The United States of America Mathematical Olympiad (USAMO) is a prestigious high school 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 resident of the United States or Canada. Only US residents and citizens may 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 500 (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.

2008

Selection for the USAMO will be made according to the following rules:

1. The goal is to select about 500 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 330 highest USAMO indices of students taking the AMC 12A or AMC 12B contest.

4. The lowest AIME score among those 330 first selected will determine a floor value. The second selection of approximately 160 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 160 young students with a score above the floor value, then approximately 160 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. In advising young students (in grade 10 or below) who desire to be selected for the USAMO whether to take the AMC 12 contest or the AMC 10 contest, please be aware of the following facts:

a. In 2007, among 506 students invited to take the USAMO, 229 were in 10th grade and below. Those students had scored 6 or greater on the AIME.

b. Among those 229 students, 87 had their AIME qualifying high score based on the AMC 12 and 142 had their AIME qualifying high score based on the AMC 10.

c. In 2007, among 8,312 students who took the AIME, 2,694 were in grades 10 and below. Of those, 998 qualified for the AIME from the AMC 12 and 1,698 qualified from the AMC 10.

2006-2007

Beginning in 2006, the USAMO will be expanded to include approximately 500 students (around 430 were actually invited, read below) due to a proposal and sponsorship from the Art of Problem Solving website:

1. The goal is to select about 500 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.*

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 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.

Source: American Mathematics Competitions

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

Recent qualification indices

Year	12th grade and below	10th grade and below	Total number of qualifiers
2008	204.0	6/15 on AIME AND 202.5+ on index	503
2007	197.5	6/15 on AIME AND 181.0+ on index	505
2006	217	8/15 on AIME	432
2005	233 (AIME I); 220.5 (AIME II)	9/15 on AIME	259
2004	210	7/15 on AIME	261
2003	226	8/15 on AIME	250
2002	210	6/15 on AIME	326
2001	213	7/15 on AIME	268
2000	164	9th grade: 7/15 on AIME 10th grade: 8/15 on AIME	239

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.

The scale of 0 to 7 goes as follows:

• 0 - No work, or completely trivial work
• 1-2 - Progress on the problem, but not completely solved
• 3-4 - All steps are present, but may lack clarity. (These scores are very rare.)
• 5-6 - Complete solution with minor errors
• 7 - Perfect solution

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 (9:00-12:00), and the other in the afternoon (1:00-4:00).

1995 and earlier

The test consisted of five problems to be solved in three and a half hours (earlier, three hours). Each problem was worth 20 points, for a perfect score of 100.

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 Mathematical Olympiad Program (also known as "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 freshmen 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 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.

Exam content

Here are the subjects on the test in different years by problem number (i.e. what subject each problem was from):

2008:

1. Number Theory
2. Geometry
3. Graph Theory
4. Combinatorics

2007:

1. Number Theory/Combinatorics
2. Geometry
3. Combinatorics
4. Graph Theory
5. Number Theory
6. Geometry

2006:

1. Number Theory
2. Algebra/Combinatorics
3. Number Theory/Algebra
4. Algebra
5. Algebra/Combinatorics
6. Geometry

2005:

1. Number Theory/Graph Theory
2. Number Theory
3. Geometry
4. Geometry/Algebra
5. Combinatorics
6. Algebra

2004:

1. Geometry/Inequalities
2. Algebra
3. Geometry
4. Combinatorics
5. Inequalities
6. Geometry

2003:

1. Number Theory
2. Geometry/Algebra
3. Algebra
4. Geometry
5. Inequalities
6. Combinatorics

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.
• MathLinks - a large community of Olympiad problems solvers from around the globe.
• How to Write a Proof
• An archive of USAMO Problems

References