First, unlike those at most other contests, almost all problems proposed at the LMO are new and original. The LMO jury, which now consists mainly of young St. Petersburg mathematicians, has been directing its best efforts toward this goal for the past 60 years. Second, the LMO is the only official competition in Russia and perhaps the world in which the final rounds are held in oral form.

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First, unlike those at most other contests, almost all problems proposed at the LMO are new and original. The LMO jury, which now consists mainly of young St. Petersburg mathematicians, has been directing its best efforts toward this goal for the past 60 years. Second, the LMO is the only official competition in Russia and perhaps the world in which the final rounds are held in oral form.

Our olympiad does not resemble an entrance examination, with mountains of papers and dull silence. Rather, it is like a series of conversations between contestants and jury members. This book presents the problems of the final two oral rounds. Its junior, the Moscow Mathematical Olympiad, was organized for the first time in , following the successful experience of the LMO.

Delone, G. Fikhtengoltz, V. Tartakovskiy, and O. Other prominent scientists who took part in bringing to life the splendid idea of B. Delone were D. Faddeev, I. Natanson, V.

Krechmar, and V. During the first several years, the Leningrad olympiad was open only to students of the senior tenth grade, but between and the contests for the lower grades were moved into the framework of the LMO. Despite this rule, some students did take part in consecutive olympiads after winning a prize at a previous contest, and in the LMO jury canceled the rule. The main goal of the LMO organizers was to encourage all Leningrad students to strive for excellence in mathematics.

We believe they were successful. Subsequently, mathematics contests were organized throughout the whole country, mainly in large industrial cities.

By the s, mathematics education in the Soviet Union had reached a high standard. In and , with the creation of the All-Russia Mathematical Olympiad and the All-Union Mathematical Olympiad, respectively, the building of the olympiad system was completed. As a consequence of this system, which was supported by local authorities and the Ministry of Education, many able students from all the Soviet republics were attracted to mathematics. The olympiad system comprised small village schools as well as special schools for mathematics and physics in large cities like Moscow, Leningrad, Kiev, Novosibirsk, Tbilisi, Erevan, Riga, Alma-Ata, Minsk, and Kharkov.

The LMO became a component of this comprehensive olympiad system, and the Leningrad and Moscow teams had equal status with teams from the various republics in the final round of the All-Union Mathematical Olympiad. The following are some of the achievements of Leningrad students in the All-Union and International Mathematical Olympiads. In the s students from Leningrad received 40 of the diplomas of the first degree at the All-Union.

Their foremost value lies in promoting knowledge of, and an interest in, mathematics among thousands of able people. The LMO takes place in four levels, or rounds: School level, for the top six grades, held at local schools in December and January. Regional level, held in each of the 22 Leningrad regions in February. Officially, only winners of the school round may compete at this level, but in practice any student can write a paper in the competition of his or her region.

This level is organized as a traditional olympiad, with papers written by 10, or 12, participants. All-city level, the main round, held in February and March. About students participate in each grade. Final level, the elimination round, held in March. About students only 34, in participate in the three senior grades. This round is oral and lasts 5 hours. In addition to being divided into levels, the LMO is divided by age groups.

Students in grades until , grades participate in the junior olympiad, while those in grades until , grades compete in the senior olympiad. Until , Soviet schools comprised grades , but since the grades are The Oral Rounds Participants in the oral rounds of the LMO receive a written list of problems, but they are not obliged to write down their solutions. Instead, any competitor with a proposed solution to one or more problems can give the solution to a jury member orally and must be prepared to answer all questions posed by the juror.

There are usually jurors, mostly students, graduates, and professors of the St. Petersburg State University. The high level of the problems, especially in the elimination round, requires the jurors to exercise extreme accuracy and precision in accepting solutions.

They traditionally work in pairs. Each score must be labeled with the initials of the jurors recording it. It so happens, sometimes, that the participants solve only a few of the proposed problems, which is not the case in many similar competitions. The overall main round consists of six or seven problems, and the complexity of the problems usually increases from the first to the last one. But not all problems are offered to all competitors. At the beginning of this round, all participants sit in preliminary classes, where the first four problems are posed to them on the blackboard or on paper.

The tradition of dividing the round into two parts relieves the labor of both contestants and jurors, the latter needing to hear the solutions to the most complicated problems from only some of the hundreds of participants.

This is very important because it takes much time to listen to an explanation of an intricate mathematical question often in poor mathematical language with logical mistakes and ambiguities and to check the solution. The first two problems - so-called consolation problems - are chosen so that a majority of the contestants can solve them.

The last one or two problems are rather difficult. Nevertheless, at least one participant usually solves all the problems in the main round. In addition, the first-prize winners of the eighth-grade olympiad are invited to participate at the ninth-grade olympiad. Those contestants who show the best results in the main round are invited to compete in the elimination round.

The written contests comprised five problems, whereas the oral contest contained six. But in the two groups of students joined in taking part in a common all-oral main round. From to , and in , the only purpose of the elimination round was to determine which contestants would represent the city team at the All-Union Olympiad, while from to the elimination round was part of the official LMO system, the awards being distributed to winners of this final round.

This is a standard form of out-of-school mathematics education in the former Soviet republics, and many mathematicians and teachers take part in such seminars. This is why the problems in the elimination round of the LMO can be so difficult. Often, at this level, some problems are not solved at all as shown in the statistical tables in Appendix B. Conclusion Finally, we wish to point out a defect and some advantages of the oral format of the LMO.

Any misjudgments made by jurors in accepting erroneous solutions cannot be adjusted after the olympiad is over: the only chance to correct an unjustified score is during the course of the olympiad. If an unjustified plus score is discovered and about to be changed to a minus, say, by another juror checking a solution, the contestant should be informed so that he or she can attempt to defend or improve the solution before the end of the olympiad.

The advantages of the LMO oral format are as follows: Direct communication between competitors and jurors teaches correct mathematical language.

This is very important, especially in the junior grades. Time is not wasted in writing solutions or in scrupulously proving well-known facts used in explanations. Scores are obtained quickly; winners can be identified immediately after the end of the olympiad.

Structure of the Book This book contains the problems and the solutions of the five Leningrad Mathematical Olympiads from through We sometimes provide two different solutions to a problem. LMO statistics for the number of contestants and the number of problems solved by contestants are provided in Appendix B.

A short glossary of special terms and an index of the authors of the problems are included at the end of the book. Acknowledgments In writing this book, we used several booklets of mathematics problems, written from to by Oleg Ivanov, Alexandr Merkuriev, Nikita Netsvetaev, Vladimir Makeev, and Dmitry Fomin.

We are pleased to express our gratitude to all these persons. Special thanks to Mark E. Petersburg and the U. We are grateful to Vladimir Kapustin, Alexander Luzhansky, and Svetlana Ryzhakova for their help in preparing this book.

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## Leningrad Mathematical Olympiads 1987-1991

Gamuro Perhaps the second solution seems long, but this is caused by our attempt to supply an essentially geometric consideration. Email Required, but never shown. Vinod marked it as to-read Apr 09, Want to Read savingâ€¦. The proof is very similar to that of the third fact above, and we leave it to the reader.

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Want to Read savingâ€¦. View online Borrow Buy Freely available Show 0 more links I managed to get a partial response, but no global answer. Home Questions Tags Users Unanswered. Mathematics Stack Exchange works best with JavaScript enabled. The problem, with two solutions, appeared in a collection that has been published by MathPro press.