A Gentle Introduction to the American Invitational Mathematics Exam is a celebration of mathematical problem solving. It is written at the level of the high school American Invitational Mathematics Exam (AIME). The book is intended, in part, as a resource for comprehensive study and practice for the AIME competition to be used by students, coaches, teachers, and mentors preparing for the exam.

However, it is also written for those who enjoy solving problems, exploring recreational mathematics, and sharpening critical thinking and problem solving skills. It is not intended just for the exceptionally brilliant or for those who are intensely competitive, even though the AIME is a competition. Anyone who enjoys thinking about interesting problems in mathematics should enjoy this book.

One of my aims inwriting this bookwas tomake the problems accessible and interesting to a wide audience and to lure in those who feel perhaps that the AIME problems are not for them. Therefore, even if you are not involved in the AIME as a student or mentor, as long as you have an interest in mathematical problem solving and a desire to explore fun mathematical problems at the high school level, you should find this book engaging. You may surprise yourself with what you can do!

The AIME is the second in a series of exams known collectively as the American Mathematics Competitions (AMC). We will describe the series of exams comprising the AMC, including the focus of this book, the AIME, and offer some of advice and encouragement. After that, we will survey a large compendium of topics and problems from past AIMEs that will form the basis for learning, discussion, and discovery.

The American Mathematics Competitions

In 1950, the Mathematical Association of America launched the AMC as a means of “strengthening themathematical capabilities of our nation's youth”. Each year, over 350,000 students in roughly 6,000 schools participate in the MAA's AMC competitions.

The AMC consists of a series of examinations covering a broad spectrum of topics. The problems appearing in them can all be solved by using precalculus methods. The exams become progressively more difficult, beginning with the AMC 10 and AMC 12 in February each year, followed by the AIME in March or early April, the United States of America Mathematical Olympiad (USAMO) later in April, and the International Mathematics Olympiad (IMO) in July.