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The USAMO (or USA Mathematical Olympiad) is a test that, since 1972, has been given to the top approximately 250 students in the AMC contests.
Each year, the top 12 scorers on the USAMO are considered for selection to the IMO team for the United States. The students are trained at the Mathematical Olympiad Summer Program (also known as "MOSP" or "MOP") in Lincoln, Nebraska, and then six are selected to the team. The next approximately 18 high scorers, usually excluding high school seniors, are also invited to MOP.
Considering that the roughly 250 participants in the USAMO each year are selected from a pool of over 200 000 participants in the AMC contests, simply participating is considered an extremely significant achievement for an American student.
Qualification for the USAMO is based on a student's combined scores for the AMC and AIME contests. This combined score, known as the USAMO Index, is equal to the student's best AMC score for that year added to 10 times the student's AIME score (such that the AMC and AIME both are worth a maximum 150 points).
In 2004, any student whose USAMO Index was at least 210.0 was invited to take the USAMO. Additionally, any student in 10th grade or below who scored at least 7 on the AIME was also invited to take the USAMO. In 2003, the minimum USAMO Index was 226.0, and the minimum AIME score for 10th-graders was 8. In 2002, the minimum USAMO Index was 210.0, and the minimum AIME score for 10th-graders was 6. Before 2002, the USAMO selection system was much more complicated and unpredictable.
From 1972 to 1995, the test consisted of five problems to be solved in three and a half hours.
From 1996 to 2001, the test consisted of two papers of three problems each, with each set of three problems to be solved in three hours. The first three problems were given in the morning while the second three problems were given in the afternoon.
Since 2002, the test has consisted of two papers of three problems each, with each set of three problems to be solved in four and a half hours. Each part is given on a separate day (usually on a Tuesday and Wednesday in late April or early May). This six-problem, nine-hour format is the same as the one used at the IMO.
Each problem requires a formal mathematical proof of a statement. Each proof is graded on a scale from 0 to 7, with 7 representing a proof that is mathematically sound. Thus a perfect score is 42 (7 points on each of 6 problems). This perfect score is rare, and achieved only by the most extraordinary of students.
In 2004, the median score of the 265 USAMO participants was 4 points (out of 42). There was exactly one perfect score, by Tiankai Liu, who at the time was a senior at Phillips Exeter Academy in New Hampshire.
In most years, students have taken the USAMO at their respective high schools. Prior to 2002, the problems were mailed to the schools in sealed envelopes, not to be opened before the appointed time on the test day. Since 2002, test problems have been posted on the AMC website (see links below) fifteen minutes prior to the official start of the test. Student responses are then faxed back to the AMC office at the end of the testing period.
In 2002, the Akamai Foundation, as a major sponsor of the American Mathematics Competitions, invited all USAMO participants to take the test at a central event at MIT in Cambridge, Massachusetts, all expenses paid. In addition, Akamai invited all 2002 USAMO participants who were not high school seniors (approximately 160 students) to take part in an enlarged MOP program. Since holding this central event every year would be prohibitively expensive, it has been discontinued.
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