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The Art of Problem Solving, Vol. 2: And Beyond

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Sam Vandervelde, in conjunction with the authors of this book, started the Mandelbrot competition in 1990. It is a four round high school mathematics competition, where the goal is to teach the participants material outside the usual high school mathematical curriculum. Topics such as number theory and proof techniques are covered and each round consists of an individual and a team test. There are two levels to the competition, beginning and advanced. For more information on the contest visit the website of the same name with a dot-org extension.
The interest the authors have in mathematics competitions is the basis for this book; they also mention several other competitions, such as Mathcounts and the American Mathematics Competitions (AMC). Therefore, the exercises at the ends of the chapters generally have a subscript such as (Mandelbrot #2) and (AHSME), noting the point of origin.
The book is designed as a problem-solving tutorial that can be used to prepare for mathematics competitions. Written at the high school level, the style is to present a small amount of explanatory text, followed by worked examples and embedded exercises. Solutions to the embedded exercises and end of chapter problems are all in a companion solutions manual.
This is an excellent book for training contestants for mathematics competitions at the high school level. The text is clear and the authors take care to make sure only the proper amount of new material is explained before they present an example and one or more exercises to work. If a student stops and understands each exercise before proceeding, when they reach the end of the book, they will be very well prepared to take on their mathematical competition.

The alternate approaches and detailed solutions bring excitement to math, up through grad level concepts. The discussion is appropriate for middle and high school students, and beyond.

If you want to improve your problem solving skills, buy this book.