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First Steps for Math Olympians: Using the American Mathematics Competitions
 
First Steps for Math Olympians
MAA Press: An Imprint of the American Mathematical Society
Now available in new edition: PRB/34
First Steps for Math Olympians
Click above image for expanded view
First Steps for Math Olympians: Using the American Mathematics Competitions
MAA Press: An Imprint of the American Mathematical Society
Now available in new edition: PRB/34
  • Book Details
     
     
    Problem Books
    Volume: 162006; 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 problem-solving 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 self-study or to give people who want to help students prepare for mathematics exams easy access to topic-oriented 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 Three-Dimensional 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: Three-Dimensional 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 162006; 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 problem-solving 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 self-study or to give people who want to help students prepare for mathematics exams easy access to topic-oriented 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 Three-Dimensional 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: Three-Dimensional 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
Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.