60-odd years of moscow mathematical
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Moscow olympiad problems
60-odd YEARS of MOSCOW MATHEMATICAL OLYMPIADS Edited by D. Leites Compilation and solutions by G. Galperin and A. Tolpygo with assistance of P. Grozman, A. Shapovalov and V. Prasolov and with drawings by A. Fomenko Translated from the Russian by D. Leites Computer-drawn figures by Version of May 10, 1997. Stockholm i ii PROBLEMS Abstract Nowadays, in the time when the level of teaching universally decreases and “pure” science does not appeal as it used to, this book can attract new students to mathematics. The book can be useful to all teachers and instructors heading optional courses and mathematical groups. It might interest university students or even scientists. But it was primarily intended for high school students who like mathematics (even for those who, perhaps, are unaware of it yet) and to their teachers. The complete answers to all problems will facilitate the latter to coach the former. The book also contains some history of Moscow Mathematical Olympiads and reflections on mathemat- ical olympiads and mathematical education in the Soviet Union (the experience that might be of help to western teachers and students). A relation of some of the problems to “serious” mathematics is mentioned. The book contains more than all the problems with complete solutions of Moscow Mathematical Olympiads starting from their beginning: some problems are solved under more general assumptions than planned dur- ing the Olympiad; there extensions are sometimes indicated. Besides, there are added about a hundred selected problems of mathematical circles (also with solutions) used for coaching before Olympiads. The Moscow Mathematical Olympiad was less known outside Russia than the “All-Union” (i.e., National, the USSR), or the International Olympiad but the problems it offers are on the whole rather more difficult and, therefore, it was more prestigious to win at. In Russia, where sports and mathematics are taken seriously, more than 1,000,000 copies of an abridged version of a part of this book has been sold in one year. This is the first book which contains complete solutions to all these problems (unless a hint is ample, in which case it is dutifully given). The abriged Russian version of the book was complied by Gregory Galperin, one of the authors of a great part of the problems offered at Moscow Mathematical Olympiads (an expert in setting olympiad-type problems) and Alexei Tolpygo, a former winner of the Moscow, National and International Olympiads (an expert in solving mathematical problems). For this complete English edition Pavel Grozman and Alexander Shapovalov (a first and a third prize winners at the 1973 and 1972 International Mathematical Olympiads, respectively) wrote about 200 new solutions each. The book is illustrated by Anatoly Fomenko, Corresponding Member of the Russian Academy of Sciences, Professor of Mathematics of Moscow University. Fomenko is very well known for his drawings and paintings illustrating the wonders of math. Figures are sketched under supervision of Victor Prasolov, Reader at the Independent University of Moscow. He is well-known as the author of several amazingly popular books on planimetry and solid geometry for high-school students. From I.M. Yaglom’s “Problems, Problems, Problems. History and Contemporaneity” (a review of MOSCOW MATHEMATICAL OLYMPIADS compiled by G. Galperin and A. Tolpygo) The oldest of the USSR Math Olympiads is the Leningrad High-school Olympiad launched in 1934 (the Moscow Math Olympiad runs since 1935). Still, for all these years the “most main” olympiad in the country was traditionally and actually the Moscow Math Olympiad. Visits of students from other towns started the expansion of the range of the Moscow Math Olympiad to the whole country, and, later, to the whole Earth: as International Olympiads. More than half-a-century-long history of MMO is a good deal of the history of the Soviet high school, history of mathematical education and interactive work with students interested in mathematics. It is amazing to trace how the level of difficulty of the problems and even their nature changed with time: problems of the first Olympiads are of the “standard-schoolish” nature (cf. Problems 1.2.B.2, 2.2.1, 3.1.1 and 4.2.3) whereas even the plot of the problems of later olympiads is often a thriller with cops and robbers, wandering knights and dragons, apes and lions, alchemists and giants, lots of kids engaged in strange activities, with just few quadratics or standard problems with triangles. Problems from the book compiled by Galperin and Tolpygo constitute a rare collection of the long work of a huge number of mathematicians of several generations; the creative potential of the (mainly anonymous) authors manifests itself in a live connection of many of the olympiads’ problems with current ideas of modern PROBLEMS, PROBLEMS, PROBLEMS. HISTORY AND CONTEMPORANEITY iii Mathematics. The abundance of problems associated with games people play, various schemes described by a finite set, or an array of numbers, or a plot, with only qualitative features being of importance, mirrors certain general trends of the modern mathematics. Several problems in this book have paradoxical answers which contradict the “natural” expectations, cf. Problems 13.1.9-10.2, 24.1.8.2, 32.7.3, 38.1.10.5, 44.7.3, and Problems 32.9.4 and 38.2.9.19 (make notice also of auxiliary queries in Hints!). iv PROBLEMS Contents Abstract ii Problems, Problems, Problems. History and Contemporaneity ii Preface 1 Forewords 4 Academician A. N. Kolmogorov’s foreword to [GT] 7 Download 1.08 Mb. Do'stlaringiz bilan baham: |
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