathematics Competitions University of Nebraska Lincoln, NE 68588-0658 U.S.A.

Ri\zvan Gelca Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A.

Ubrary of Congress Cataloging-In-Publication Data

Andreescu, Titu, 1956-. Mathematical Olympiad challenges / Titu Andreescu, Rlizvan Gelca.

p.cm. Includes bibliographical references.

ISBN-l3: 978-0-81764155-9 e-ISBN-l3: 978-1-4612-2138-8

001: 10.1007/978-1-4612-2138-8

1. Mathematics-Problems, exercises, etc. I. Gelca, Rlizvan, 1%7-11. Title. QA43.A52 2000 510'.76-dc21 99-086229

CIP

Mathematics Subject Classifications 2000: ooA05, ooA07, OS-XX, 08-XX, ll-XX, 51-XX

Printed on acid-free paper ©2ooo Birkhiiuser Boston

m® Birkhiiuser a(J>.J

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhiiuser Boston, c/o Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, 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 of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN-l3: 978-0-81764190-0 (Hardcover)

ISBN-13: 978-0-81764155-9 (Softcover)

SPIN 10766705

SPIN 10748545

Reformatted from authors' files by TEXniques, Inc, Cambridge, MA. Cover design by Jeff Cosloy, Newton, MA.

9 8 765 432 1

Contents

Foreword Mark Saul

Preface Titu Andreescu and Riizvan Gelca

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 ......... .

2 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 + l/x ........ . 2.9 Matrices ......... . 2.10 The Mean Value Theorem

xi

xiii

1

3 4 6 9

13 16 20 22 24 26 29

33 34 36 38 41 43 47 50 52 54 55

vi

3 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 Equations with Unknowns as Exponents 3.7 Numerical Functions 3.8 Invariants ............... . 3.9 Pell Equations . . . . . . . . . . . . . . 3.10 Prime Numbers and Binomial Coefficients.

II Solutions

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 ......... .

2 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 + l/x ........ . 2.9 Matrices.......... 2.10 The Mean Value Theorem

Contents

59 60 62 64 66 69 72 74 77 80 84

87

89 90 93 99

106 114 124 129 136 140 145

151 152 156 159 164 166 172 176 183 188 191

Contents

3 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 Equations with Unknowns as Exponents 3.7 Numerical Functions 3.8 Invariants . . . . . . . . . . . . . . . . 3.9 Pell Equations ............. . 3.10 Prime Numbers and Binomial Coefficients.

A Appendix A: Definitions and Notation A.l Glossary of Terms. . A.2 Glossary of Notation . . . . . . .

About the Authors

vii

197 198 201 206 209 216 221 226 233 237 244

251 252 257

259

M atematicii, matematicii, matematicii, atfta matematicii? Nu, mai multii. 1

Grigore Moisil

1 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 which refuses to yield. If the solver is lucky, there will be the moment of insight which heralds the start of a successful solution. Like a well-crafted work of fiction, a good Olympiad problem tells a story of mathematical creativity which captures a good part of the real experience, and leaves the participant wanting still more.

And the present 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. While 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.

xii Foreword

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 practice material which 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 mathematics. 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 too--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 motivation. 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

At the beginning of the twenty-first century, elementary mathematics is under­going two major changes. The first one 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. Mathematical Olympiad Challenges reflects both trends. It gathers essay-type, non-routine, 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, in seminars, and in the books and articles we write, to train our students to be better problem-posers and problem-solvers than we are."

The problems we selected are definitely not exercises. Our definition of an exercise is that you look at it and you know immediately how to complete it. It is just a question of doing the work, whereas by a problem, we mean a more. intricate question for which at first one has probably no clue to how to approach it, but by perseverance and inspired effort one can transform it into a sequence of exercises. We have chosen mainly Olympiad problems, because they are beautiful, interesting, fun to solve, and they best reflect mathematical ingenuity and elegant arguments.

Mathematics competitions have a long-standing tradition. More than 100 years ago, Hungary and Romania instituted their first national competitions in mathematics. The Eotvos Contest in elementary mathematics has been open to Hungarian students in their last years of high school since 1894. The Gazeta Matematidi contest, named after the major Romanian mathematics journal for high school students, was founded in 1895. Other countries soon followed, and by 1938 as many as 12 countries were regularly organizing national mathemat­ics contests. In 1959, Romania had the initiative to host the first International Mathematical Olympiad (IMO). Only seven European countries participated. Since then, the number has grown to more than 80 countries, from all conti­nents. The United States joined the IMO in 1974. Its greatest success came in 1994, when all six USA team members won a gold medal with perfect scores, a unique performance in the 40-year history of the IMO.

Within the United States there are several national mathematical compe-

xiv Preface

titions, such as the AMC ..... 8 (formerly the American Junior High School Mathematics Examination), AMC ..... 10 (the American Mathematics Contest for students in grades 10 or below) and AMC ..... 12 (formerly the Ameri­can High School Mathematics Examination), the American Invitational Math­ematics Examination (AIME), the United States Mathematical Olympiad (US­AMO), the W. L. Putnam Mathematical Competition, and a number of regional contests such as the American Regions Mathematics League (ARML). Every year more than 600,000 students take part in these competitions. The prob­lems from this book are of the type that usually appear in the AIME, USAMO, IMO and the W. L. Putnam competition, and similar contests from other coun­tries, such as Austria, Bulgaria, Canada, China, France, Germany, Hungary, India, Ireland, Israel, Poland, Romania, Russia, South Korea, Ukraine, United Kingdom, and Vietnam. Also included are problems from international com­petitions such as the: IMO, Balkan Mathematical Olympiad, Ibero-American Mathematical Olympiad, Asian-Pacific Mathematical Olympiad, Austrian-Po­lish Mathematical Competition, Tournament of the Towns, and selective ques­tions from problem books and from the following journals: Kvant (Quantum), Revista Matematica din Timi§oara (Timi§oara Mathematics Review), Gazeta Matematica (Mathematics Gazette), Matematika v Skole (Mathematics in School), American Mathematical Monthly, and Matematika Sofia. Over 60 problems were created by the authors and have yet to be circulated.

Mathematical Olympiad Challenges is written as a textbook to be used in advanced problem solving courses, or as a reference source for people inter­ested in tackling challenging mathematical problems. The problems are clus­tered in thirty sections, grouped in three chapters: Geometry and Trigonometry, Algebra and Analysis, and Number Theory and Combinatorics. The placement of geometry at the beginning of the book is unusual but not accidental. The reason behind this choice is well reflected in the words of V. I. Arnol'd: "Our brain has two halves: one is responsible for the multiplication of polynomials and languages, and the other half is responsible for orientating figures in space and all things important in real life. Mathematics is geometry when you have to use both halves." (Notices of the AMS, April 1997).

Each section is self-contained, independent of the others, and focuses on one main idea. All sections start with a short essay discussing basic facts, and one or more representative examples. This sets the tone for the whole unit. Next, a number of carefully chosen problems are listed, to be solved by the reader. The solutions to all problems are given in detail in the second part of the book. After each solution we provide the source of the problem, if known. Even if successful in approaching a problem, the reader is advised to study the solution given at the end of the book. As problems are generally listed in increasing order of difficulty, solutions to initial problems might suggest

Preface xv

illuminating ideas for completing later ones. At the very end we include a glossary of definitions and fundamental properties used in the book.

Mathematical Olympiad Challenges has been successfully tested in classes taught by the authors at the Illinois Mathematics and Science Academy, the University of Michigan, the University of Iowa, and in the training of the USA International Mathematical Olympiad Team. In the end, we would like to ex­press our thanks to Gheorghe Eckstein, Chetan Balwe, Mircea Grecu, Zuming Feng, Zvezdelina Stankova-Frenkel and Florin Pop for their suggestions, and especially to Svetoslav Savchev for carefully reading the manuscript and for contributions that improved many solutions in the book.

Titu Andreescu, American Mathematics Competitions Razvan Gelca, University of Michigan

MATHEMATICAL OLYMPIAD

CHALLENGES