This interesting collection of problems and solutions had its origins in Romanian Sunday Math Circles that were intended to train students for mathematical competitions such as the International Mathematical Olympiads. Eventually the meetings expanded their purpose to that of “disseminating beautiful mathematics at an elementary level.”
The topics discussed in the book are traditional for the original purpose of such circles: number theory, algebra and combinatorics, geometry, and analysis; and the problems come from many of the usual sources: country and international Olympiads, American competitions such as the Putnam, and journals such as Kvant and the American Mathematical Monthly. In addition to having the problems in each section arranged in increasing order of difficulty, the authors have tried to ensure that the problems fall into three categories — basic exercises in the tools and methods being illustrated, problems of medium difficulty suitable for mathematical competition training, and difficult problems that are bridges to research — in the proportions 25%, 50%, and 25%, respectively. Problems in this third category are accompanied by comments and key references for further reading.
Enhancing the book’s usefulness, the authors have appended a glossary of important terms and formulas related to triangles, a treatment of Pell’s equation, an index of mathematical results, an index of mathematical terms, and an index of topics and methods (Classical arithmetic functions and applications, … , Graphs and applications, … , Linear functionals and operators). Some of these problems will be new to even the most avid of problem collectors; but even familiar problems are given solutions which the enthusiast may not have seen — perhaps a solution different from the “official” Putnam problem solution or a variant of an Olympiad problem solution.
Although Boju and Funor work in Canada and France, respectively, they have honored their Romanian roots by producing this book. It seems as if Romanian problem books will eventually become dense in the set of all problem collections, and devotees of mathematical challenges should rejoice.
Henry Ricardo (firstname.lastname@example.org) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book A Modern Introduction to Differential Equations was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.