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Mathematical Olympiad Challenges

Titu Andreescu and Razvan Gelca
Publication Date: 
Number of Pages: 
[Reviewed by
Herbert E. Kasube
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This fascinating volume contains 344 "challenging" problems partitioned into three major categories: 114 in geometry and trigonometry, 112 in algebra and analysis and 118 in number theory and combinatorics. Actually, the problems themselves comprise only the first 85 pages of this volume. What follow are the solutions.

In the preface, the authors make the distinction between an "exercise" and a "problem"; the latter requiring more time and thought to solve. The questions contained in this volume are definitely problems by this definition. The authors have gathered many problems that have appeared as part of the Mathematical Olympiad and offer a glimpse of the challenge faced by the competitors.

As a regional director for the American Mathematics Competitions (AMC-8,-10 and -12) this reviewer is often asked why students should participate in these competitions. My typical response is that such competition stimulates mathematics students to think "outside the box". The problems one sees on these examinations are not your typical textbook exercises and the problems in this volume represent a good sample.

Where might this text be used? One important audience is the high school teacher who wishes to stimulate interest in mathematics by challenging students with some difficult (but accessible) problems. This might be to help them prepare for a mathematics competition or it may just be for fun.

While these problems originate from a competition among high school students, the problems contained within will be a challenge to some of our brightest college mathematics students, and even for college faculty. I could foresee this volume being used in a number of ways at the college level.

First, one could use it as a text and/or resource for a problem-solving seminar taken by mathematics majors and minors.

Secondly, an institution could use these problems to help prepare a team for the Putnam competition.

Finally, many institutions sponsor a "Problem of the Week". Here are some great candidates!

What did I like about this book? The format of dividing the problems into three broad categories can be helpful to faculty who are looking for problems for specific courses. I am also pleased that complete solutions were given, but the solution is not immediately after the problem. In fact, one could just read the solution section and learn a lot about mathematical problem solving. The problems are well chosen with credit given to the original proposer.

What did I not like? I have only one small concern. The problems are (possibly) too good. Perhaps the inclusion of a few "warm-up" problems would have been appropriate. I want to emphasize that this is a minor drawback and it should not influence ones decision to use this nook. After all, such warm-up problems can be exercises that are found in our "usual" textbooks.

The authors are experienced problem solvers and coaches of mathematics teams. This expertise shows through and the result is a volume that would be a welcome addition to any mathematician's bookshelf.

Herbert E. Kasube ( is associate professor of mathematics at Bradley University with a particular interest in the history of mathematics. He also serves as regional director for the AMC-8 and AMC-10/12.

The table of contents is not available.