The Math Olympiads in the United States are run by the American Math Competitions, also known as the AMC (www.unl.edu/amc). Hundreds of thousands of students take these exams, some for fun, some as a school requirement, and many because they love math and the spirit of competing.

The AMC selection process is divided into several different competitions ranging from a middle school to high school level. The ultimate aim of these is to select a six-member team to represent the US at the International Mathematics Olympiad (IMO). The different steps are (briefly) outlined below.

AMC 8 (American Mathematics Competition 8)
Format: 25 question multiple choice
Time Limit: 40 minutes
Target Audience: 7th - 8th grades
Full score: 25 points (1 for each question)
Qualification Process: None

This competition is held each November. It is not directly part of the selection process for the AMC, but is intended to provide a middle-school counterpart to the AMC 10 and AMC 12.

AMC 10 (American Mathematics Competition 10)
Format: 25 question multiple choice
Time Limit: 75 minutes
Target Audience: 9th - 10th grades
Full score: 150 points (6 for correct solution, 2.5 for no answer, 0 for incorrect solution)
Qualification Process: None

This competition is held each February. It forms the first step in the selection process for 10th graders and below, and provides an easier path to qualify for the AIME than the AMC 12 (but only for younger students, since older students are ineligible). Qualifying scores for the next round vary.

AMC 12 (American Mathematics Competition 12)
Format: 25 question multiple choice
Time Limit: 75 minutes
Target Audience: 9 - 12
Full score: 150 points (6 for correct solution, 2.5 for no answer, 0 for incorrect solution)
Qualification Process: None

This competition is held each February. It forms the main part of the initial selection process and is taken by the large majority of contestants. However, it covers more topics than the AMC 10 and contains more mathematical sophistication. Qualifying score for the next round is usually 100 points out of 150.

AIME (American Invitational Mathematics Examination)
Format: 15 question fill-in-the-blank (integer answers between 000 and 999)
Time Limit: 3 hours
Target Audience: 9th - 12th
Full score: 15 points (1 for each question)
Qualification Process: Performing well on the AMC 10 / AMC 12 / USAMTS

This is the second competition in the AMC sequence. Students scoring well on the AMC 10, AMC 12, or USAMTS are invited to take it in March. Covering a much wider array of mathematical ideas, the AIME is designed to segue students into problems closer to Olympiad level. Qualifying scores for the USAMO vary (usually from 7-9 points).

USAMO (USA Mathematical Olympiad)
Format: 6 question proof
Time Limit: 2 days, 4.5 hours/day
Target Audience: 9th - 12th
Full score: 42 points (7 points/problem)
Qualification Process: AMC 10 / AMC 12 / USAMTS + AIME

This is the third competition in the AMC sequence and can be reached only by the top 200-400 math students in the United States through a combination of AMC/AIME scores. The few problems are all proofs. Moving on to the next round, the MOSP, Math Olympiad Summer Program, requires a score of high teens to low 20s on the USAMO.

MOSP/TST (Math Olympiad Summer Program/Team Selection Test)
Format: 6 question proof
Time Limit: 2 days, 4.5 hours/day
Target Audience: 9th - 12th
Full score: 42 points (7 points/problem)
Qualification Process: AMC 10 / AMC 12 / USAMTS + AIME + USAMO

The training camp is reserved for the top math students in the nation, who have a shot at competing at the International Math Olympiad (IMO). In the past few years, many 9th graders were selected to this camp apart from the regular selection process. The Team Selection Test is the final test to determine the 6 members on the USA Team for the IMO.