This book was not intended to be an exhaustive treatment of elementary mathematics if ever such a book appears, it will be a sad day for mathematics. What will most likely surprise the reader when browsing just the table of contents is the absence of any chapters on geometry. Allow us to discuss in brief the structure of the book. We strove to select only those problems, easy and hard, that best illustrate the ideas we wanted to exhibit. Viii tine exercises that engender familiarity with the joy of the difficult problems in which we find the truly beautiful ideas. The selection of problems was made by weighing the need for rou. We will assume of the reader a certain familiarity with classical theorems of elementary mathematics, which we will use freely. Whenever their proofs are particularly useful, they are provided. Certainly, more subtle results like quadratic reciprocity and existence of primitive roots are related to the basic results in linear algebra or mathematical analysis. Thus, the theoretical portions are short, while the emphasis is squarely placed on the problems. Even so, as it will be easily discovered, many of the problems in this book are very difficult. In this context, the audience is more diverse than one might think. They serve as an invitation to mathematics beyond competitions, regarded by many to be the "true mathematics". And no, classical does not mean easy! These mathematical beauties are more than just proof that elementary mathematics can produce jewels. While the book is rife with problems collected from various mathematical competitions and journals, one cannot neglect the classical results of mathematics, which naturally exceed the level of time-constrained competitions. It takes more effort to realize that these are not the only potential beneficiaries of this work. A mere perusal of this book should be sufficient to identify its target audience: students and coaches preparing for mathematical Olympiads, national or international. It could be a new problem, a comment we considered pertinent, or a solution that escaped our rationale until this predictive moment, when we were supposed to bring it under the scrutiny of a specialist in the field. The inevitability of this question does not facilitate the answer, because after five years of writing and rewriting we still had something to add. Preface What can a new book of problems in elementary mathematics possibly contribute to the vast existing collection of journals, articles, and books? This was our main concern when we decided to write this book.
Math isn't the art of answering mathematical questions, it is the art of asking the right questions, the questions that give you insight, the ones that lead you in interesting directions, the ones that connect with lots of other interesting questions - the ones with beautiful answers. 9 8 7 6 5 4 3 2 1 Cover design by Iury Ulzutuev The use in this publication of tradenames, 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. 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. This work may not be translated or copied in whole or in part without the written permission of the publisher (XYZ Press, LLC, 1721 Monaco Drive, Allen TX, 75002, USA) and the authors except for brief excerpts in connection with reviews or scholarly analysis.
Library of Congress Control Number: 2008924026 ISBN-13: 978-0-979 © 2008 XYZ Press, LLC All rights reserved. Gabriel Dospinescu Ecole Normale Superieure Departement de Mathematiques 45, rue d'Ulm Paris, F-75230 France Titu Andreescu University of Texas at Dallas School of Natural Sciences and Mathematics 800 W Campbell Road Richardson, TX 75080 USA
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