Libro de imo

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Problem Books in Mathematics

Edited by P. Winkler

ˇ ´ Dusan Djukic ´ Vladimir Jankovic ´ Ivan Matic ´ Nikola Petrovic

The IMO Compendium
A Collection of Problems Suggested for the International Mathematical Olympiads: 1959–2004

With 200 Figures

ˇ ´ Dusan Djukic Department of Mathematics University of Toronto Toronto ON, M5S3G3 Canada djdusan@EUnet.yu ´ Ivan Matic Department ofMathematics Berkeley, CA USA matic@matf.bf.ac.yu Series Editor: Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755-3551 USA Peter.winkler@dartmouth.edu

´ Vladimir Jankovic Department of Mathematics University of Belgrade 11000 Belgrade Serbia and Montenegro vjankovic@matf.bg.ac.yu ´ Nikola Petrovic Institute of Physics 11000 Belgrade Serbia and Montenegronzpetr@eunet.yu

Mathematics Subject Classification (2000): 00A07 Library of Congress Control Number: 2005934915 ISBN-10: 0-387-24299-6 ISBN-13: 978-0387-24299-6 © 2006 Springer Science+Business Media, Inc. 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, Inc., 233 Spring Street, New York, NY10013, 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 notidentified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springer.com (MVY)

Preface

The International Mathematical Olympiad (IMO) is nearing its fiftieth anniversary and has already created a very rich legacy and firmly established itself as the most prestigiousmathematical competition in which a high-school student could aspire to participate. Apart from the opportunity to tackle interesting and very challenging mathematical problems, the IMO represents a great opportunity for high-school students to see how they measure up against students from the rest of the world. Perhaps even more importantly, it is an opportunity to make friends and socialize withstudents who have similar interests, possibly even to become acquainted with their future colleagues on this first leg of their journey into the world of professional and scientific mathematics. Above all, however pleasing or disappointing the final score may be, preparing for an IMO and participating in one is an adventure that will undoubtedly linger in one’s memory for the rest of one’s life. It is tothe high-school-aged aspiring mathematician and IMO participant that we devote this entire book. The goal of this book is to include all problems ever shortlisted for the IMOs in a single volume. Up to this point, only scattered manuscripts traded among different teams have been available, and a number of manuscripts were lost for many years or unavailable to many. In this book, all manuscriptshave been collected into a single compendium of mathematics problems of the kind that usually appear on the IMOs. Therefore, we believe that this book will be the definitive and authoritative source for high-school students preparing for the IMO, and we suspect that it will be of particular benefit in countries lacking adequate preparation literature. A high-school student could spend an enjoyableyear going through the numerous problems and novel ideas presented in the solutions and emerge ready to tackle even the most difficult problems on an IMO. In addition, the skill acquired in the process of successfully attacking difficult mathematics problems will prove to be invaluable in a serious and prosperous career in mathematics. However, we must caution our aspiring IMO participant on the use of...
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