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50Th IMO – 50 Years of International Mathematical Olympiads − Hans-Dietrich Gronau • Hanns-Heinrich Langmann Dierk Schleicher Editors

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50Th IMO – 50 Years of International Mathematical Olympiads − Hans-Dietrich Gronau • Hanns-Heinrich Langmann Dierk Schleicher Editors 50th IMO – 50 Years of International Mathematical Olympiads − Hans-Dietrich Gronau • Hanns-Heinrich Langmann Dierk Schleicher Editors 50thIMO–50Years of International Mathematical Olympiads 123 Editors Hans-Dietrich Gronau Hanns-Heinrich Langmann Universität Rostock Bildung & Begabung gemeinnützige GmbH Institut für Mathematik Bundeswettbewerb Mathematik Ulmenstr. 69 Kortrijker Str. 1 18051 Rostock 53177 Bonn Germany Germany [email protected] [email protected] Dierk Schleicher Jacobs University Research I Campus Ring 1 28759 Bremen Germany [email protected] ISBN 978-3-642-14564-3 e-ISBN 978-3-642-14565-0 DOI 10.1007/978-3-642-14565-0 Springer Heidelberg Dordrecht London New York Mathematics Subject Classification (2010): 00A09 c Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer.Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface From 10 to 22 July 2009 Germany hosted the 50th International Mathematical Olympiad (IMO). This jubilee anniversary event took place on the campus of Jacobs University Bremen. For the first time the number of participating countries exceeded 100, with 104 countries from all continents sending a delegation consisting of six students, a leader and a deputy leader. Celebrating the 50th anniversary of the IMO provides an ideal opportunity to look back over the past five decades and to review its development from modest beginnings to become a worldwide event. This book is aimed at students in the IMO age group and seeks to demonstrate that mathematics is an active and lively subject, thousands of years old and yet young and fresh as ever. We have invited several IMO pioneers to tell us about the development of IMO since it first began: they reveal that initially nobody thought it would develop into an annual event, let alone grow as big as it is today. We include three accounts from IMO pioneers: Mircea Becheanu from Roma- nia, speaking on behalf of the country that created the IMO and hosted the initial Olympiad event; István Reiman from Hungary who personally experienced the early years; and Wolfgang Engel from (then East) Germany who witnessed the impact the Olympiads have had from the outset on mathematical activities for students, par- ticularly in Germany. These reports give a vivid and personal description of what gradually became an international success story. Also included is data about the 50 annual Olympiads which illustrates the devel- opment of the competition from just seven countries in 1959 to 104 countries, with 565 participants, in 2009. We list the most successful contestants, the results of the 50 Olympiads and the names of all 112 countries that have ever taken part. In mathematics, talent is often developed and demonstrated at an early age. Thus it is impressive to see that many of the world’s leading research mathematicians were among the most successful IMO participants in their youth. To illustrate this we include a list showing winners of prestigious research awards who participated in the IMO when they were young. In addition to the mathematics competition an important aspect of the Olympiads is that students from all over the world meet and exchange ideas. It is both a personal v vi Preface highlight of their lives that they treasure for a long time and a place where interna- tional friendships develop. Some of the personal or amusing anecdotes of previous IMOs are included in the last chapter of the book. The outstanding event of the 2009 Olympiad was the celebration of the IMO’s 50th anniversary. Six of the world’s leading mathematicians, all of them highly suc- cessful former IMO participants, were invited as guests of honour to give presenta- tions: Béla Bollobás, Timothy Gowers, László Lovász, Stanislav Smirnov, Terence Tao and Jean-Christophe Yoccoz. Their contributions are also included in this book. We would like to express our sincere gratitude to all those who supported the organisation of the 50th IMO, making it a memorable and unique event for all par- ticipants. Finally, we wish to thank the Springer Verlag for publishing this book. We are indebted to Clemens Heine for his support and collaboration. Rostock, Bonn, Bremen Hans-Dietrich Gronau May 2010 Hanns-Heinrich Langmann Dierk Schleicher Welcome Message Dear IMO participants, This year’s IMO has a special significance: we celebrate the 50th anniversary of the IMO. In the beginnings it started as a one-time event for students from the countries which at the time were said to form the “socialist block” or “Soviet block”, to wit: Bulgaria, Czechoslovakia, German Democratic Republic, Hungary, Poland, Romania and the Soviet Union. It is remarkable that of those 7 countries, 3 do not exist any more. But the IMO, which, encouraged by the success of the first competition became a yearly event, flourishes. Some key dates: 1967 — the first West-European coun- tries (United Kingdom, France, Italy, Sweden) take part in the 9th IMO in Cetinje, Yugoslavia. (Finland’s one-time participation in 1965 had no immediate continua- tion.) The Netherlands followed suit in 1969, Austria in 1970. Two non-European socialist countries participated in the 60’s and early 70’s: Mongolia since 1964 and Cuba since 1971. The 16th IMO in 1974 in Erfurt (East Germany) saw a true break- through: the teams of the USA and of Vietnam participated for the first time. Lienz (Austria) was the venue of the first “Western” IMO in 1976, Washington, DC of the first non-European IMO in 1981. Australia organized the 1988 IMO in Canberra, the first Asian IMO took place in 1990 in Beijing, China. Argentina was the venue of the 1997 IMO; we are looking forward to an IMO on African soil in the not-too-far-away future. My sincere hope is that more and more countries will join the IMO community; old participating countries will see more and more benefit from the IMO for their own mathematical education system and mathematical talent search; and the IMO will continue to be a success story in the next 50 years. Bremen, July 2009 József Pelikán Chairman of the IMO Advisory Board vii viii Welcome Message József Pelikán Hungary József Pelikán took part in 4 IMOs in his student days: In 1963 he won a silver medal, and in 1964- 65-66 three gold medals. He was also awarded twice with a special prize for an outstanding solution of a problem. Since 1971 he has been teaching at E˝otv˝osLoránd University, Budapest, Hungary. He participated in the organization of the 1970 IMO held in Keszthely, Hun- gary, and was one of the chief organizers of the 1982 IMO in Budapest, Hungary. He has been the leader of the Hungarian IMO team since 1988. He was an elected member of the IMO Advisory Board between 1992 and 2002, and has been the Chairman of the IMO Advisory Board since 2002. Contents Part I 50th International Mathematical Olympiad – Germany 2009 1 Committees ................................................ 5 1.1 IMO Advisory Board ...................................... 5 1.2 Steering Committee ....................................... 5 1.3 Organizers ............................................... 6 1.4 Problem Selection Committee ............................... 6 1.5 Jury ..................................................... 7 1.6 Coordinators ............................................. 8 1.7 Invigilators ............................................... 9 1.8 Guides .................................................. 10 1.9 50th IMO Anniversary Committee ........................... 12 1.10 Volunteers ............................................... 12 2 Regulations ................................................ 13 3 Carl Friedrich Gauss, Curvature, and the Cover Art of the 50th IMO 21 4 Programme ................................................ 27 4.1 IMO Programme Overview ................................. 27 4.2 Programme Details ........................................ 28 4.3 Daily News .............................................. 32 5 Participants ................................................ 47 5.1 National Teams ........................................... 48 5.2 Observer Countries ........................................100 ix x Contents 6 Mathematical Competition ................................... 101 6.1 Problems ................................................103 6.2 Solutions ................................................113 6.2.1 Problem 1 – Number Theory – Australia ..............113 6.2.2 Problem 2 – Geometry – Russia .....................116 6.2.3 Problem 3 – Algebra – United States .................118 6.2.4 Problem 4 – Geometry – Belgium ....................120 6.2.5 Problem 5 – Algebra – France .......................122
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