Problems from the Book

This is one of roughly two dozen Olympiad preparation problem books published by XYZ Press and distributed by the AMS. XYZ Press is a publishing company established by Andreescu, the co-author of this book and many others published under that imprint. The title of this book, though apparently not explained in the book itself, is obviously a nod to the famous dictum of Paul Erdős that God maintains The Book, in which are located the best and most elegant proofs of mathematical results. (See, for example, Proofs from THE BOOK, by Aigner and Ziegler.)

This entry in the XYZ series is divided into 23 chapters, each one averaging about 25–30 pages. Chapter topics range from individual results (e.g., the Cauchy-Schwarz inequality and the law of quadratic reciprocity) to general topics (e.g., a chapter entitled “primes and squares” and another on the use of complex numbers to solve combinatorial problems). Number theory, algebra and combinatorics are very heavily represented among the various problems; there are also some problems in analysis, but geometry problems are largely omitted. (I did notice one or two of them sneaking in and winding up getting solved by algebraic means.)

Each chapter follows the same general pattern. First, necessary background results are stated and sometimes proved (a proof is provided, for example, of the law of quadratic reciprocity). Then there are roughly a dozen solved problems, followed by a lengthy selection of “Practice Problems”, solutions to which are not provided. (However, another XYZ book by the authors, Straight from the Book, was published in 2012; although unseen by me, its description states that it contains, among other things, solutions to the first 12 chapters of the first edition of the book now under review. I was surprised to see that Straight from the Book is not listed in the Bibliography to this text.)

Obviously, judging the quality or elegance of a mathematical problem is a highly subjective matter of individual taste. Perhaps because of this, or because I have seen a lot of problem books lately, or because I am not involved in preparing students for competitions and am more interested in material that I can use in lectures in my upper-level courses, or perhaps because the reference to The Book raised my expectations, I found myself a little underwhelmed at this collection of problems.

To me, a “book result” should be a short, snappy, memorable piece of mathematics with a particularly elegant proof. Many of the problems in this book struck me as rather technical results that make good problem-solving challenges but did not strike me as particularly memorable or of intrinsic interest in themselves — proving inequalities or identities, for example, that are so complicated that they cannot be comfortably reproduced here, or proving that some horrible-looking number is an integer, is irrational, or has some other property.  A week from now, I doubt I will be able to remember the results of most of the problems that I read in this text.

The solutions given for the examples in the book are often clever, yet, for the most part, very difficult, and often consist of fairly involved or lengthy calculations. It should definitely be stressed that this is not a book for neophytes. Experienced problem-solvers with a substantial background in undergraduate mathematics will probably get quite a lot out of this book, but I don’t see it as having the same level of appeal as, say, Number Theory: Concepts and Problems, another book co-authored by Andreescu in the XYZ series.

Mark Hunacek (mhunacek@iastate.edu) teaches mathematics at Iowa State University.