Mathematical Olympiads: problems and solutions from around the world: 2000-2001 represents a continuation of the compilation of Mathematical Olympiad Problems from the 1998-1999, and 1999-2000 competitions published by the Mathematical Association of America. The authors have collected olympiad problems from the national contests of 22 different countries, together with 5 regional contests from 2000 and the national contest of 16 countries and 6 regional contests from 2001. Problems from 2000 are published with solutions, but the solutions for 2001 problems are notincluded.

Besides the main part containing the problems and solutions, the book contains the usual useful appendices: a glossary of basic mathematical identities and definitions, and an index of problems classified lexicographically by subject area, country of origin, and year.

The following is just a small sample of the problems discussed in the book:

- Let M = {1,2,...,40}. Find the smallest positive integer n for which it is possible to partition M into n subsets such that whenever a, b and c (not necessarily distinct) are in the same subset, a is not equal to b+c.

- In the plane are given 2000 congruent triangles of area 1, which are images of a single triangle under different translations. Each of these triangles contains the centroids of all the others. Show that the area of the union of these triangles is less than 22/9.

The authors have an efficient and clear approach to proofs and explanations. It is unfortunate, however, that the book contains more than 70 geometry problems but no figure is given at all. This complicates, sometimes, the comprehension of the solutions.

All in all this book is very well written, full of interesting problems and I warmly recommend it to anyone interested in mathematical competitions, or just in nice problems.

Mohammed Aassila is a mathematics professor whose research area is analysis. He is interested in mathematics competitions and is the author of two books on the subject: 300 Défis Mathématiques and Olympiades Internationales de Mathématiques.