Contest Problem Book II: Annual High School Contests of the Mathematical Association of America, 1961-1965

Charles T. Salkind, "Contest Problem Book II: Annual High School Contests of the Mathematical Association of America, 1961-1965" 
From the preface ...
Mathematical Association of America | 1966 | ISBN: 0883856174 | 112 pages | PDF | 2,1 MB 

The thesis that selective problem solving can be a vital factor in learning mathematics needs no extended defense. It is implicit in the suggestion by some curriculum experts that problems be made the central point of topical development. A good problem, like the acorn, has in it the potential for grand development.
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Charles T. Salkind 

This text compiles the (American) Annual High School Mathematics Examinations (AHSME), now called the American Mathematics Competition, that were administered from 1961-1965. It includes the questions, their answers, their solutions, and a listing of the problems classified by subject. This book is briefer than the others in this series. That may be because 1966 was the year in which a penalty was introduced for wrong answers in order to discourage random guessing.

Each examination consists of forty multiple choice problems based on the American high school curriculum at the time. The problems are meant to be done without a calculator. The principal topics are algebra and geometry, although there are questions on set theory and logic. The examination consists of three parts, which are successively more difficult. The first part, consisting of twenty questions, contains short problems that are meant to test for conceptual understanding. The next two parts each contain ten problems. These problems are designed to make you think more deeply about the subject matter. The problems range in difficulty from routine problems to ones that require considerable ingenuity to solve.

The format of the book enables you to work through the problems, check your answers, and correct any mistakes you make before examining the solutions that Salkind provides. Reading his solutions is instructive for a couple reasons. Salkind's solutions tend to be elegant, and he sometimes presents alternate methods of solving a problem. That said, if you are not familiar with the results that Salkind assumes, you may find his solutions cryptic.

Working through this text will help you develop your problem solving skills. If you are preparing for the American Mathematics Competition, you may want to use this as a source of practice problems. However, keep in mind that both the format of the examination and the curriculum have changed since these examinations were administered. There are problems dealing with non-decimal bases and logic which will be unfamiliar to current high school students.

Update (23 March 2009): To prepare for the current format of the AMC, you should work through The Contest Problem: Book VIII (MAA Problem Book Series) in order to prepare for the AMC 10 or The Contest Problem Book IX (Maa Problem Books) (Bk. 9) to prepare for the AMC 12. Top scorers on those examinations qualify for the American Invitational Mathematics Examination (AIME). Of the volumes in this series, only The Contest Problem Book V: American High School Mathematics Examinations (AHSME) / American Invitational Mathematics Examinations (AIME) 1983-1988 (Anneli Lax New Mathematical Library) contains problems from the AIME.