The annual American Mathematics Competitions are a program of the Mathematical Association of America for pre-college students. Great care is taken to create new and interesting problems for these competitions. After each examination, all the problems and their solutions are made public in individual pamphlets so students can learn mathematics by practicing for future contests, and to give teachers new and exciting problems for their classes.

The oldest of the American Mathematics Competitions is the American High School Mathematics Examination [AHSME], first given in 1949. This book presents the problems and solutions from the 40th through 45th annual AHSMEs, the examinations administered 1989 through 1994. All the statements of the problems appear in this book exactly as they appeared on the examinations.

The solutions in this book include all the official solutions originally made available after the competitions, many enhanced. In addition, this book includes many alternative solutions that have never been published before.

How To Use This Book

Students who will participate in the American Mathematics Competitions and many other mathematics contests can use the problems in this book for practice. The rules for the AHSME appear on the page xix, preceding the statements of the AHSME problems so students can try the examinations under simulated contest conditions. Answers follow the questions from each competition, together with the percentage of times honor roll students used each answer choice. A few comments calling attention to unusually attractive ‘distractors’ are also included. One or more complete solutions to each problem are given in the section following all the examination questions and answers.

The study of mathematics is the same, whether it be for a mathematics class, for one's pure enjoyment in gaining new insights, or for competitions. The use of certain mathematical tools may be encountered more frequently in mathematics competition problems than in text book problems. However, responsible organizers of mathematics competitions try to align the topics they emphasize reasonably well with current curricula. For example, statistics was becoming a more important topic in the secondary schools during the years of the competitions in this book. Note the development from problems 24 on the 41st AHSME and 16 on the 42nd AHSME which are about weighted averages, problems that are actually more algebraic than statistical, to problem 24 on the 45th AHSME which presumes knowledge of fundamental statistical terms.

Tools for Mathematics Competitions

The tools for doing well on the AHSME and AIME are all the topics in a pre-calculus curriculum including some elementary probability, statistics, discrete mathematics, and number theory. The creative aspect of problem solving on competitions lies in the skill of knowing which of these tools to select for which problems. Facility with the use of these tools and their selection can be gained by entering mathematics competitions and by practicing on contests such as those in this book.

Good competitors on timed tests practice to the point where they recognize the use of certain sequences of these tools. They can write down the results without any intermediate writing, almost as if they have the result memorized. One very simple example might be the formula for the area of an equilateral triangle of side s.