Mathematics, along with great orchestral music, thick novels, and classic

ballet, is among the great achievements of Russian culture. Since Catherine

the Great summoned Euler to St. Petersburg, the Russian school of

mathematics has contributed a stream of ideas to the field.

During the Soviet period of Russian history this stream became a flood.

Soviet mathematicians contributed enormously to the fields of number theory,

analysis, and probability. Entire journals were started in the West to

keep track of Soviet advances in mathematics.

Some of this increase in mathematical activity was due to a conscious

investment on the part of the Soviets in science and technology. Communism

was "scientific'' socialism, and the role of science and technology in the

betterment of the human condition was supreme in Soviet dogma.

But the totalitarian nature of the Soviet regime had a second effect on the

mathematical life of the country, which stimulated mathematical achievement in

an entirely unconscious manner. Young people of high intellectual ability,

seeking a field in which they could contribute, were drawn to mathematics by

the relative lack of politics in that field. It was very difficult, during

the Soviet era, to do research in the humanities while dodging the

intellectual barriers posed by the totalitarian state. In the sciences,

biologists were hampered by the inviolable supremacy of Lysenco's views,

including a rejection of most of genetic theory. In physics, chemistry, and

medicine, the need for laboratories and equipment, and the tendency of

government to direct research into particular areas led to a heavy political

presence in the field.

Mathematicians found themselves relatively free from these trammels. They

needed no laboratories. The applications of much of their work were

sufficiently general or far removed from their own lives to forestall serious

political meddling. While political pressure was never entirely absent, and

career paths might be cleared or obstructed by politics, mathematicians

enjoyed a freedom to explore their intellectual environment that other

scientists were denied. And so young people were attracted to mathematics,

and found role models and mentors among working mathematicians.

All these various circumstances led also to another key feature of Soviet

mathematical life: the close connections between research and education,

between university and high school, between professor and teacher.

Mathematicians at the highest levels of their profession found the time to

work with young people in elementary and high school. Such world-class

mathematicians as Kolmogorov, Dinkin, and Gelfond founded schools and

journals, wrote articles, and gave talks to students on the college and

pre-college levels. The state boarding schools for mathematics and science

attached to several Russian universities stand as monuments to this culture,

as does the Russian-language journal Kvant, a publication in

mathematics and physics for high school students (the American journal

Quantum seeks to transplant a piece of this culture to American soil).

The Leningrad Olympiads must be seen against this background. As the

preface will explain, they are the product of much thought by some of the

most able mathematical minds to be found anywhere. The labor-intensive form

of the olympiad program, and especially of the oral round, could only have

evolved in an environment rich in mathematics, and possessing a tradition of

cooperation and communication between people doing mathematics on every level,

from the most abstruse research material down through elementary schools.

It is estimated that more than 60% of the mathematics faculty of Leningrad

State University would have a hand in writing, grading, or judging each

Olympiad. Even in the (former) Soviet Union, this figure is astonishing.

The Leningrad Olympiads are among the most exciting and interesting

mathematical events in the world on the pre-college level. The problems in

this volume represent significant mathematical thought by some of the most

able minds of their country. It is rare, in any country, to find a

group of mathematicians so able, and so motivated, to provide for the future

of their own profession.

Mark Saul

Bronxville Schools

Bronxville, NY

July 1993