The USAMO is restricted to approximately 375 (250 prior to 2006) participants each year. To keep this quota constant, the AMC Committee uses a selection process, which has seen a number of revisions in the exam's history.
Post-2006
Beginning in 2006, the USAMO will be expanded to include approximately 375 students (around 430 were actually invited, read below) due to a proposal and sponsorship from the Art of Problem Solving web site:
1. The goal is to select about 375 of the top scorers from this years’s AIME and AMC 12A, AMC 12B, AMC 10A and AMC 10B contests to participate in the USAMO.
2. Selection will be based on the USAMO index which is defined as 10 times the student’s AIME score plus the student’s score on the AMC 12 or the AMC 10.
3. The first selection will be the approximately 240 highest USAMO indices of students taking the AMC 12A or AMC 12B contest.
4. The lowest AIME score among those 240 first selected will determine a floor value. The second selection of approximately 120 USAMO participants will be among students in the 10th grade and below who received an AIME score at least as high as the floor value. If there are more than 120 young students with a score above the floor value, then approximately 120 students will be selected from this group by using the USAMO index.
5. The student with the highest USAMO index from each state, territory, or U.S. possession not already represented in the selection of the first and second groups will be invited to take the USAMO.
6. To adjust for variations in contest difficulty, the number of students selected from A & B contests will be proportional to the number of students who took the A & B Contests.
7. The selection process is designed to favor students who take the more mathematically comprehensive AMC 12A and AMC 12B contests.*
* Statement 7 above (quoted from the AMC web site) has recently come under controversy. During the selection for the 2006 USAMO, students who qualified by the floor value (in grades seven through ten) were qualified based on AMC scores as well (see * below) as their AIME scores, yet no distinction was made between the AMC 12 contest and the generally easier AMC 10 contest, giving those who took the AMC 10 an advantage over those who took the AMC 12. Students in grades seven through ten who were in the first selection of qualifiers (see 3. above) would still have qualified even if they had taken the AMC 10, except in the rare case that they set the floor themselves, making the AMC 12 still non-advantageous.
2002-2005
Since 2002, the following set of guidelines have been adopted for use in determining each year's USAMO participants:
1. The goal is to select about 250 of the top scorers from the prior AIME and AMC 12A, AMC 12B, AMC 10A and AMC 10B contests to participate in the USAMO.
2. Selection will be based on the USAMO index which is defined as 10 times the student’s AIME score plus the student’s score on the AMC 12 or the AMC 10.
3. The first selection (consisting of participants from all grade levels) will be the approximately 160 highest USAMO indices of students taking the AMC 12A or AMC 12B contest.
4. The lowest AIME score among those 160 first selected will determine a floor value. The second selection of USAMO participants will be from the highest USAMO indices among students in grades seven through ten who got an AIME score at least as high as the floor value. To note, during 2002-2005 period, this included all students in grades seven through ten.
5. The student with the highest USAMO index from each state, territory, or U.S. possession not already represented in the selection of the first and second groups will be invited to take the USAMO.
6. To adjust for variations in contest difficulty, the number of students selected from A & B contests will be proportional to the number of students who took the (A & B) Contests.
7. The selection process is designed to favor students who take the more mathematically comprehensive AMC 12A and AMC 12B contests.
2001 and earlier
Prior to 2001, the following guidelines were used:
* First Group: The top 120 students.
* Second Group: The next 20 students in grades 11 and below.
* Third Group: The next 20 students in grades 10 or below.
* Fourth Group: The next 20 students in grades 9 or below.
* Fifth Group: One student from each state, one student from the combined U.S.A. Territories, and one student from the APO/FPO schools- if not represented in the first four groups.
In 2006, the number of USAMO qualifiers was only to be 375 and a tiebreaker was distributed amongst those with AIME scores of 8, however, due to reconsideration by the CAMC, anyone with a floor of 8 qualified, making the total 432.