# A Moscow Math Circle: Week-by-week Problem Sets

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*Sergey Dorichenko*

A co-publication of the AMS and the Mathematical Sciences Research Institute

Moscow has a rich tradition of successful math circles, to the extent
that many other circles are modeled on them. This book presents
materials used during the course of one year in a math circle
organized by mathematics faculty at Moscow State University, and also
used at the mathematics magnet school known as Moscow School Number
57.

Each problem set has a similar structure: it combines review
material with a new topic, offering problems in a range of difficulty
levels. This time-tested pattern has proved its effectiveness in
engaging all students and helping them master new material while
building on earlier knowledge.

The introduction describes in detail how the math circles at Moscow
State University are run. Dorichenko describes how the early sessions
differ from later sessions, how to choose problems, and what sorts of
difficulties may arise when running a circle. The book also includes
a selection of problems used in the competition known as the
Mathematical Maze, a mathematical story based on actual lessons with
students, and an addendum on the San Jose Mathematical
Circle, which is run in the Russian style.

In the interest of fostering a greater awareness and appreciation of
mathematics and its connections to other disciplines and everyday life, MSRI
and the AMS are publishing books in the Mathematical Circles Library series as
a service to young people, their parents and teachers, and the mathematics profession.

Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI).

#### Readership

Undergraduate students interested in math circles, clever math problems, and high school education.

#### Reviews & Endorsements

[T]his is an excellent resource for those interested in math circles, including students and parents . . . For those interested in starting and running a math circle, I think it is an invaluable resource.

-- Vicentiu D. Radulescu, Zentralblatt MATH

#### Table of Contents

# Table of Contents

## A Moscow Math Circle: Week-by-week Problem Sets

- Cover Cover11
- Title page iii4
- Contents v6
- Foreword ix10
- Preface xi12
- Acknowledgments xxiii24
- Problem set 0 126
- Problem set 1 328
- Problem set 2 530
- Problem set 3 732
- Problem set 4 934
- Problem set 5 1136
- Problem set 6 1338
- Problem set 7 1540
- Problem set 8 1742
- Problem set 9 1944
- Problem set 10 2146
- Problem set 11 2348
- Problem set 12 2550
- Problem set 13 2752
- Winter competition 2954
- Problem set 14 3257
- Problem set 15 3459
- Problem set 16 3661
- Problem set 17 3863
- Problem set 18 4065
- Problem set 19 4267
- Problem set 20 4469
- Problem set 21 4671
- Problem set 22 4873
- Problem set 23 5075
- Problem set 24 5277
- Problem set 25 5479
- Problem set 26 5681
- Problem set 27 5883
- Problem set 28 6085
- Solutions to problem set 0 6388
- Solutions to problem set 1 6691
- Solutions to problem set 2 6994
- Solutions to problem set 3 7398
- Solutions to problem set 4 78103
- Solutions to problem set 5 83108
- Solutions to problem set 6 87112
- Solutions to problem set 7 90115
- Solutions to problem set 8 95120
- Solutions to problem set 9 98123
- Solutions to problem set 10 102127
- Solutions to problem set 11 106131
- Solutions to problem set 12 114139
- Solutions to problem set 13 118143
- Solutions to the winter competition 121146
- Solutions to problem set 14 130155
- Solutions to problem set 15 134159
- Solutions to problem set 16 140165
- Solutions to problem set 17 144169
- Solutions to problem set 18 147172
- Solutions to problem set 19 153178
- Solutions to problem set 20 159184
- Solutions to problem set 21 164189
- Solutions to problem set 22 168193
- Solutions to problem set 23 173198
- Solutions to problem set 24 176201
- Solutions to problem set 25 179204
- Solutions to problem set 26 182207
- Solutions to problem set 27 185210
- Solutions to problem set 28 188213
- Mathematical maze 193218
- Two and two is more than four; A story 205230
- Addendum: The San Jose experience 215240
- Problem set SJ1 219244
- Solutions to problem set SJ1 221246
- Problem set SJ2 227252
- Solutions to problem set SJ2 229254
- Problem set SJ3 235260
- Solutions to problem set SJ3 237262
- Back Cover Back Cover1266