I’ve always enjoyed reading mathematics problem-solving books. So it comes as no surprise that I enjoyed this book as well. Not just because of the collection of problems, but also because of their sheer scope and depth. This is a great collection which is extremely well-organized!
The book is a compilation of advanced problems (from the Putnam exams and various Mathematical Olympiads, etc), arranged by topic. After a brief chapter on the different methods of mathematical proof, the topics covered are Algebra, Real Analysis, Geometry and Trigonometry, Number Theory, and Combinatorics and Probability. These chapters cover just over 300 pages, with well over 1000 problems. The remaining 450 or so pages contain complete solutions to all of the problems.
This extraordinary book can be read for fun. However, it can also serve as a textbook for preparation for the Putnam (or other advanced mathematical competitions), for an advanced problem-solving course, or even as an overview of undergraduate mathematics. Due to the level of the mathematics, it would be difficult to use in a course in which the students had not already been exposed to the topic. I would not advise using the Real Analysis chapter to introduce an undergraduate to that topic. But it could certainly serve as a great review for senior-level students. If you enjoy these sorts of collections of mathematics problems, then this book really is a must-have.
I leave you with an example from the chapter on geometry: Prove that the plane cannot be covered by the interior of finitely many parabolas. Want to see the proof? Get the book!
Donald L. Vestal is Assistant Professor of Mathematics at South Dakota State University. His interests include number theory, combinatorics, spending time with his family, and working on his hot sauce collection. He can be reached at Donald.Vestal(AT)sdstate.edu.
Preface.- Methods.- Algebra.- Real Analysis.- Geometry and Trigonometry.- Number Theory.- Combinatorics and Probabilities.- Solutions.- Definitions and Notations.