
Book DetailsProblem BooksVolume: 16; 2006; 307 pp
Reprinted edition available: PRB/34
Any high school student preparing for the American Mathematics Competitions should get their hands on a copy of this book!
A major aspect of mathematical training and its benefit to society is the ability to use logic to solve problems. The American Mathematics Competitions (AMC) have been given for more than fifty years to millions of high school students. This book considers the basic ideas behind the solutions to the majority of these problems, and presents examples and exercises from past exams to illustrate the concepts. Anyone taking the AMC exams or helping students prepare for them will find many useful ideas here. But people generally interested in logical problem solving should also find the problems and their solutions interesting.
This book will promote interest in mathematics by providing students with the tools to attack problems that occur on mathematical problemsolving exams, and specifically to level the playing field for those who do not have access to the enrichment programs that are common at the top academic high schools. The book can be used either for selfstudy or to give people who want to help students prepare for mathematics exams easy access to topicoriented material and samples of problems based on that material. This is useful for teachers who want to hold special sessions for students, but it is equally valuable for parents who have children with mathematical interest and ability.
As students' problem solving abilities improve, they will be able to comprehend more difficult concepts requiring greater mathematical ingenuity. They will be taking their first steps towards becoming math Olympians!

Table of Contents

cover

copyright page

title page

Contents

Preface

A Brief History of the American Mathematics Competitions

My Experience with the American Mathematics Competitions

The Basis and Reason for this Book

Structure of the Book

Acknowledgments

1 Arithmetic Ratios

1.1 Introduction

1.2 Time and Distance Problems

1.3 Least Common Multiples

1.4 Ratio Problems

Examples for Chapter 1

Exercises for Chapter 1

2 Polynomials and their Zeros

2.1 Introduction

2.2 Lines

2.3 Quadratic Polynomials

2.4 General Polynomials

Examples for Chapter 2

Exercises for Chapter 2

3 Exponentials and Radicals

3.1 Introduction

3.2 Exponents and Bases

3.3 Exponential Functions

3.4 Basic Rules of Exponents

3.5 The Binomial Theorem

Examples for Chapter 3

Exercises for Chapter 3

4 Defined Functions and Operations

4.1 Introduction

4.2 Binary Operations

4.3 Functions

Examples for Chapter 4

Exercises for Chapter 4

5 Triangle Geometry

5.1 Introduction

5.2 Definitions

5.3 Basic Right Triangle Results

Special Right Triangles

5.4 Areas of Triangles

5.5 Geometric Results about Triangles

Examples for Chapter 5

Exercises for Chapter 5

6 Circle Geometry

6.1 Introduction

6.2 Definitions

6.3 Basic Results of Circle Geometry

6.4 Results Involving the Central Angle

Examples for Chapter 6

Exercises for Chapter 6

7 Polygons

7.1 Introduction

7.2 Definitions

7.3 Results about Quadrilaterals

7.4 Results about General Polygons

Examples for Chapter 7

Polygons 81Exercises for Chapter 7

8 Counting

8.1 Introduction

8.2 Permutations

8.3 Combinations

8.4 Counting Factors

Examples for Chapter 8

Exercises for Chapter 8

9 Probability

9.1 Introduction

9.2 Definitions and Basic Notions

9.3 Basic Results

Examples for Chapter 9

Exercises for Chapter 9

10 Prime Decomposition

10.1 Introduction

10.2 The Fundamental Theorem of Arithmetic

Examples for Chapter 10

Exercises for Chapter 10

11 Number Theory

11.1 Introduction

11.2 Number Bases and Modular Arithmetic

11.3 Integer Division Results

11.4 The Pigeon Hole Principle

Examples for Chapter 11

Exercises for Chapter 11

12 Sequences and Series

12.1 Introduction

12.2 Definitions

Examples for Chapter 12

Exercises for Chapter 12

13 Statistics

13.1 Introduction

13.2 Definitions

13.3 Results

138 First Steps for Math OlympiansExamples for Chapter 13

Exercises for Chapter 13

14 Trigonometry

14.1 Introduction

14.2 Definitions and Results

14.3 Important Sine and Cosine Facts

14.4 The Other Trigonometric Functions

Examples for Chapter 14

Exercises for Chapter 14

15 ThreeDimensional Geometry

15.1 Introduction

15.2 Definitions and Results

Examples for Chapter 15

Exercises for Chapter 15

16 Functions

16.1 Introduction

16.2 Definitions

16.3 Graphs of Functions

16.4 Composition of Functions

Examples for Chapter 16

Exercises for Chapter 16

17 Logarithms

17.1 Introduction

17.2 Definitions and Results

Examples for Chapter 17

Exercises for Chapter 17

18 Complex Numbers

18.1 Introduction

18.2 Definitions

18.3 Important Complex Number Properties

Examples for Chapter 18

Exercises for Chapter 18

Solutions to Exercises

Solutions for Chapter 1: Arithmetic Ratios

Solutions for Chapter 2: Polynomials

Solutions for Chapter 3: Exponentials and Radicals

Solutions for Chapter 4: Defined Functions and Operations

Solutions for Chapter 5: Triangle Geometry

Solutions for Chapter 6: Circle Geometry

Solutions for Chapter 7: Polygons

Solutions for Chapter 8: Counting

Solutions for Chapter 9: Probability

Solutions for Chapter 10: Prime Decomposition

Solutions for Chapter 11: Number Theory

Solutions for Chapter 12: Sequences and Series

Solutions for Chapter 13: Statistics

Solutions for Chapter 14: Trigonometry

Solutions for Chapter 15: ThreeDimensional Geometry

Solutions for Chapter 16: Functions

Solutions for Chapter 17: Logarithms

Solutions for Chapter 18: Complex Numbers

Epilogue

Sources of the Exercises


Additional Material

Reviews

... This book should be included in the library of anyone involved in preparing high school students to the AMC. Highly recommended.
J. T. Noonan, CHOICE 
Provides a wealth of opportunities for students to become experienced problem solvers.
David Webb, Penn State University 
An impressive problem solving primer.The book presents a wide variety of problems and problem solving strategies.
Richard Gibbs, Fort Lewis College


RequestsReview Copy – for publishers of book reviewsAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
Reprinted edition available: PRB/34
Any high school student preparing for the American Mathematics Competitions should get their hands on a copy of this book!
A major aspect of mathematical training and its benefit to society is the ability to use logic to solve problems. The American Mathematics Competitions (AMC) have been given for more than fifty years to millions of high school students. This book considers the basic ideas behind the solutions to the majority of these problems, and presents examples and exercises from past exams to illustrate the concepts. Anyone taking the AMC exams or helping students prepare for them will find many useful ideas here. But people generally interested in logical problem solving should also find the problems and their solutions interesting.
This book will promote interest in mathematics by providing students with the tools to attack problems that occur on mathematical problemsolving exams, and specifically to level the playing field for those who do not have access to the enrichment programs that are common at the top academic high schools. The book can be used either for selfstudy or to give people who want to help students prepare for mathematics exams easy access to topicoriented material and samples of problems based on that material. This is useful for teachers who want to hold special sessions for students, but it is equally valuable for parents who have children with mathematical interest and ability.
As students' problem solving abilities improve, they will be able to comprehend more difficult concepts requiring greater mathematical ingenuity. They will be taking their first steps towards becoming math Olympians!

cover

copyright page

title page

Contents

Preface

A Brief History of the American Mathematics Competitions

My Experience with the American Mathematics Competitions

The Basis and Reason for this Book

Structure of the Book

Acknowledgments

1 Arithmetic Ratios

1.1 Introduction

1.2 Time and Distance Problems

1.3 Least Common Multiples

1.4 Ratio Problems

Examples for Chapter 1

Exercises for Chapter 1

2 Polynomials and their Zeros

2.1 Introduction

2.2 Lines

2.3 Quadratic Polynomials

2.4 General Polynomials

Examples for Chapter 2

Exercises for Chapter 2

3 Exponentials and Radicals

3.1 Introduction

3.2 Exponents and Bases

3.3 Exponential Functions

3.4 Basic Rules of Exponents

3.5 The Binomial Theorem

Examples for Chapter 3

Exercises for Chapter 3

4 Defined Functions and Operations

4.1 Introduction

4.2 Binary Operations

4.3 Functions

Examples for Chapter 4

Exercises for Chapter 4

5 Triangle Geometry

5.1 Introduction

5.2 Definitions

5.3 Basic Right Triangle Results

Special Right Triangles

5.4 Areas of Triangles

5.5 Geometric Results about Triangles

Examples for Chapter 5

Exercises for Chapter 5

6 Circle Geometry

6.1 Introduction

6.2 Definitions

6.3 Basic Results of Circle Geometry

6.4 Results Involving the Central Angle

Examples for Chapter 6

Exercises for Chapter 6

7 Polygons

7.1 Introduction

7.2 Definitions

7.3 Results about Quadrilaterals

7.4 Results about General Polygons

Examples for Chapter 7

Polygons 81Exercises for Chapter 7

8 Counting

8.1 Introduction

8.2 Permutations

8.3 Combinations

8.4 Counting Factors

Examples for Chapter 8

Exercises for Chapter 8

9 Probability

9.1 Introduction

9.2 Definitions and Basic Notions

9.3 Basic Results

Examples for Chapter 9

Exercises for Chapter 9

10 Prime Decomposition

10.1 Introduction

10.2 The Fundamental Theorem of Arithmetic

Examples for Chapter 10

Exercises for Chapter 10

11 Number Theory

11.1 Introduction

11.2 Number Bases and Modular Arithmetic

11.3 Integer Division Results

11.4 The Pigeon Hole Principle

Examples for Chapter 11

Exercises for Chapter 11

12 Sequences and Series

12.1 Introduction

12.2 Definitions

Examples for Chapter 12

Exercises for Chapter 12

13 Statistics

13.1 Introduction

13.2 Definitions

13.3 Results

138 First Steps for Math OlympiansExamples for Chapter 13

Exercises for Chapter 13

14 Trigonometry

14.1 Introduction

14.2 Definitions and Results

14.3 Important Sine and Cosine Facts

14.4 The Other Trigonometric Functions

Examples for Chapter 14

Exercises for Chapter 14

15 ThreeDimensional Geometry

15.1 Introduction

15.2 Definitions and Results

Examples for Chapter 15

Exercises for Chapter 15

16 Functions

16.1 Introduction

16.2 Definitions

16.3 Graphs of Functions

16.4 Composition of Functions

Examples for Chapter 16

Exercises for Chapter 16

17 Logarithms

17.1 Introduction

17.2 Definitions and Results

Examples for Chapter 17

Exercises for Chapter 17

18 Complex Numbers

18.1 Introduction

18.2 Definitions

18.3 Important Complex Number Properties

Examples for Chapter 18

Exercises for Chapter 18

Solutions to Exercises

Solutions for Chapter 1: Arithmetic Ratios

Solutions for Chapter 2: Polynomials

Solutions for Chapter 3: Exponentials and Radicals

Solutions for Chapter 4: Defined Functions and Operations

Solutions for Chapter 5: Triangle Geometry

Solutions for Chapter 6: Circle Geometry

Solutions for Chapter 7: Polygons

Solutions for Chapter 8: Counting

Solutions for Chapter 9: Probability

Solutions for Chapter 10: Prime Decomposition

Solutions for Chapter 11: Number Theory

Solutions for Chapter 12: Sequences and Series

Solutions for Chapter 13: Statistics

Solutions for Chapter 14: Trigonometry

Solutions for Chapter 15: ThreeDimensional Geometry

Solutions for Chapter 16: Functions

Solutions for Chapter 17: Logarithms

Solutions for Chapter 18: Complex Numbers

Epilogue

Sources of the Exercises

... This book should be included in the library of anyone involved in preparing high school students to the AMC. Highly recommended.
J. T. Noonan, CHOICE 
Provides a wealth of opportunities for students to become experienced problem solvers.
David Webb, Penn State University 
An impressive problem solving primer.The book presents a wide variety of problems and problem solving strategies.
Richard Gibbs, Fort Lewis College