# American Mathematics Competitions

The **American Mathematics Competitions** (AMC) consist of a series of increasingly difficult tests for students in middle school and high school. The AMC sets the standard in the United States for talented high school students of mathematics. The AMC curriculum is both comprehensive and modern. AMC exams are so well designed that some top universities such as MIT now ask students for their AMC scores. "AMC" is also used as an abbreviation for American Math Contest, used to refer to the AMC 8, AMC 10, and AMC 12.

## Contents

## AMC Contests

In order of increasing difficulty, AMC competitions are

- AMC 8 — for students grades 8 and under.
- AMC 10 — for students grades 10 and under.
- AMC 12 — for students grades 12 and under.
- American Invitational Mathematics Examination (AIME) — high scorers from the AMC 10/12 exams.
- United States of America Mathematics Olympiad (USAMO) — high AIME and AMC scorers.

The top students on the USAMO are invited to participate in the Mathematical Olympiad Summer Program, where students train for possible inclusion on the U.S. IMO team.

## History

The AMC contest started in 1950, but unlike the modern AMC, it had 50 questions, was only available in New York, and was called the Annual High School Contest. Two years later, the Annual High School Contest was available nationwide. The amount of questions also decreased from 50 to 35 in a span of 17 years. In 1974, the Annual High School Contest was re-named the Annual High School Mathematics Examination. In 1983, the AHSC was renamed the American High School Examination, and the AIME was also introduced. In 1985, the American Junior High School Examination (Now known as the AMC 8) was introduced. In 2000, the AHSME was split into the AMC 10 and 12, reduced to only 25 questions, and 2 years later, the A and B version of the AMC's were introduced.

## Curriculum

AMC tests mathematical problem solving with arithmetic, algebra, counting, geometry, number theory, and probability, with far more cross-over between the subject areas than in nearly all classrooms. For example, most classrooms only have divisibility rules and little tidbits of number theory, and consider number theory as not a whole branch of mathematics but just a bunch of short cuts. The AMCs use number theory in much deeper (although elementary, without analysis) ways. Tests vary widely in difficulty. All three of the tests are designed such that no background in calculus, analysis, or any other higher mathematics is needed to take the exams.

## Chain

The AMC tests are the first in a series of test to select the American International Mathematical Olympiad team. High scoring students on the AMC 10 or 12 are invited to take the American Invitational Mathematics Examination. Students who have a high AMC index, or a high score on both the AMCs and the AIME, are invited to take the United States of America Mathematics Olympiad, the national Olympiad of the United States. There, many high scorers go to the Math Olympiad Summer Program, which is divided into three "colors" depending on how high one scored. The highest color, black, consists of twelve students, six of whom will form the United States' IMO team.

## Resources

### Links

### Recommended reading

- Introduction to Counting & Probability by Dr. David Patrick. Information
- Introduction to Geometry by Richard Rusczyk. Information
- The Art of Problem Solving Volume I by Sandor Lehoczky and Richard Rusczyk. Information.
- The Art of Problem Solving Volume II by Sandor Lehoczky and Richard Rusczyk. Information.

### Preparation Classes

- Art of Problem Solving offers many helpful online classes on topics covered by the AMC exams.
- AoPS holds many free Math Jams, some of which are devoted to discussing problems on the various AMC exams.
- EPGY offers AMC contest preparation classes.