50th IMO - 50 Years of International Mathematical OlympiadHans-Dietrich Gronau
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50th IMO - 50 Years of International Mathematical Olympia is a pleasant, entertaining, and informative volume. This lovely 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" (Preface). I am pretty sure the book will be enjoyed by a wider audience.
The book lists the organizational structure of the olympiad, its regulations and schedule; the latter in a formal table format followed by the colorful newspaper-style Daily News. Chapter 3 explains the source for the official olympiad poster that showed Gauss' 1820 map of the Kingdom of Hanover. This map - together with Gauss' likeness and an image of a sextant - appears also on the 10 Deutsche Mark bill. Along the way, the chapter outlines several of Gauss' contributions, including the Gaussian curvature being an intrinsic property of a surface and Gauss-Bonnet Theorem on the value of the integral over the surface of the Gaussian curvature (which is 2π for closed oriented surfaces homeomorphic to a sphere.)
Chapter 5 lists all 104 participating country teams, with group photographs of the participants holding national flags; a good quality execution - as all other illustrations in the book. Given the upheavals of the 1990s, I did not have a chance to observe some of the flags before.
Chapter 6 gives the olympiad problems in five official languages (English, French, German, Russian, Spanish) and their multiple (2-4) solutions. There is also statistics that reflects the relative difficulty of the problems - the percent of the participants who solved them. The chapter also shows all the medalists and their scores and then additionally each participant's per problem performance.
Chapter 7 is a remarkable collection of articles by six former IMO medalists, all world's leading mathematicians who were invited as guests of honor to give presentations before the participants. The 70 pages chapter forms a fascinating book in its own right. Béla Bollobás writes about pursuit and evasion games; Timothy Gowers about the Ramsey theory; László Lovász outlines the the developments of Graph Theory over the last 4 years; Stanislav Smirnov solves a couple of IMO problems and discusses the self-avoiding walks, with the idea of comparing the research and olympiad problems; Terence Tao writes about the structure and randomness in the prime numbers; and Jean-Christophe Yoccoz about applications of Dynamic Systems to Number Theory.
The following chapter shows cumulative statistics of various countries over the 50 years of olympiads, and country performances at every olympiad from 1959 on. Then come three accounts that place the IMO enterprise in historic perspective. In the first, Mircea Becheanu from Romania delves into the circumstances that brought about the first olympiad; István Reiman from Hungary relates his personal experiences from the early olympiads; Wolfgang Engel describes the effect the olympiads had on student activities, particularly in Germany.
Finally, the book lists several of the most prestigious awards in mathematics, along with the recipients who participated at IMOs. In particular, including the 51st year 2010, 12 of the IMO participants have been awarded the Fields Prize. This may not appear very impressive as the total number of IMO participants is near 13000. However, as could easily be found on the Web, during these years (1960-2010) only 44 mathematicians have been awarded the Fields Prize. In 2006, two out of four prizes have been awarded to IMO alumni (Grigori Perelman who declined the prize and Terence Tao); in 2010, the number was three out of four (Elon Lindenstrauss, Ngô Bao Châu, Stanislav Smirnov).
To sum up, this is a beautiful book and a very interesting read for every one who is interested in mathematics and the IMOs - even if marginally.
50th IMO - 50 Years of International Mathematical Olympiad, Hans-Dietrich Gronau, Hanns-Heinrich Langmann, Dierk Schleicher (Editors). Springer. Softcover, 241 p. 131 illus. in color, $29.95. ISBN 978-3-642-14564-3
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