Valentin Boju Louis Funar The Math Problems Notebook Birkhauser¨ Boston • Basel • Berlin Valentin Boju Louis Funar MontrealTech Institut Fourier BP 74 Institut de Technologie de Montreal URF Mathematiques P.O. Box 78575, Station Wilderton Universite´ de Grenoble I Montreal, Quebec, H3S 2W9 Canada 38402 Saint Martin d’Heres cedex [email protected] France [email protected] Cover design by Alex Gerasev. Mathematics Subject Classification (2000): 00A07 (Primary); 05-01, 11-01, 26-01, 51-01, 52-01 Library of Congress Control Number: 2007929628 ISBN-13: 978-0-8176-4546-5 e-ISBN-13: 978-0-8176-4547-2 Printed on acid-free paper. c 2007 Birkhauser¨ Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhauser¨ Boston, c/o 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 forbid- den. 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. 987654321 www.birkhauser.com (TXQ/SB) The Authors Valentin Boju was professor of mathematics at the University of Craiova, Romania, until his retirement in 2000. His research work was primarily in the field of geome- try. He further promoted biostatistics and biomathematics as a discipline within the Department of Medicine of the University of Craiova. He left Romania for Canada, where he has taught at MontrealTech since 2001. He was actively involved in coach- ing college students in problem-solving and intuitive mathematics. In 2004, he was honored with the title of Officer of the Order “Cultural Merit,” Category “Scientific Research.” MontrealTech Institut de Technologie de Montréal P.O. Box 78575 Station Wilderton Montréal, Quebec, H3S 2W9 Canada e-mail: [email protected] Louis Funar has been a researcher at CNRS, at the Fourier Institute, University of Grenoble, France, since 1994. During his college years, he participated three times in the International Mathematics Olympiads with the Romanian team in the years 1983–1985, winning bronze, gold, and silver medals. His main research interests are in geometric topology and mathematical physics. Institut Fourier, BP 74 Mathématiques UMR 5582 Université de Grenoble I 38402 Saint-Martin-d’Hères cedex France e-mail: [email protected] To all schoolteachers, particularly to my wonderful parents, Evdochia and Nicolae Boju, who genuinely believed in the education of the younger generation. To my parents, Maria and Ioan Funar, who laid the foundation for our collaboration, as a late reward for their hard work and help, for their enthusiasm, determinedness, and strength during all these years. Preface The authors met on a Sunday morning about 25 years ago in Room 113. One of us was a college student and the other was leading the Sunday Math Circle. This circle, within the math department of the University of Craiova, gathered college students who possessed a common passion for mathematics. Most of them were participating in the mathematical competitions in vogue at that time, namely the Olympiads. Others just wanted to have good time. There were similar math circles everywhere in the country, in which high-school students were committed to active training for math competitions. The highest stan- dard was achieved by the selective training camps organized at the national level, which were led by professional coaches and mathematicians who were able to stimu- late and elicit high performance. To mention a few among the leaders, we recall Dorin Andrica, TomaAlbu, TituAndreescu, Mircea Becheanu, Ioan Cuculescu, Dorel Mihet, and Ioan Tomescu. The Sunday Math Circle shifted partially from its purpose of Olympiad training by following freely the leader’s ideas and thus becoming a place primarily intended for disseminating beautiful mathematics at an elementary level. The fundamental texts were the celebrated book of Richard Courant and Herbert Robbins with the mysterious title What is mathematics?, and the book of David Hilbert and Stefan Cohn–Vossen entitled Geometry and the Imagination. The participants soon realized the differences in both scale and novelty between the competition-type math problems that they encountered every day and the mathematics built up over hundreds of years and engaged in by professional mathematicians. Competition problems have to be solved in a short amount of time, and people compete against each other. These might be highly nontrivial, unconventional prob- lems requiring deep insights and a lot of imagination, but still they have the advantage that one knows in advance that they have a solution. In the real mathematical world, problems often had to wait not merely years but sometimes centuries to be solved. A working mathematician could go for months or years trying to solve a particular prob- lem that had not been solved before and take the risk of bitterly failing. Moreover, the sustained effort needed for accomplishing such a task would have to take into account a delicate balance between the accumulation of knowledge, methods, and tools and viii Preface the creation from scratch of a path leading to a solution. This intellectual adventure is filled with suspense and frustration, since one does not know for sure what one is trying to prove or whether it is indeed true. Real mathematics seemed to many of us to be too far away from competition problems. The philosophy of the Sunday Math Circle was that, in contrast to what we might think, the border between the two kinds of mathematics could be so vague and flexible that at times one could cross it at an early age. The history of mathematics abounds in examples in which a fresh mind was able to find an unexpected solution that specialists had been unable to find. One needs, however, the right training and the unconventional sense of finding the inspiring problem. Eventually, once a solution has been found, mathematicians are then willing to try to understand it even better, and other solutions follow in time, each one simpler and clearer than the previous one. In some sense, once solved, even the hardest problems start losing slowly their aura of difficulty and eventually become just problems. Problems that are today part of the curriculum of the average high-school student were difficult research problems three hundred years ago, solved only by brilliant mathematicians. This phenomenon demonstrates the evolution that language and science have undergone since then. The authors conceived the present book with the nostalgia of the “good old times" of the Sunday Math Circles. We wanted something that carries the mark of that philos- ophy, namely, a number of challenging math problems for Olympiads with a glimpse of related problems of interest for the mathematician. The present text is a collection of problems that we think will be useful in training students for mathematical com- petitions. On the other hand, we hope that it might fulfill our second goal, namely, that of awakening interest in advanced mathematics. Thus its audience might range from college students and teachers to advanced math students and mathematicians. The problems in each section are in increasing order of difficulty, so that the reader give some of the problems a try. We wanted to have 25% easy problems concerning basic tools and methods and consisting mainly of instructional exercises. The beginner might jump directly to the solutions, where a concentrate of the general theory and basic tricks can be found. The largest chunk contains about 50% problems of medium difficulty, which could be useful in training for mathematical competitions from local to international levels. The remaining 25% might be considered difficult problems even for the experienced problem-solver. These problems are often accompanied by comments that put the results in a broader perspective and might incite the reader to pursue the research further. The problems serve as an excuse for introducing all sorts of generalizations and closely related open problems, which are spread among the solutions. Some of these are truly outstanding problems that resisted the efforts of mathematicians over the centuries, such as the congruent numbers conjecture and the Riemann hypothesis, which are among seven Millennium Prize Problems that the Clay Mathematics Insti- tute recorded as some of the most difficult issues with which mathematicians were struggling at the turn of the second millennium and offered a reward of one million dollars for a solution to each one. In mathematics the frontier of our knowledge is still open, and it abounds in important unsolved problems, many of which can be Preface ix understood at the undergraduate level. The reader might be soon driven to the edge of that part of mathematics where he or she could undertake original research. We drew inspiration from the spirit of the famous books by R. K. Guy and his collaborators. Nevertheless, our aim was not to build up a collection of open questions in elementary mathematics but rather to offer a journey among the basic methods in problem-solving. Developing intuition and strengthening the most popular techniques in mathematical competitions are equally part of the goal. In the meantime, we offer the enthusiastic reader a brief glimpse of mathematical research, a place where problems yet unsolved long for deliverance. The present collection of problems evolved from a notebook in which the second- named author collected the most interesting and unconventional problems that he encountered during his training for mathematical competitions in the 1980s.