This book is a celebration of mathematical problem solving at the level of the high school American Invitational Mathematics Examination. There is no other book on the market focused on the AIME. It is intended, in part, as a resource for comprehensive study and practice for the AIME competition for students, teachers, and mentors. After all, serious AIME contenders and competitors should seek a lot of practice in order to succeed. However, this book is also intended for anyone who enjoys solving problems as a recreational pursuit. The AIME contains many problems that have the power to foster enthusiasm for mathematics–the problems are fun, engaging, and addictive. The problems found within these pages can be used by teachers who wish to challenge their students, and they can be used to foster a community of lovers of mathematical problem solving!

There are more than 250 fully-solved problems in the book, containing examples from AIME competitions of the 1980’s, 1990’s, 2000’s, and 2010’s. In some cases, multiple solutions are presented to highlight variable approaches. To help problem-solvers with the exercises, the author provides two levels of hints to each exercise in the book, one to help stuck starters get an idea how to begin, and another to provide more guidance in navigating an approach to the solution.

Topics include algebra, combinatorics, probability, number theory, sequences and series, logarithms, trigonometry, complex numbers, polynomials, and geometry. Many AIME problems involve using several of these mathematical disciplines simultaneously. To make the book indeed “gentle,” the author has prepared well-motivated, detailed, clear solutions to all examples and exercises in the book. He has not always opted for the shortest solution, but rather, he has presented solutions that a typical student might say “I could have come up with that!” The book avoids excessive reliance on heavy machinery, deep theorems, or undue abstraction.

Preface

List of Abbreviations and Symbols

1. Algebraic Equations

2. Combinatorics

3. Probability

4. Number Theory

5. Sequences and Series

6. Logarithmic and Trigonometric Functions

7. Complex Numbers and Polynomials

8. Plane Geometry

9. Spatial Geometry

10. Hints for the Exercises

11. Solutions to Exercise Sets

Answers to All Exercises

About the Author

Index

### About the Author

Scott A. Annin received his B.S. in mathematics and physics from the University of Nebraska in 1995. He earned his Ph.D. in mathematics from the University of California at Berkeley in 2002, specializing in noncommutative algebra. He has been a professor at California State University in Fullerton ever since. Annin has received several teaching awards at Fullerton, including university-wide honors in 2008 and 2015. In 2009, he received the Mathematical Association of America’s Henry L. Alder Award for Outstanding Teaching by a Beginning Faculty Member in Mathematics. In addition to being a former AIME contestant himself (while a student at Lincoln East High School in Lincoln, Nebraska), Annin had led workshops for many AIME contestants. He has also increased awareness of the AMC program among high school teachers at both the state and national levels at math education and MAA meetings. Annin has also co-authored the widely used undergraduate mathematics text *Differential Equations and Linear Algebra* with Stephen Goode.