Leningrad Mathematical Olympiads 1987-1991

Authors' Preface

The Leningrad Mathematical Olympiad (LMO) is perhaps a unique
phenomenon in the world of mathematics competitions.

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.

Some Historical Notes

Founded in 1934, the LMO is the oldest mathematics competition in Russia. Its
junior, the Moscow Mathematical Olympiad, was organized for the first time in
1935, following the successful experience of the LMO.

The organizing committee of the 1934 LMO consisted of well-known Soviet
mathematicians, professors at Leningrad State University,
B. N. Delone, G. M. Fikhtengoltz, V. A. Tartakovskiy, and O. K. Zhitomirskiy.
Other prominent scientists who took part in bringing to life the splendid idea of
B. N. Delone were D. K. Faddeev, I. P. Natanson, V. A. Krechmar, and V. I. Smirnov.

During the first several years, the Leningrad olympiad was open only
to students of the senior (tenth) grade, but between 1938 and 1940 the
contests for the lower grades were moved into the framework of the LMO.

At first, olympiad winners were not allowed to take part in the
following year's competition because the contest organizers were afraid
the LMO would become an exclusive, competitive sport instead of fostering
a system of mathematical enrichment for all Leningrad students. Despite
this rule, some students did take part in consecutive olympiads after winning
a prize at a previous contest, and in 1939 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 1960s, mathematics education in the Soviet
Union had reached a high standard.

In 1961 and 1967, 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 1980s
students from Leningrad received 40 of the 129 diplomas of the first degree at
the All-Union. During the same decade they occupied 21 of 58 places on
the Soviet Union's team at the International Olympiad; from 1987 to 1991 at least
half of the Soviet contestants were from Leningrad (although the city's
population was only 4\% of the Soviet urban population). Prominent Leningrad
mathematicians who took part in the LMO are Mikhail Gromov, who won several
diplomas of the first degree at the LMO, Alexey Aleksandrov, Yuri Burago,
Jakov Eliashberg, Viatcheslav Kharlamov, Yuri Matiyasevitch, Alexandr
Merkuriev, Garald Natanson, Vladimir Peller, Mikhail Solomyak, Andrey
Suslin, Grigory Tseitin, Vladimir Turaev, Nina Uraltseva, Anatoly
Vershik, Oleg Viro, Victor Zalgaller, and many others.

Yet the value of mathematics olympiads is found not only in honors
and the "discovery'' of outstanding mathematicians. Their foremost value
lies in promoting knowledge of, and an interest in, mathematics among
thousands of able people.

Structure of the LMO

The LMO takes place in four levels, or rounds:

  1. School level, for the top six grades, held at local schools in
    December and January.

  2. 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 10,000 or 12,000

  3. All-city level, the main round, held in February and March. About
    90--130 students participate in each grade. These trials are oral and last
    3\textfraction{1/2}--4 hours.

  4. Final level, the elimination round, held in March. About 80--100
    students (only 34, in 1991) 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 6-8 (until 1989, grades 5-7) participate in
the junior olympiad, while those in grades 9-11 (until 1989, grades 8-10)
compete in the senior olympiad. (Until 1989, Soviet schools comprised
grades 1-10, but since 1990 the grades are 1-11.)

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 40-60 jurors, mostly students, graduates, and
professors of the St. Petersburg State University. The jurors record a
score of plus (+) or minus (-) for a right or wrong solution, and the
contestants have three chances to solve each problem, resulting in a
possible score of plus, minus-plus, minus-minus-plus, or minus-minus-minus
for each problem. 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.

The contestants' standing is always based on how many problems
they have solved. It so happens, sometimes, that the participants solve
only a few of the proposed problems, which is not the case in
many similar competitions.

Main Round

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. Two or
three additional problems are given in an "outer'' class only, after three
preliminary problems have been solved.

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.

The following students are invited by the LMO jury to take part in
the main round: the winners of the second-level (regional) contests of
the current year; all students who received diplomas of the first
three degrees of the previous year's junior
olympiad; and all students who received diplomas of the first two degrees
of the previous year's senior olympiad.
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.

From 1984 to 1990, the top two grades in special schools for
mathematics and physics took the main round of the senior olympiad
separately from their age-mates in regular schools - from 1984 to 1988
with written solutions in a separate "intermediate'' contest; and in 1989
and 1990 with written solutions as against their age-mates' oral
solutions. The written contests comprised five problems, whereas the oral
contest contained six. But in 1991 the two groups of students joined in
taking part in a common all-oral main round.

Elimination Round

Unlike the main round, the elimination final round is not divided into
preliminary and "outer'' parts, and it consists of eight or nine problems
instead of six or seven.

From 1962 to 1983, and in 1991, the only purpose of the elimination round
was to determine which contestants would represent the city team at the All-Union
Olympiad, while from 1984 to 1990 the elimination round was part of the
official LMO system, the awards being distributed to winners of this
final round.

Most of the contestants in the elimination round are students of
special mathematics and physics schools who have taken part outside
of school in an informal "math circle,'' a kind of seminar devoted to
problem solving or to investigating certain areas of elementary and
higher mathematics that are not covered in the school curriculum. 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).


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:

  1. Direct communication between competitors and jurors teaches correct
    mathematical language. This is very important, especially in the junior

  2. There is the possibility of correcting mistakes during the olympiad
    and even of changing one's point of view on a question.

  3. Time is not wasted in writing solutions or in scrupulously proving
    well-known facts used in explanations.

  4. 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 1987 through 1991.
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.


In writing this book, we used several booklets of
mathematics problems, written from 1987 to 1991 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. Saul, who created a "mathematical bridge''
connecting St. Petersburg and the U.S.A.

We are grateful to Vladimir Kapustin, Alexander Luzhansky, and Svetlana
Ryzhakova for their help in preparing this book.

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