# First Steps for Math Olympians: Using the American Mathematics Competitions

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*J. Douglas Faires*

MAA Press: An Imprint of the American Mathematical Society

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!

#### Reviews & Endorsements

… 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

# Table of Contents

## First Steps for Math Olympians: Using the American Mathematics Competitions

- cover cover11
- copyright page ii3
- title page iii4
- Contents vii8
- Preface xiii14
- 1 Arithmetic Ratios 124
- 2 Polynomials and their Zeros 932
- 3 Exponentials and Radicals 1942
- 4 Defined Functions and Operations 2952
- 5 Triangle Geometry 3760
- 6 Circle Geometry 5578
- 7 Polygons 7194
- 8 Counting 85108
- 9 Probability 97120
- 10 Prime Decomposition 109132
- 11 Number Theory 115138
- 12 Sequences and Series 127150
- 13 Statistics 135158
- 14 Trigonometry 143166
- 15 Three-Dimensional Geometry 155178
- 16 Functions 167190
- 17 Logarithms 179202
- 18 Complex Numbers 187210
- Solutions to Exercises 197220
- Solutions for Chapter 1: Arithmetic Ratios 197220
- Solutions for Chapter 2: Polynomials 202225
- Solutions for Chapter 3: Exponentials and Radicals 206229
- Solutions for Chapter 4: Defined Functions and Operations 209232
- Solutions for Chapter 5: Triangle Geometry 214237
- Solutions for Chapter 6: Circle Geometry 219242
- Solutions for Chapter 7: Polygons 225248
- Solutions for Chapter 8: Counting 230253
- Solutions for Chapter 9: Probability 235258
- Solutions for Chapter 10: Prime Decomposition 241264
- Solutions for Chapter 11: Number Theory 245268
- Solutions for Chapter 12: Sequences and Series 249272
- Solutions for Chapter 13: Statistics 255278
- Solutions for Chapter 14: Trigonometry 260283
- Solutions for Chapter 15: Three-Dimensional Geometry 266289
- Solutions for Chapter 16: Functions 273296
- Solutions for Chapter 17: Logarithms 280303
- Solutions for Chapter 18: Complex Numbers 284307

- Epilogue 291314
- Sources of the Exercises 295318