Mathematical Olympiad Treasures

Titu Andreescu  Bogdan Enescu Mathematical Olympiad Treasures Second Edition Titu Andreescu School of Natural Sciences and Mathematics University of Texas at Dallas Richardson, TX 75080 USA [email protected] Bogdan Enescu Department of Mathematics “BP Hasdeu” National College Buzau 120218 Romania [email protected] ISBN 978-0-8176-8252-1 e-ISBN 978-0-8176-8253-8 DOI 10.1007/978-0-8176-8253-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011938426 Mathematics Subject Classification (2010): 00A05, 00A07, 05-XX, 11-XX, 51-XX, 97U40 © Springer Science+Business Media, LLC 2004, 2011 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), 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 is part of Springer Science+Business Media (www.birkhauser-science.com) Preface Mathematical Olympiads have a tradition longer than one hundred years. The first mathematical competitions were organized in Eastern Europe (Hungary and Romania) by the end of the 19th century. In 1959 the first International Mathematical Olympiad was held in Romania. Seven countries, with a total of 52 students, attended that contest. In 2010, the IMO was held in Kazakhstan. The number of participating countries was 97, and the number of students 517. Obviously, the number of young students interested in mathematics and mathematical competitions is nowadays greater than ever. It is sufficient to visit some mathematical forums on the net to see that there are tens of thousands registered users and millions of posts. When we were thinking about writing this book, we asked ourselves to whom it will be addressed. Should it be the beginner student, who is making the first steps in discovering the beauty of mathematical problems, or, maybe, the more advanced reader, already trained in competitions. Or, why not, the teacher who wants to use a good set of problems in helping his/her students prepare for mathematical contests. We have decided to t