Scott A. Annin's *A Gentle Introduction to the American Invitational Mathematics Exam* provides assistance for students preparing for the American Invitational Mathematics Examination (AIME) and for those who are mentoring such students. Following a description of the examination, the book has nine chapters that discuss basic information from the topics found in high school mathematics that are of particular value for success on the AIME. Each chapter contains illustrative problems, taken primarily from past American Mathematics Competition (AMC) problems and AIME problems and a set of problems for the reader to solve. The last two chapters of the book contain two levels of hints for the problems as well as full solutions. If I were taking the examination or helping someone prepare for it, I would use this book.

A word about the AIME itself: This competitive 15-question examination is based on pre-calculus material. It is offered to students who have scored well on either the AMC 10 exam (open to all students in grade 10 or below) or the AMC 12 exam (open to all students in grade 12 or below) in a particular year. Roughly 500 of those who score highest on the AIME are in turn invited to compete in the United States of America Mathematical Olympiad, from which six students are selected to represent the USA at the International Mathematics Olympiad.

The table of contents provides an outline of the topics that Annin considers. Generally information is presented without detailed developments, appropriate for his purposes. Definitions and theorems are given in a concise fashion. Readers are encouraged to look in standard high school texts for more information if it would be difficult to summarize everything within a particular topic.

Of course, a challenge for Annin is to decide what information can be covered in a brief manner that will be of value to the student — after all, he needs to provide the problem-worthy techniques and topics from all of high school mathematics in very few pages. Annin's experience leading workshops for AIME contestants guides his decisions, and it seems to me that his selections are quite appropriate.

There are a few peculiarities. Completing the square is not explained. Annin admits that knowledge of this technique is valuable for contestants, but he does not suggest looking in a text (as he does with topics from trigonometry, for example) nor does he provide a brief discussion of it, even though he does provide what seems to me to be more generally understood information, such as distance equals rate times time and the definition of prime numbers. Another peculiarity is that a definition of even and odd functions is given, but I don't see that the notion is put to any use.

There are very few misprints. The only point that might cause some concern to a reader is on page 160, in the third line from the bottom, where \(x\) need not be an integer, but rather it is \(x^2\) that must be integral.

The solutions Annin provides to the example problems in the text are insightful and valuable to contestants. The problems offered for the readers to solve are interesting and of increasing difficulty in each section. The hints are on target, and the complete solutions to the problems are also done well.

The author suggests the book will be valuable for someone who enjoys solving problems. It's true that most of the problems require some insight and cleverness. Many also require a good bit of calculation or case analysis — one good idea might suffice to suggest a solution, but to complete the problem can take a page or two of computation.

I plan to consider the use of this book in a capstone course for mathematics secondary education majors. The problems in the book often require the integration of material from a variety of topics in high school mathematics. For example, a solution to a problem might well require ideas from geometry, combinatorics, and number theory. I expect that almost all preservice teachers will find the problems on an AIME both interesting and challenging.

Thank you, Dr. Annin, for writing this book. It will serve the community well.

Joel Haack is Professor of Mathematics at the University of Northern Iowa.