Derek Holton is well known for his teaching and for his success in creating mathematics competition preparation programs in New Zealand. In 1996, he was presented with the Paul Erdös Award by the World Federation of National Mathematics Competitions for his contributions to mathematical enrichment in his country.
The book under review is an amalgamation of the first 8 of 15 booklets originally produced to prepare students to compete for the six positions on New Zealand’s International Mathematical Olympiad (IMO) team. The material covers discrete mathematics (sophisticated counting, the pigeonhole principle, basic graph theory), number theory (divisibility, Fermat’s Little Theorem, arithmetic progressions), plane geometry (squares, rectangles/parallelograms, triangles, circles), and a chapter on Cartesian geometry, including conics. The final chapter consists of six problems posed for the IMO by various countries and some related exercises. Solutions are provided for all problems and exercises.
Late in the book, there is a chapter on proof, focusing on proofs by contradiction and mathematical induction. At first, the position of this discussion seemed odd, coming after several chapters of problems for which readers are asked to provide proofs. However, this discussion is an overview of the reasons that proof is necessary and it makes the student aware that there are set patterns of proof.
The problems in this book are, in general, kinder and gentler than those in several other competition preparation manuals―for example, the Olympiad-level collection by Andreescu and Gelca or that of A. Gardiner. On the other hand, so is the exposition. Holton seems to be assuming a lower level of student background than many authors of such preparation material. Leavened with historical comments and humor (humour), Holton’s treatment is colloquial and could be used for general classroom enrichment as well as for competition preparation. I recommend Holton’s book as an introduction to problem solving and the construction of proofs and I look forward to the next volume in this Mathematical Olympiad Series.
Henry Ricardo (email@example.com) has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.