United States of America Mathematical Olympiad

The United States of America Mathematical Olympiad (USAMO) is a prestigious mathematics competition held annually in the United States. Since its debut in 1972, it has served as the final round of the AMC series of contests. Top scorers on the USAMO usually represent the United States at the International Mathematical Olympiad.

Eligibility

In order to be eligible to take the USAMO, a participant must be either a citizen of the United States or a legal permanent resident. This rule was put in place after at least one instance of a Canadian student winning the USAMO in the early 90s, and the rule has been relaxed again in recent years to allow Canadians and international students to take the test. It has never been the case that anyone but a resident or citizen could join the American IMO team. In addition, all participants, regardless of geographic location, must meet qualification indices determined by previous rounds of the AMC contests. Entry to the USAMO is by invitation only.

Participant selection process

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.

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.

Source: American Mathematics Competitions

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.

Source: American Mathematics Competitions

Post-2006

Beginning in 2006, the USAMO will be expanded to include approximately 375 students due to a proposal and sponsorship from the Art of Problem Solving website:

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 diffi culty, 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.*

Source: American Mathematics Competitions

• Statement 7 above (quoted from the AMC website) 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 distiction 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.

Recent qualification indices

Year	11th grade and above	10th grade and below
2006	217.5	8/15*
2005	233 (AIME I); 220.5 (AIME II)	9/15 on AIME
2004	210	7/15 on AIME
2003	226	8/15 on AIME
2002	210	6/15 on AIME

• 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 regardless making the total 431.

Test format and scoring

Post-2002

Since 2002, the USAMO has been a six-question, nine-hour mathematical proof examination spread out over two days. (The IMO uses the same format.) On each day, four and a half hours are given for three questions.

Each question is graded on a scale from 0 to 7, with a score of 7 representing a proof that is mathematically sound. Thus, a perfect score is 42 points. The number of perfect papers each year has varied depending on test difficulty. Regardless, the top 12 scorers are all named contest winners.

1996 to 2001

The test consisted of two three-problem sets. Three hours were given for each set; one set was given in the morning, and the other in the afternoon.

1995 and earlier

The test consisted of five problems to be solved in three and a half hours.

Test procedures

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. In 2004 and 2005, however, funding was found to send 30 rising sophomores to MOP as well, in a program popularly called "Red MOP."

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 Program (also known as "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.

See also

• American Mathematics Competitions
• List of mathematics competitions

External links

• The Official AMC Home Page - Contains rules, problems, qualifiers, and winners for each year since 1999
• An archive of all USAMO problems, as well as other olympiad-style math problems
• The IMO Compendium - huge collection of problems from mathematical competitions, and the most complete collection of IMO shortlists and longlists.