Product Details This book contains unconventional problems in algebra, arithmetic, elementary number theory, and trigonometry. Most of the problems first appeared in competitive examinations sponsored by the School Mathematical Society of the Moscow State University and the Mathematical Olympiads held in Moscow. Although most of the problems presuppose only high school mathematics, they are not easy; some are of uncommon difficulty and will challenge the ingenuity of any research mathematician. Nevertheless, many are well within the reach of motivated high school students and even advanced seventh and eighth graders. The problems are grouped into twelve separate sections. Among these are: the divisibility of integers, equations having integer solutions, evaluating sums and products, miscellaneous algebraic problems, the algebra of polynomials, complex numbers, problems of number theory, distinctive inequalities, difference sequences and sums, and more.

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Contest format[ edit ] Both Olympiads had the same format of the contests. Students would come in teams representing their location, e. Each contest could have parts.

For instance, the Republican round of University Olympiads on physics could have three parts: theory, lab and computer modeling. All students were given the same set of problems to solve. They would work on solutions strictly individually - no teamwork was allowed - and then they were scored by judges. Team scores were simply sum of individual member scores.

Earlier rounds could take just one round, while the later rounds could span for a week having several parts. Olympiads in schools general education [ edit ] Overview[ edit ] Contests were conducted on many subject of Soviet school curriculum such as Mathematics, Physics, Chemistry, Biology, and others. These competitions were organized separate for every school grade.

Depending on the subject and geographical region the highest round of the Olympiads varied from the All-Union level in Mathematics, Physics, and Chemistry to the Regional level in some other disciplines.

In addition, at certain time Moscow carried out joint "Olympiads in Linguistics and Mathematics" Олимпиада по языкознанию и математике. After each such Olympiad its problems were printed in the Science and Life Наука и Жизнь popular science magazine. There were numerous other Olympiads in Moscow including interdisciplinary "Lomonosov tour".

Besides, there were correspondence Olympiads, in particular, Olympiads carried out by some newspapers, journals and universities. One of the important correspondence Olympiads was organized by the Kvant magazine.

Also, there were team contests organized for schools to compete for District, City or Regional honors. There also was a "Physics fight" contest organized by the Moscow State University.

In maths, there were contests organized for cities to compete for Republican and All-Union honors. School round[ edit ] Every school was supposed to have a school round competition. Judges were from subject teachers. The winners from this round could compete in the next round, representing their schools. Each grade could send students to the next round.

This round was usually conducted in the beginning of the school year. District Raion round[ edit ] This round was for schools of the administrative division called "Raion" district , a district of a larger city or of an oblast.

Participants would come in teams, but both teams and individual members were recognized and awarded. The winners form teams representing their areas, consisting of students from every grade.

Usually, area rounds on each subjects took place in different days, so one student could participate in competitions on several subjects. This round was usually conducted in the first half of the school year.

City round[ edit ] This round was for students of the big cities, which had several areas Raions. The winners from the previous round could participate. City round was organized by GorONO, i. Again, the winners would form a team and take part in the next round representing their city.

Depending on a demographic situation, in some places this round was skipped. Regional Oblast round[ edit ] This round was for students of the whole region oblast. Regional round was organized by OblONO, i. Again, the winners would form a team and take part in the next round representing their region Oblast.

They were joined by the winners of the Kvant magazine competition and of the republican and All-Union olympiads of the previous year. This round was usually conducted in the second half of the school year. Republican round[ edit ] This round was a major round, since it recognized the best students of the 15 Republics of the Soviet Union , which are now Independent Countries.

The winners from the previous round could participate in teams and individually. Republican round was organized by Republican Ministries of Education. The winners would form a team and take part in the next round representing their republic. In Russia the competition was conducted separately in four zones and was known as the zonal round. Moscow, Leningrad, a few specialized mathematical schools, and the schools of the transportation ministry system did not compete at the republican level and sent their teams directly to the All-Union round.

It recognized the best students of the Soviet Union in each subject for every grade. In America, it would be on a national level. This round was organized by Soviet Ministry of Education. This round was usually conducted at the end of the school year. Awards[ edit ] The winners were awarded with the diplomas. Material prizes were minor and usually included scientific books that were otherwise difficult to obtain. Notable winners[ edit ] Vladimir Drinfeld who was later awarded a Fields medal for the development of quantum groups is considered by many as the most outstanding "mathematical sportsman" in the history of the All-Union Mathematical Olympiads.

His first scientific publication was based on a generalization of an Olympic problem. Many other winners of the Mathematical Olympiad became outstanding mathematicians and physicists. Yuri Matiyasevich who solved the 10th Hilbert problem in was the absolute winner of the Olympiad.

Grisha Perelman also had an exceptional Olympic record. All three of these national olympiad winners were also selected for the USSR team to the International Mathematical Olympiad and obtained gold medals, with Perelman and Drinfeld achieving perfect scores. Olympiads in universities higher education [ edit ] Overview[ edit ] Contests were conducted on several subject of Soviet higher education curriculum such as Math, Physics, Programming.

These Olympiads had several rounds. So, there were University, Republican and All-Union rounds. There was one contest for all students regardless of their year in the university. University round[ edit ] Every university was supposed to have its own competition. Judges were from faculty staff. The winners from this round could compete in the next round, representing their university. Republican round[ edit ] This round was a major round, since it recognized the best university students in each of the republics.

Teams had up to a dozen students each. All-Union round[ edit ] This round was a final round for Soviet university students. It recognized the best students of the Soviet Union in each subject. Teams had members. This round was usually conducted in the beginning of the next school year. The winners were awarded with diplomas and minor material prizes in some cases.

Moscow Olympiads in Linguistics and Mathematics[ edit ] An interesting experiment was olympiads in linguistics and mathematics, at which students were challenged to solve problems in both seemingly non-related domains.

It was argued that problems in linguistics often require logical reasoning akin to that required in mathematics. After the olympiads, the problems and solutions were published in the Science and Life popular science journal. Further reading[ edit ] D.

Shklarsky, N. Chentzov, I.

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