Joint review of

When I was a school boy in Moscow in the 1960s, there were two mathematical olympiads that addressed different groups of school children. The Moscow City Mathematical Olympiad was intended to attract a broad stratum of middle and high school students. The Moscow State University Mathematical Olympiad was intended for brighter students with interest in mathematics. Problems at the City Olympiad required fluency in the school curriculum and some of them a little ingenuity; all the problems at the University Olympiad went beyond the curriculum and could only be solved with talent and practiced ingenuity. The winners of the latter received certificates and mathematical books — the higher the achievement, the more books would be awarded. A special committee selected the team for the International Olympiad from among the University Olympiad winners. The winners of the City Olympiad received recognition in the form of official certificates and respect from their school mates and administration. The participants in the University Olympiad would usually take part in one or two math circles; there were many, including one run by the graduate students at the Mathematics Department of the Moscow State University. The City Olympiad had two streams, one for the middle, the other for the high school. The University Olympiad, while open to all, was intended for high school students.

In the US, the first Annual High School Mathematics Examination (AHSME) was offered in 1950. First confined to the Metropolitan New York area, it became a national contest in 1957. From the beginning, the contest had a practically impossible goal: to accommodate as many students of various abilities as possible. This goal dictated the format of the contest. In 1950 the contest consisted of 50 problems split into three sections. The problems in the first one were almost like ordinary homework; in the second the problems required a thorough knowledge of high school mathematics, in the third some ingenuity was helpful.

The parameters of the contest — time allocation, number of problems, number of sections, point weights, have been changed every several years until it split into several (now designated AMC for “American Mathematics Competitions”), AMC8, AMC10, and AMC12. Student who performed well at AMC10 and AMC12 are invited to participate (since 1983) in the AIME — the American Invitational Mathematics Examination. The next step (since 1972) is USAMO — the USA Mathematical Olympiad. For the brightest, a summer camp and an extra test opens the road to the International Mathematical Olympiad, in which the USA team began to participate in 1974.

The *Contest Problem Books* under review cover the AHSME from 1950 through 1982. With rare exceptions, AHSME problems were rather easier than those that would be included in a Moscow City Olympiad. However, due to a more stringent time framework and a greater number of problems, a participant needed to demonstrate real mastery of the school curriculum to entertain any hope of winning. In the first 10 years the examination was conducted only three students achieved a perfect score.

Over the years the number of problems was reduced. In the 1982 contest there were only 30 problems (but there were years with 50, 40, and 35 problems.) On average, the difficulty of the problems has increased. Interestingly, so has the scope of the problems (apparently following changes in the curriculum). A curious indicator is the list of symbols in the books, which it kept expanding over time. All of the books except for II contain a List of Symbols. There are 15 in the first one and more than 40 in the fourth. The fourth book contains symbols from set theory, sums, determinants, finite differences.

The books share the same format: besides the List of Symbols there are problems organized by year with complete solutions. There is also a thematic classification of problems, which is a huge convenience for a student who would like to test his/her skills in a specific area and for a teacher to quickly find problems related to the current topic.

It goes without saying that the books provide a convenient tool for preparation for future AMC competitions. To that end, they could be used by ambitious or homeschooled students or by teachers and coaches. They may also be recommended to middle school math circles. Many math circles go way beyond the curriculum, probably under the assumption that there is nothing exciting in school mathematics. The contest books contain fine counterexamples! More than that, if the task is to find several solutions to a problem, then even the more routine problems may form a starting point for an engaging activity.

Alex Bogomolny is a freelance mathematician and educational web developer. He regularly works on his website Interactive Mathematics Miscellany and Puzzles and blogs at Cut The Knot Math. Follow Alex on twitter.