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.