Softcover ISBN:  9781470460105 
Product Code:  MCL/29 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $41.25 
eBook ISBN:  9781470475895 
Product Code:  MCL/29.E 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $41.25 
Softcover ISBN:  9781470460105 
eBook: ISBN:  9781470475895 
Product Code:  MCL/29.B 
List Price:  $110.00 $82.50 
MAA Member Price:  $99.00 $74.25 
AMS Member Price:  $82.50 $61.88 
Softcover ISBN:  9781470460105 
Product Code:  MCL/29 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $41.25 
eBook ISBN:  9781470475895 
Product Code:  MCL/29.E 
List Price:  $55.00 
MAA Member Price:  $49.50 
AMS Member Price:  $41.25 
Softcover ISBN:  9781470460105 
eBook ISBN:  9781470475895 
Product Code:  MCL/29.B 
List Price:  $110.00 $82.50 
MAA Member Price:  $99.00 $74.25 
AMS Member Price:  $82.50 $61.88 

Book DetailsMSRI Mathematical Circles LibraryVolume: 29; 2023; 197 ppMSC: Primary 00; 05; 52; 60; 94; 97
This book is a translation from Russian of Part III of the book Mathematics via Problems: From Olympiads and Math Circles to Profession. Part I, Algebra, and Part II, Geometry, have been published in the same series.
The main goal of this book is to develop important parts of mathematics through problems. The authors tried to put together sequences of problems that allow high school students (and some undergraduates) with strong interest in mathematics to discover such topics in combinatorics as counting, graphs, constructions and invariants in combinatorics, games and algorithms, probabilistic aspects of combinatorics, and combinatorial geometry.
Definitions and/or references for material that is not standard in the school curriculum are included. To help students that might be unfamiliar with new material, problems are carefully arranged to provide gradual introduction into each subject. Problems are often accompanied by hints and/or complete solutions.
The book is based on classes taught by the authors at different times at the Independent University of Moscow, at a number of Moscow schools and math circles, and at various summer schools. It can be used by high school students and undergraduates, their teachers, and organizers of summer camps and math circles.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, SLMath (formerly 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 copublished with the Mathematical Sciences Research Institute (MSRI).
ReadershipHigh school students and their mentors and teachers.

Table of Contents

Cover

Title page

Contents

Foreword

Problems, exercises, circles, and olympiads

Why this book and how to use it

Englishlanguage references

Introduction

What this book is about and whom it is for

Learning by doing problems

Parting words By A.Ya.Kanel–Belov

Olympiads and mathematics

Research problems for high school students

How this book is organized

Resources and literature

Acknowledgements

Numbering and notation

Notation

Bibliography

Chapter 1. Counting

1. How many ways? (1) By A.A.Gavrilyuk and D.A.Permyakov

Suggestions, solutions, and answers

2. Sets of subsets (2) By D.A.Permyakov

Suggestions, solutions, and answers

3. The principle of inclusionexclusion (2) By D.A.Permyakov

Suggestions, solutions, and answers

Chapter 2. Finite sets

1. The pigeonhole principle (1) By A.Ya.KanelBelov

Part 1

Part 2 (2)

Suggestions, solutions, and answers

2. The extremal principle (2) By A.Ya.KanelBelov

Suggestions, solutions, and answers

3. Periodicity I (2) By A.Ya.KanelBelov

Suggestions, solutions, and answers

4. Periodicity II (2) By P.A.Kozhevnikov

Suggestions, solutions, and answers

5. Finite and countable sets (2) By P.A.Kozhevnikov

Suggestions, solutions, and answers

Comments about the solutions of problems 2.5.1 and 2.5.2

Chapter 3. Graphs By D.A.Permyakov and A.B.Skopenkov

1. Graphs (2)

Suggestions, solutions, and answers

2. Counting in graphs (2)

Suggestions, solutions, and answers

3. Paths in graphs (2)

Suggestions, solutions, and answers

Chapter 4. Constructions and invariants

1. Constructions (1) By A.V.Shapovalov

Suggestions, solutions, and answers

2. Invariants I (1) By A.Ya.KanelBelov

Suggestions, solutions, and answers

3. Invariants II (1) By A.V.Shapovalov

Suggestions, solutions, and answers

4. Colorings

4.A. Tilings (1) By A.Ya.KanelBelov

4.B. Tables (2) By D.A.Permyakov

Suggestions, solutions, and answers

5. Semiinvariants (1) By A.V.Shapovalov

Suggestions, solutions, and answers

Chapter 5. Algorithms

1. Games (1) By D. A.Permyakov, M. B.Skopenkov, and A.V.Shapovalov

Symmetric strategy

Game on outracing

Accumulation of advantages

Joke games

Growing a tree of positions

Passing the move

Miscellany

Suggestions, solutions, and answers

2. Information problems (2) By A.Ya.KanelBelov

Suggestions, solutions, and answers

3. Error correction codes (2) By M.B.Skopenkov

Suggestions, solutions, and answers

4. Boolean cube (2) By A.B.Skopenkov

Hints

Suggestions, solutions, and answers

5. Expressibility for functions of the algebra of logic By A.B.Skopenkov

Examples and definitions (1)

Post’s theorem (2*)

Hints

Suggestions, solutions, and answers

6. Complexity of summation By Yu.G.Kydryashov and A.B.Skopenkov

Introductory problems (2)

Definitions and examples (3*)

Asymptotic estimates (4*)

Suggestions, solutions, and answers

Chapter 6. Probability By A.B.Skopenkov and A.A.Zaslavsky

1. Classical definition of probability (1)

Suggestions, solutions, and answers

2. A more general definition of probability (1)

Suggestions, solutions, and answers

3. Independence and conditional probability (1)

Suggestions, solutions, and answers

4. Random variables (3)

Suggestions, solutions, and answers

5. Bernoulli trials (3)

Suggestions, solutions, and answers

6. Random walks and electrical circuits (3) By A.A.Zaslavsky, M.B.Skopenkov, and A.V.Ustinov

Biased random walk*

Physical interpretation

Existence and uniqueness of voltage

Conductance of circuits

The variational principle

Twodimensional random walk

Threedimensional random walks

Suggestions, solutions, and answers

Chapter 7. Combinatorial geometry

1. Rug runners and napkins (2) By P.A.Kozhevnikov

Onedimensional geometry, or \enquote{rug runners}

Twodimensional geometry, or \enquote{napkins on the table}

Three dimensions

Suggestions, solutions, and answers

2. Helly’s theorem (2) By A.V.Akopyan

Suggestions, solutions, and answers

3. Lattice polygons (2) By V.V.Prasolov and M.B.Skopenkov

3.A. Area of a polygon on grid paper (2)

3.B. Dual lattice polygons (3*)

Suggestions, solutions, and answers

4. Pigeonhole principle on a line (3) By A.Ya.KanelBelov

Suggestions, solutions, and answers

5. The pigeonhole principle and its application to geometry (3) By I.V.Arzhantsev

The area of a figure

The pigeonhole principle for areas

The theorems of Blichfeldt and Minkowski

Dirichlet’s theorem on approximation of irrational numbers

Suggestions, solutions, and answers

6. Phase spaces (3) By A.Ya.KanelBelov

7. Linear variation (3) By A.Ya.KanelBelov

Suggestions, solutions, and answers

8. Compose a square (3*) By M.B.Skopenkov, O.A.Malinovskaya, S.A.Dorichenko, and F.A.Sharov

Leading questions

Rectangles from squares.

From cutting to roots of polynomials

What’s next

Suggestions, solutions, and answers

9. Is it possible to make a cube from a tetrahedron? (3) By M.V.Prasolov and M.B.Skopenkov

Reduction to a plane geometry problem

Solution of the plane geometry problem

Suggestions, solutions, and answers

Bibliography

Index

Back Cover


Additional Material

RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Requests
This book is a translation from Russian of Part III of the book Mathematics via Problems: From Olympiads and Math Circles to Profession. Part I, Algebra, and Part II, Geometry, have been published in the same series.
The main goal of this book is to develop important parts of mathematics through problems. The authors tried to put together sequences of problems that allow high school students (and some undergraduates) with strong interest in mathematics to discover such topics in combinatorics as counting, graphs, constructions and invariants in combinatorics, games and algorithms, probabilistic aspects of combinatorics, and combinatorial geometry.
Definitions and/or references for material that is not standard in the school curriculum are included. To help students that might be unfamiliar with new material, problems are carefully arranged to provide gradual introduction into each subject. Problems are often accompanied by hints and/or complete solutions.
The book is based on classes taught by the authors at different times at the Independent University of Moscow, at a number of Moscow schools and math circles, and at various summer schools. It can be used by high school students and undergraduates, their teachers, and organizers of summer camps and math circles.
In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, SLMath (formerly 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 copublished with the Mathematical Sciences Research Institute (MSRI).
High school students and their mentors and teachers.

Cover

Title page

Contents

Foreword

Problems, exercises, circles, and olympiads

Why this book and how to use it

Englishlanguage references

Introduction

What this book is about and whom it is for

Learning by doing problems

Parting words By A.Ya.Kanel–Belov

Olympiads and mathematics

Research problems for high school students

How this book is organized

Resources and literature

Acknowledgements

Numbering and notation

Notation

Bibliography

Chapter 1. Counting

1. How many ways? (1) By A.A.Gavrilyuk and D.A.Permyakov

Suggestions, solutions, and answers

2. Sets of subsets (2) By D.A.Permyakov

Suggestions, solutions, and answers

3. The principle of inclusionexclusion (2) By D.A.Permyakov

Suggestions, solutions, and answers

Chapter 2. Finite sets

1. The pigeonhole principle (1) By A.Ya.KanelBelov

Part 1

Part 2 (2)

Suggestions, solutions, and answers

2. The extremal principle (2) By A.Ya.KanelBelov

Suggestions, solutions, and answers

3. Periodicity I (2) By A.Ya.KanelBelov

Suggestions, solutions, and answers

4. Periodicity II (2) By P.A.Kozhevnikov

Suggestions, solutions, and answers

5. Finite and countable sets (2) By P.A.Kozhevnikov

Suggestions, solutions, and answers

Comments about the solutions of problems 2.5.1 and 2.5.2

Chapter 3. Graphs By D.A.Permyakov and A.B.Skopenkov

1. Graphs (2)

Suggestions, solutions, and answers

2. Counting in graphs (2)

Suggestions, solutions, and answers

3. Paths in graphs (2)

Suggestions, solutions, and answers

Chapter 4. Constructions and invariants

1. Constructions (1) By A.V.Shapovalov

Suggestions, solutions, and answers

2. Invariants I (1) By A.Ya.KanelBelov

Suggestions, solutions, and answers

3. Invariants II (1) By A.V.Shapovalov

Suggestions, solutions, and answers

4. Colorings

4.A. Tilings (1) By A.Ya.KanelBelov

4.B. Tables (2) By D.A.Permyakov

Suggestions, solutions, and answers

5. Semiinvariants (1) By A.V.Shapovalov

Suggestions, solutions, and answers

Chapter 5. Algorithms

1. Games (1) By D. A.Permyakov, M. B.Skopenkov, and A.V.Shapovalov

Symmetric strategy

Game on outracing

Accumulation of advantages

Joke games

Growing a tree of positions

Passing the move

Miscellany

Suggestions, solutions, and answers

2. Information problems (2) By A.Ya.KanelBelov

Suggestions, solutions, and answers

3. Error correction codes (2) By M.B.Skopenkov

Suggestions, solutions, and answers

4. Boolean cube (2) By A.B.Skopenkov

Hints

Suggestions, solutions, and answers

5. Expressibility for functions of the algebra of logic By A.B.Skopenkov

Examples and definitions (1)

Post’s theorem (2*)

Hints

Suggestions, solutions, and answers

6. Complexity of summation By Yu.G.Kydryashov and A.B.Skopenkov

Introductory problems (2)

Definitions and examples (3*)

Asymptotic estimates (4*)

Suggestions, solutions, and answers

Chapter 6. Probability By A.B.Skopenkov and A.A.Zaslavsky

1. Classical definition of probability (1)

Suggestions, solutions, and answers

2. A more general definition of probability (1)

Suggestions, solutions, and answers

3. Independence and conditional probability (1)

Suggestions, solutions, and answers

4. Random variables (3)

Suggestions, solutions, and answers

5. Bernoulli trials (3)

Suggestions, solutions, and answers

6. Random walks and electrical circuits (3) By A.A.Zaslavsky, M.B.Skopenkov, and A.V.Ustinov

Biased random walk*

Physical interpretation

Existence and uniqueness of voltage

Conductance of circuits

The variational principle

Twodimensional random walk

Threedimensional random walks

Suggestions, solutions, and answers

Chapter 7. Combinatorial geometry

1. Rug runners and napkins (2) By P.A.Kozhevnikov

Onedimensional geometry, or \enquote{rug runners}

Twodimensional geometry, or \enquote{napkins on the table}

Three dimensions

Suggestions, solutions, and answers

2. Helly’s theorem (2) By A.V.Akopyan

Suggestions, solutions, and answers

3. Lattice polygons (2) By V.V.Prasolov and M.B.Skopenkov

3.A. Area of a polygon on grid paper (2)

3.B. Dual lattice polygons (3*)

Suggestions, solutions, and answers

4. Pigeonhole principle on a line (3) By A.Ya.KanelBelov

Suggestions, solutions, and answers

5. The pigeonhole principle and its application to geometry (3) By I.V.Arzhantsev

The area of a figure

The pigeonhole principle for areas

The theorems of Blichfeldt and Minkowski

Dirichlet’s theorem on approximation of irrational numbers

Suggestions, solutions, and answers

6. Phase spaces (3) By A.Ya.KanelBelov

7. Linear variation (3) By A.Ya.KanelBelov

Suggestions, solutions, and answers

8. Compose a square (3*) By M.B.Skopenkov, O.A.Malinovskaya, S.A.Dorichenko, and F.A.Sharov

Leading questions

Rectangles from squares.

From cutting to roots of polynomials

What’s next

Suggestions, solutions, and answers

9. Is it possible to make a cube from a tetrahedron? (3) By M.V.Prasolov and M.B.Skopenkov

Reduction to a plane geometry problem

Solution of the plane geometry problem

Suggestions, solutions, and answers

Bibliography

Index

Back Cover