10. IMO Compendium 1959-2009.pdf - Problem Books in Mathematics Series Editor Peter Winkler Department of Mathematics Dartmouth College Hanover NH 03755

10. IMO Compendium 1959-2009.pdf - Problem Books in...

This preview shows page 1 out of 825 pages.

You've reached the end of your free preview.

Want to read all 825 pages?

Unformatted text preview: Problem Books in Mathematics Series Editor Peter Winkler Department of Mathematics Dartmouth College Hanover, NH 03755 USA [email protected] For other titles published in the series, go to ˇ Djukic´ • Vladimir Jankovi´c Dusan Ivan Matic´ • Nikola Petrovi´c The IMO Compendium A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009 Second Edition Dušan Djukić Department of Mathematics University of Toronto Toronto Ontario, M5S3G3 Canada [email protected] Vladimir Janković Department of Mathematics University of Belgrade Studentski Trg 16 11000 Belgrade Serbia [email protected] Ivan Matić Department of Mathematics Duke University Durham, North Carolina 27708 USA [email protected] Nikola Petrović Science Department Texas A&M University PO Box 23874 Doha Qatar [email protected] ISSN 0941-3502 ISBN 978-1-4419-9853-8 e-ISBN 978-1-4419-9854-5 DOI 10.1007/978-1-4419-9854-5 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011926996 © Springer Science+Business Media, LLC 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 ( ) Preface The International Mathematical Olympiad (IMO) exists for more than 50 years and has already created a very rich legacy and firmly established itself as the most prestigious mathematical 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 with students 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 to the 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 manuscripts have 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 enjoyable year 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 this book. Any book of problems, no matter how large, quickly depletes itself if the reader merely glances at a problem and then five minutes later, having determined that the problem seems unsolvable, glances at the solution. VI Preface The authors therefore propose the following plan for working through the book. Each problem is to be attempted at least half an hour before the reader looks at the solution. The reader is strongly encouraged to keep trying to solve the problem without looking at the solution as long as he or she is coming up with fresh ideas and possibilities for solving the problem. Only after all venues seem to have been exhausted is the reader to look at the solution, and then only in order to study it in close detail, carefully noting any previously unseen ideas or methods used. To condense the subject matter of this already very large book, most solutions have been streamlined, omitting obvious derivations and algebraic manipulations. Thus, reading the solutions requires a certain mathematical maturity, and in any case, the solutions, especially in geometry, are intended to be followed through with pencil and paper, the reader filling in all the omitted details. We highly recommend that the reader mark such unsolved problems and return to them in a few months to see whether they can be solved this time without looking at the solutions. We believe this to be the most efficient and systematic way (as with any book of problems) to raise one’s level of skill and mathematical maturity. We now leave our reader with final words of encouragement to persist in this journey even when the difficulties seem insurmountable and a sincere wish to the reader for all mathematical success one can hope to aspire to. Belgrade, November 2010 Dušan Djuki´c Vladimir Jankovi´c Ivan Mati´c Nikola Petrovi´c Over the previous years we have created the website: . There you can find the most current information regarding the book, the list of detected errors with corrections, and the results from the previous olympiads. This site also contains problems from other competitions and olympiads, and a collection of training materials for students preparing for competitions. We are aware that this book may still contain errors. If you find any, please notify us at [email protected] If you have any questions, comments, or suggestions regarding both our book and our website, please do not hesitate to write to us at the above email address. We would be more than happy to hear from you. Preface VII Acknowledgements The making of this book would have never been possible without the help of numerous individuals, whom we wish to thank. First and foremost, obtaining manuscripts containing suggestions for IMOs was vital in order for us to provide the most complete listing of problems possible. We obtained manuscripts for many of the years from the former and current IMO team leaders of Yugoslavia / Serbia, who carefully preserved these valuable papers throughout the years. Special thanks are due to Prof. Vladimir Mi´ci´c, for some of the oldest manuscripts, and to Prof. Zoran Kadelburg. We also thank Prof. Djordje Dugošija and Prof. Pavle Mladenovi´c. In collecting shortlisted and longlisted problems we were also assisted by Prof. Ioan Tomescu from Romania, Hà Duy Hưng from Vietnam, and Zhaoli from China. A lot of work was invested in cleaning up our giant manuscript of errors. Special thanks in this respect go to David Kramer, our copy-editor, and to Prof. Titu Andreescu and his group for checking, in great detail, the validity of the solutions in this manuscript, and for their proposed corrections and alternative solutions to several problems. We also thank Prof. Abderrahim Ouardini from France for sending us the list of countries of origin for the shortlisted problems of 1998, Prof. Dorin Andrica for helping us compile the list of books for reference, and Prof. Ljubomir ˇ c for proofreading part of the manuscript and helping us correct several errors. Cuki´ We would also like to express our thanks to all anonymous authors of the IMO problems. Without them, the IMO would obviously not be what it is today. It is a pity that authors’ names are not registered together with their proposed problems. In an attempt to change this, we have tried to trace down the authors of the problems, with partial success. We are thankful to all people who were so kind to help us in our investigation. The names we have found so far are listed in Appendix C. In many cases, the original solutions of the authors were used, and we duly acknowledge this immense contribution to our book, though once again, we regret that we cannot do this individually. In the same vein, we also thank all the students participating in the IMOs, since we have also included some of their original solutions in this book. We thank the following individuals who discussed problems with us and helped us with correcting the mistakes from the previous edition of the book: Xiaomin Chen, Orlando Döhring, Marija Jeli´c, Rudolfs Kreicbergs, Stefan Mehner, Yasser Ahmady Phoulady, Dominic Shau Chin, Juan Ignacio Restrepo, Arkadii Slinko, Harun Šiljak, Josef Tkadlec, Ilan Vardi, Gerhard Woeginger, and Yufei Zhao. The illustrations of geometry problems were done in WinGCLC, a program created by Prof. Predrag Janiˇci´c. This program is specifically designed for creating geometric pictures of unparalleled complexity quickly and efficiently. Even though it is still in its testing phase, its capabilities and utility are already remarkable and worthy of highest compliment. Finally, we would like to thank our families for all their love and support during the making of this book. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The International Mathematical Olympiad . . . . . . . . . . . . . . . . . . . . . . 1.2 The IMO Compendium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 2 Basic Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Groups and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Vectors in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Barycenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.7 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.8 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.9 Formulas in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Divisibility and Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Exponential Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Quadratic Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Farey Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Counting of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 6 7 9 10 12 12 13 14 14 15 16 17 17 18 19 19 20 21 22 22 22 23 X 3 Contents Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 IMO 1959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 IMO 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 IMO 1961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 IMO 1962 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 IMO 1963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 IMO 1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 IMO 1965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 IMO 1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Some Longlisted Problems 1959–1966 . . . . . . . . . . . . . . . . . . 3.9 IMO 1967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 IMO 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 IMO 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 IMO 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 IMO 1971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 IMO 1972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 IMO 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 IMO 1974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 27 29 29 30 30 31 31 32 32 33 33 34 34 35 35 35 41 41 41 49 49 49 53 53 53 61 61 62 68 70 70 71 76 79 79 79 83 86 86 87 89 89 90 Contents 3.16.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 IMO 1975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18 IMO 1976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.18.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 IMO 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 IMO 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21 IMO 1979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.21.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.22 IMO 1981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.22.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.22.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23 IMO 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.23.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
View Full Document

  • Winter '18
  • winter viver
  • The Land, International Mathematical Olympiad, imos

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture