"This is a sample of rich Russian mathematical culture written by professional mathematicians with great experience in working with high school students..... Problems are on very simple levels, but building to more complex and advanced work... [contains] solutions to almost all problems; methodological notes for the teacher... developed for a peculiarly Russian institution (the mathematical circle), but easily adapted to American teachers' needs, both inside and outside the classroom." --- from the Translator's notes. What kind of book is this? It is a book produced by a remarkable cultural circumstance in the former Soviet Union which fostered the creation of groups of students, teachers, and mathematicians called "mathematical circles". The work is predicated on the idea that studying mathematics can generate the same enthusiasm as playing a team sport -- without necessarily being competitive. This book is intended for both students and teachers who love mathematics and want to study its various branches beyond the limits of school curriculum. It is also a book of mathematical recreations and, at the same time, a book containing vast theoretical and problem material in main areas of what authors consider to be "extracurricular mathematics". The book is based on a unique experience gained by several generations of Russian educators and scholars.