It is hard to say anything about this book without saying first some things about the author…so I will start there. Terence Tao is only thirty–one years old, but has done more work than many mathematicians three times his age. He was born in Australia in 1975, has competed in the International Mathematical Olympiad in 1987, 1988 and 1989 and won a bronze, silver, and gold medal respectively. He was the younger competitor ever to win a gold medal at this very tough competition. In 1996, at the age of 21, Terence Tao got his PhD in mathematics from Princeton University. Four years later, in 2000, he was the youngest full professor in the Mathematics Department at the University of California, Los Angeles. He has published over a hundred of mathematics papers already.

The year 2006 has been especially good for Terence Tao: he was awarded a Fields Medal at the International Congress of Mathematicians, as well as a “genius grant” of $500,000 from the MacArthur Foundation. When awarded the Fields Medal, Tao was called “a supreme problem solver whose spectacular work has had an impact across several mathematical areas”. The MacArthur Foundation described Tao as “a mathematician bringing technical brilliance and profound insight to a host of seemingly intractable problems in such areas as partial differential equations, harmonic analysis, combinatorics, and number theory”.

And now, finally, about the book! This is a second edition, the first one having been published fifteen years ago, when the author was half his present age (i.e. fifteen)! As the author mentions in the preface to the second edition, “in many places the exposition has a certain innocence, or even naivety. I have been reluctant to temper too much with this, as my younger self was almost certainly more attuned to the world of the high-school problem solver than I am now.” This is probably very good: it gives the book a very special charm.

The book is a wonderful read for anybody interested in challenging mathematics problems at the high-school level, but is invaluable for the high-school students interested in participating in the mathematical Olympiads and these students’ coaches. The material is organized as follows:

- Strategies in problem solving;
- Examples in number theory;
- Examples in algebra and analysis;
- Euclidean geometry;
- Analytic geometry;
- Sundry examples.

There are a handful of really wonderful books that can introduce a young high-school student to the beauty of mathematics. This is definitely one of them. Besides, this book is probably going to be known as the first book written by one of the best mathematicians of the twenty-first century.

Mihaela Poplicher is an associate professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.