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.

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 10,000 or 12,000

participants.

- 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.

- 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.)

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.

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.

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:

- Direct communication between competitors and jurors teaches correct

mathematical language. This is very important, especially in the junior

grades.

- There is the possibility of correcting mistakes during the olympiad

and even of changing one's point of view on a question.

- 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.

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.