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 table format, followed by the colorful newspaper-style Daily News. Chapter 3 explains the source for the official olympiad poster: it showed Gauss’s 1820 map of the Kingdom of Hanover. This map — together with Gauss’s likeness and an image of a sextant — appears also on the 10 Deutsche Mark bill. Along the way, the chapter outlines several of Gauss’s contributions, including the Gaussian curvature being an intrinsic property of a surface and the 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. The reproduction is of excellent quality — as are all other illustrations in the book. Given the upheavals of the 1990s, I did not have a chance to see some of these flags before.
Chapter 6 gives the olympiad problems in the five official languages (English, French, German, Russian, Spanish) and their multiple (2–4) solutions. There are also statistics that reflect the relative difficulty of the problems, namely the percentage of the participants who solved each of 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 leading mathematicians who were invited as guests of honor to give presentations to the participants. The 70-page chapter forms a fascinating book in its own right. Béla Bollobás writes about pursuit and evasion games; Timothy Gowers about 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 self-avoiding walks, with the idea of comparing research problems and olympiad problems; Terence Tao writes about the structure and randomness in the prime numbers; and Jean-Christophe Yoccoz about applications of Dynamical Systems to Number Theory.
The following chapter gives cumulative statistics on various countries over the 50 years of mathematical olympiads and country performances at every olympiad from 1959 on. Then come three accounts that place the IMO enterprise in historical 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 have 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 Medal. This may not appear very impressive as the total number of IMO participants is near 13,000. However, as could easily be found on the Web, during these years (1960–2010) only 44 mathematicians have been awarded the Fields Medal. In 2006, two out of four prizes were 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.
Alex Bogomolny is a freelance mathematician and educational web developer. He regularly works on his website Interactive Mathematics Miscellany and Puzzles and blogs at Cut The Knot Math.