60-odd YEARS Of MOSCOW MATHEMATICAL OLYMPIADS

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60-odd YEARS ofMOSCOW MATHEMATICALOLYMPIADSEdited by D. LeitesCompilation and solutions by G. Galperin and A. Tolpygowith assistance of P. Grozman, A. Shapovalov and V. Prasolovand with drawings by A. FomenkoTranslated from the Russian by D. LeitesComputer-drawn figures byVersion of May 10, 1997. Stockholmi

iiPROBLEMSAbstractNowadays, in the time when the level of teaching universally decreases and “pure” science does notappeal 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 facilitatethe latter to coach the former.The book also contains some history of Moscow Mathematical Olympiads and reflections on mathematical olympiads and mathematical education in the Soviet Union (the experience that might be of help towestern 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 Olympiadsstarting from their beginning: some problems are solved under more general assumptions than planned during the Olympiad; there extensions are sometimes indicated. Besides, there are added about a hundredselected 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 difficultand, therefore, it was more prestigious to win at. In Russia, where sports and mathematics are takenseriously, 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, inwhich case it is dutifully given).The abriged Russian version of the book was complied by Gregory Galperin, one of the authors of agreat part of the problems offered at Moscow Mathematical Olympiads (an expert in setting olympiad-typeproblems) 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 andAlexander Shapovalov (a first and a third prize winners at the 1973 and 1972 International MathematicalOlympiads, respectively) wrote about 200 new solutions each.The book is illustrated by Anatoly Fomenko, Corresponding Member of the Russian Academy ofSciences, Professor of Mathematics of Moscow University. Fomenko is very well known for his drawings andpaintings illustrating the wonders of math.Figures are sketched under supervision of Victor Prasolov, Reader at the Independent Universityof Moscow. He is well-known as the author of several amazingly popular books on planimetry and solidgeometry for high-school students.From I.M. Yaglom’s “Problems, Problems, Problems. History and Contemporaneity”(a review of MOSCOW MATHEMATICAL OLYMPIADScompiled by G. Galperin and A. Tolpygo)The oldest of the USSR Math Olympiads is the Leningrad High-school Olympiad launched in 1934 (theMoscow Math Olympiad runs since 1935). Still, for all these years the “most main” olympiad in the countrywas traditionally and actually the Moscow Math Olympiad. Visits of students from other towns startedthe expansion of the range of the Moscow Math Olympiad to the whole country, and, later, to the wholeEarth: 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 isamazing to trace how the level of difficulty of the problems and even their nature changed with time: problemsof 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, wanderingknights and dragons, apes and lions, alchemists and giants, lots of kids engaged in strange activities, withjust few quadratics or standard problems with triangles.Problems from the book compiled by Galperin and Tolpygo constitute a rare collection of the long workof 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 CONTEMPORANEITYiiiMathematics. The abundance of problems associated with games people play, various schemes described bya finite set, or an array of numbers, or a plot, with only qualitative features being of importance, mirrorscertain 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 alsoof auxiliary queries in Hints!).

ivPROBLEMS

ContentsAbstractProblems, Problems, Problems. History and ContemporaneityiiiiPrefaceForewordsAcademician A. N. Kolmogorov’s foreword to [GT]147Part 1: Problems9IntroductionPrerequisites and notational conventions1111Selected lectures of mathmathematics circlesDirichlet’s principleNondecimal number systemsIndefinite second-order equations17171820MOSCOW MATHEMATICAL OLYMPIADS 1 – 59Olympiad 1 (1935)Olympiad 2 (1936)Olympiad 3 (1937)Olympiad 4 (1938)Olympiad 5 (1939)Olympiad 6 (1940)Olympiad 7 (1941)Olympiad 8 (1945)Olympiad 9 (1946)Olympiad 10 (1947)Olympiad 11 (1948)Olympiad 12 (1949)Olympiad 13 (1950)Olympiad 14 (1951)Olympiad 15 (1952)Olympiad 16 (1953)Olympiad 17 (1954)Olympiad 18 (1955)Olympiad 19 (1956)Olympiad 20 (1957)Olympiad 21 (1958)Olympiad 22 (1959)Olympiad 23 (1960)Olympiad 24 (1961)Olympiad 25 (1962)Olympiad 26 (1963)Olympiad 27 (1964)Olympiad 28 (1965)Olympiad 29 (1966)Olympiad 30 626567707477808485v

viCONTENTSOlympiad 31 (1968)Olympiad 32 (1969)Olympiad 33 (1970)Olympiad 34 (1971)Olympiad 35 (1972)Olympiad 36 (1973)Olympiad 37 (1974)Olympiad 38 (1975)Olympiad 39 (1976)Olympiad 40 (1977)Olympiad 41 (1978)Olympiad 42 (1979)Olympiad 43 (1980)Olympiad 44 (1981)Olympiad 45 (1982)Olympiad 46 (1983)Olympiad 47 (1984)Olympiad 48 (1985)Olympiad 49 (1986)Olympiad 50 (1987)Olympiad 51 (1988)Olympiad 52 (1989)Olympiad 53 (1990)Olympiad 54 (1991)Olympiad 55 (1992)Olympiad 56 (1993)Olympiad 57 (1994)Olympiad 58 (1995)Olympiad 59 (1996)Olympiad 60 (1997)Selected problems of Moscow mathematical circlesHints to selected problems of Moscow mathematical circlesAnswers to selected problems of Moscow mathematical 157Historical remarks177A little problem191BibliographySuggested books for further readingRecreational mathematics195195196

PrefaceI never liked Olympiads.The reason is I am a bad sportsman: I hate to lose. Sorry to say, I realize that at any test there usuallyis someone who can pass the test better, be it a soccer match, an exam, or a competition for a promotion.Whatever the case, skill or actual knowledge of the subject in question often seem to be amazingly lessimportant than self-assurance.Another reason is that many of the winners in mathematical Olympiads that I know have, unfortunately,not been very successful as mathematicians when they grew up unless they continued to study like hell (whichmeans that those who became good mathematicians were, perhaps, not very successful as human beings;however, those who did not work like hell were even less successful). Well, life is tough, but nevertheless itis sometimes very interesting to live and solve problems.To business.Regrettable as it is, an average student of an ordinary school and often, even the1 teacher, has no ideathat not all theorems have yet been discovered.For better or worse, the shortest way for a kid to discover mathematics as science, not just a cook bookfor solving problems, is usually through an Olympiad: it is a spectacular event full of suspense, and a goodplace to advertise something really worth supporting like a math group or a specialized mathematical school.(Olympiads, like any sport, need sponsors. Science needs them much more but draws less.)On the other hand, there are people who, though slow-witted at Olympiads, are good at solving problemsthat may take years to solve, and at inventing new theorems or even new theories.One should also be aware of the fact that today’s mathematical teaching all over the world is on theaverage at a very low level; the textbooks that students have to read and the problems they have to solveare very boring and remote from reality,whatever that might mean. As a Nobel prize winner and remarkablephysicist Richard Feynman put it2 , most (school) textbooks are universally lousy.This is why I tried to do my best to translate, edit and advertise this book — an exception from thepattern (for a list of a few very good books on elementary mathematics see Bibliography and paragraph H.5of Historical remarks; regrettably, some of the most interesting books are in Russian).If you like the illustrations in this book you might be interested in the whole collection of Acad.3A. Fomenko’s drawings (A. Fomenko, Mathematical impressions, AMS, Providence, 1991) and the mathematics (together with works of Dali, Breughel and Esher) that inspired Fomenko to draw them.***This is the first complete compilation of the problems from Moscow Mathematical Olympiads withsolutions of ALL problems. It is based on previous Russian selections: [SCY], [Le] and [GT]. The firsttwo of these books contain selected problems of Olympiads 1–15 and 1–27, respectively, with painstakinglyelaborated solutions. The book [GT] strives to collect formulations of all (cf. Historical remarks) problemsof Olympiads 1–49 and solutions or hints to most of them.For whom is this book? The success of its Russian counterpart [Le], [GT] with their 1,000,000 copiessold should not decieve us: a good deal of the success is due to the fact that the prices of books, especiallytext-books, were increadibly low ( 0.005 of the lowest salary.) Our audience will probably be more limited.1 We usually use a neutral “(s)he” to designate indiscriminately any homo, sapiens or otherwise, a Siamease twin of eithersex, a bearer of any collection of X and Y chromosomes, etc. In one of the problems we used a “(s)he” speaking of a wisecockroach. Hereafter editor’s footnotes.2 Feynman R. Surely you’re joking, Mr. Feynman. Unwin Paperbacks, 1989.3 There were several scientific degrees one could get in the USSR: that of Candidate of Science is roughly equivalent to aPh.D., that of Doctor of Science is about 10 times as scarce. Scarcer still were members of the USSR Academy of Sciences.Among mathematicians there were about 100 Corresponding Members — in what follows abbreviated to CMA — and about20 Academicians; before the inflation of the 90’s they were like gods. (This is why the soviet authors carefully indicate thescientists’ ranks.)1

2PREFACEHowever, we address it to ALL English-reading teachers of mathematics who could suggest the book to theirstudents and libraries: we gave understandable solutions to ALL problems.Do not ignore fine print, please. Though not as vital, perhaps, as contract clauses, Remarks and Extensions, i.e.,generalizations of the problems, might be of no less interest than the main text.Difficult problems are marked with an asterisk .Whatever the advertisements inviting people to participate in a Moscow Mathematical Olympiad say,some extra knowledge is essential and taken for granted. The compilers of [Le] and [SCY], not so pressed tosave space, earmarked about half the volume to preparatory problems. We also provide sufficient Prerequisites. Most of the problems, nevertheless, do not require any special background.The organizers of Olympiads had no time to polish formulations of problems. Sometimes the solutionsthey had in mind were wrong or trivial and the realization of the fact dawned at the last minute. It wasthe task of the “managers” (responsible for a certain grade) and the Vice Chairperson of the Organizingcommittee to be on the spot and clarify (sometimes considerably). Being unable to rescue the reader on thespot, I have had to alter some formulations, thus violating the Historical Truth in favour of clarity.While editing, I tried to preserve the air of Moscow mathematical schools and circles of the period and,therefore, decided to season with historical reminiscences and clarifying footnotes. We also borrowed Acad.Kolmogorov’s foreword to [GT] with its specific pompous style. One might think that political allusionsare out of place here. However, the stagnation and oppression in politics and social life in the USSR was areason that pushed many bright (at least in math) minds to mathematics.The story A little problem1 and Historical remarks describe those times. Nowadays the majority of themlive or work in America or Europe. I hope that it is possible to borrow some experience and understandthe driving forces that attracted children to study math (or, more generally, to mathematical schools, fromwhere many future physicists, biologists, etc., or just millionaires, also emerged). It was partly the way theystudied and later taught, that enabled them to collect a good number of professorial positions in leadingUniversities all over the world (or buy with cash a flat on Oxford street, London).What is wrong with the educational system in the USA or Europe, that American or European studentscannot (with few exceptions) successively compete with their piers from the USSR? This question is sointeresting and important that The Notices of American Mathematical Society devoted the whole issue (v.40, n.2, 1993) to this topic, see also the collection of reminiscences in: S. Zdravkovska, P. Duren (eds.), Thegolden years of Moscow mathematics, AMS, Providence, 1993.There were several features that distinguished mathematical circles and mathematical olympiads. Thebetter ones were almost free of official bureaucratic supervision: all circles, olympiads, even regular lecturesat mathematical schools (a lot of hours!) were organized by volunteers who often worked “the second shift”gratis for weeks or years (sic!); their only reward being moral satisfaction. There was freedom of dress code,possibility for children to address the leader of a circle, a Professor, by the first name (unheard of at regularschools), and the possibility for students who ran the circles and olympiads to ridicule the governing Rulesin problems, without endangering the whole enterprise, by sticking the head out too far.One of the problems (32.2.9.4 on “democratic elections”) was even published recently in a politicalmagazin Vek XX i mir (20-th Century and the World, no. 10, 1991) with a discussion of its timelyness andrealistic nature.We should realize, however, that graduates of mathematical schools, though freer in thinking, were oftenhandicapped by overestimation of professional (especially mathematical) skills of a person as opposed tohumane qualities.***This compilation seemingly exhausts the topic: problems of the 70’s are often more difficult than interesting; owing to the general lack of resources Moscow Mathematical Olympiads became less popular. About15 years ago similar lack of enthusiasm gripped famous Moscow mathematical schools. A way to revitalizemathematical education was suggested by one of the principal organizers of Moscow mathematical schools,Nikolaj Nikolaevich (Kolya) Konstantinov. It was similar to the most effective modern scientific way ofgetting rid of stafillacocus in maternity wards in our learned times: burn down the whole damned house.Konstantinov organized several totally new mathematical schools and a so-called Tournament of Towns (asa rival to counterbalance the Olympiads). The tournament became an international event several years ago;for the first collection see [T].***1 Thisstory was published during an abortive thaw in 60’s; its author was unable to publish since.

PREFACE3I thank those who helped me: I. Bernstein, L. Makar-Limanova and Ch. Devchand; V. Pyasetsky, V. Prasolov andI. Shchepochkina. Pavel Grozman and Alexander Shapovalov had actually (re)written about 150 solutions each, Grozmanmade about a 1000 clarifying comments.I also thank N. N. Konstantinov who introduced me to mathematics.Dimitry LeitesStockholm University, Department of Mathematics, Roslagsv. 101, 106 91 Stockholm, Sweden

4PREFACEForewordsMainly for the teacher. The problems collected in this book were originally designed for a competition, that is, to be solved in five hours time during an Olympiad. Many mathematicians in Russia werequite unhappy about this. They argued against this mixture of sport and science: many winners later didnot achieve nearly so much in their studies as in this really very specific kind of “mathematical sport”. Viceversa, many people who could never succeed under stress proved later to be among the most talented andproductive. It is true also that real mathematics deals mostly with problems taking months and years, nothours, to make a step forward.Still, for many schoolchildren, the idea of a competition is very attractive, and they can take part justfor its sake and so discover how diverse and interesting Mathematics (not just math) can be. Afterwards onecan find a lot of more productive mathematical activities than competitions: reading mathematical booksis just one. But there should be the very first step, and Olympiads, as well as Olympiad style problems inschool mathematical clubs and such, help to make it.One can use this book as the source of problems to organize an Olympiad-like competition on one’sown, or for the group or individual studies. In Moscow the same group of the University1 professors andpostgraduate students that launched the Olympiads (see Historical Remarks) also established a tradition of“mathematical circles” — weekly gatherings of schoolchildren at the University, where they can attend alecture, solve some problems, report their progress and get advice. Many of the problems first proposed atthe Olympiad later became the “circles’ folklore” and taught several generations.To use these problems in this way is probably much better, because it is up to a student to choose: eitherto compete with others for the number of problems solved, or just to besiege a single difficult one. Thus,different psychological types can be properly treated without hurting anybody. (A failure at the Olympiadcan be a cause for a grave psychological disturbance in the whole future life.)Some problems are tremendously difficult2 ; only few individuals could solve such problems. As you maylearn from Historical Remarks, there were several problems with not a single correct solution presented tothe Organizing Committee (while the Committee only knew a wrong solution). Therefore, never mind if youtry to crack some of these hard nuts and fail: so did many others. Try it again later or look up Solutions:perhaps you just misunderstood the formulation. Just do not try a new problem on your pupils beforeexamining it yourself properly: it may save a teacher a lot of trouble.You may encounter some difficulties trying to explain solutions to your pupils due to the curriculumdifferences in the U.S. and S.U. You can find feeble consolation in the fact that your colleagues in Russiaexperience the same difficulties: three more or less r

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, an

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