There are 3 levels of American Mathematics Competition. AMC 8 and AMC 10, and AMC 12.
The AMC 8 is a 25-question multiple-choice contest for middle-school students in grades 8 and below who are under 14.4 years of age on the day of the contest. The material covered is the middle school mathematics curriculum. Topics include probability, estimation, percentages, spatial visualization, everyday applications and reading and interpreting graphs. The AMC 8 is for students in the sixth through eighth grade, although accelerated fourth and fifth graders can also take part. The AMC 8 is 25 questions in length and is multiple choice with no penalty for guessing. The contest takes only 40 minutes. A student's score is the number of problems correctly answered. The AMC 8 is administered in schools in November.
The AMC 12 covers high school mathematics, and is for students in high school who are under 19.5 years of age on the day of the contest. The AMC 10 covers mathematics normally associated with grades 9 and 10 and is for students under 17.5 years of age on the day of the contest who are not enrolled in grades 11, 12 or equivalent. Both contests are given in a convenient 75-minute interval, is 25 questions in length, with approximately 12 questions in common to both contests. The AMC 10 and AMC 12 are administered in schools in February.
Taking the AMC 10/12 will pave the way for your kid to take the USA Junior Mathematical Olympiad competition (USAJMO) and the USAMathematics Olympiad competition (USAMO), which can eventually lead to joining the United States International Mathematics Olympiad team.
The path to the USAMO and USAJMO begins with either the AMC 10 or AMC 12. Approximately the top 2.5% of AMC 10 students and top 5% of AMC 12 students qualify to take the American Invitation Mathematics Examination (AIME). Students with the top 270 10*AIME + AMC 12 scores are then invited to take the USAMO. The 230 students with the 10*AIME + AMC 10 scores are then invited to take the USAJMO. Taking it a step farther, the top 12 students named USAMO winners take the Team Selection Test (TST) to determine the 6 member United States International Mathematics Olympiad team.
For more information, visit http://www.maa.org/math-competitions/about-amc.