United States of America Mathematical Olympiad

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The United States of America Mathematical Olympiad (USAMO) is a highly selective 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. The United States of America Junior Mathematical Olympiad (USAJMO) was introduced in 2010 to recognize top scorers based on their AMC10-based index. Qualifying for the USAMO is considered one of the most prestigious awards for high school students in the United States, with only 264 students qualifying in 2013 out of over 350,000 students competing.[1] Top scorers on the USAMO are invited to the Mathematical Olympiad Summer Program and represent the United States at the International Mathematical Olympiad.

Eligibility[edit]

In order to be eligible to take the USAMO, a participant must be either a U.S. citizen or a legal resident of the United States or Canada.[2] Only U.S. 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[edit]

The USAMO (and the USAJMO since 2010) is restricted to approximately 500 (250 prior to 2006) participants combined 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.

2011[edit]

In 2011 we will have slightly revised qualification rules for the USA Mathematical Olympiad and USA Junior Mathematical Olympiad. The goal is to select approximately 500 students total for the two Olympiads, split approximately 270 for the USAMO and 230 for the USAJMO respectively. Selection for the 2011 USA Mathematical Olympiad (USAMO) and 2011 USA Junior Mathematical Olympiad (USAJMO) will be made according to the following rules:[3]

1.U.S. citizens and students residing in the United States and Canada (with qualifying scores) are eligible to take the USAMO and USAJMO.

2.Selection to the USAMO will be based on the USAMO index which is defined as AMC 12 Score + 10 * AIME Score. Selection to the USAJMO will be based on the USAJMO index which is defined as AMC 10 Score + 10 * AIME Score.

3.Only AMC 12 A or AMC 12 B takers who are U.S. citizens and students residing in the United States and Canada will be eligible for the USAMO.

4.Only AMC 10 A or AMC 10 B takers who are U.S. citizens and students residing in the United States and Canada will be eligible for the USAJMO. This automatically limits Junior Math Olympiad participation to 10th graders and below. Students who take ONLY the AMC 10 test, whether AMC 10 A or AMC 10 B or both, will NOT be eligible for the USAMO regardless of their score on the AMC 10 or the AIME.

5.The approximately 260-270 individual students with the top AMC 12 based USAMO indices will be invited to take the USAMO. These indices will be selected from the pool of AMC 12 takers with an AIME score.

6.The approximately 230-240 individual students with the top AMC 10 based USAMO indices will be invited to take the USAJMO. These indices will be selected from the pool of AMC 10 takers with an AIME score after removing students who also took an AMC 12 test and qualified for the USAMO in rule 5. This means young students MUST take the USAMO if they qualify through an AMC 12 index.

7.We will select the student with the numerically largest index, whether AMC 10 based USAJMO index or AMC 12 based USAMO index, from each US state not already represented in either the USAMO or the USAJMO. The student will be invited to the USAMO if the numerically highest index in the state is AMC 12 based, and invited to the USAJMO if the index is AMC 10 based.

2010[edit]

Starting in 2010, the USA Mathematical Olympiad is split into two parts. The USA Mathematical Olympiad will be administered to approximately 270 students, mostly selected from top ranking AMC12 participants. The AMC10 only participants will take part in USA Junior Mathematical Olympiad.[4]

1.Selection to the USAMO and JMO will be based on the USAMO index which is defined as AMC score + 10 * AIME score.

2.Only AMC 12A or AMC 12B takers are eligible for the USAMO (with the slight exception mentioned in item 5 below).

3.Only AMC 10A and AMC 10B takers are eligible for the JMO. (This automatically limits Junior Math Olympiad participation to 10th graders and below.)

4.Approximately the top 260 AMC12 based USAMO indices will be invited to the USAMO.

5.In order to find unrecognized young talent, AMC 10 takers who score 11 or more on the AIME will be invited to the USAMO. (In 2008 and 2009 this was 5 or 6 students).

6.Select the top index from any state not already represented in the USAMO.

7.Approximately the top 220-230 students with AMC10 based USAMO indices and not already selected to the USAMO via an AMC12 based index will be invited to the JMO.

Source: [3]

2008[edit]

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 year'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, 2,696 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[edit]

Beginning in 2006, the USAMO was 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 year'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[edit]

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[edit]

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[edit]

Year AMC 12 AMC 10 Total number of qualifiers
2014 211.5 for USAMO 211.0 for USAJMO 266 USAMO; 231 USAJMO
2013 209.0 for USAMO 210.5 for USAJMO 264 USAMO; 231 USAJMO
2012 204.5 for USAMO 204.0 for USAJMO 268 USAMO; 233 USAJMO
2011 188.0 (AIME I); 215.5 (AIME II) for USAMO 179.0 (AIME I); 196.5 (AIME II) for USAJMO 282 USAMO; 222 USAJMO
2010 208.5 (USAMO); 204.5 (USAMO—11th and 12th) 188.5 (USAJMO) or 11/15 on AIME (USAMO) 328 USAMO; 235 USAJMO
2009 201.0 7/15 on AIME AND 215.0+ on index 514
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 212 (12th); 204 (11th) 9th grade: 7/15 on AIME AND 164+ on index; 10th grade: 8/15 on AIME AND 174+ on index 239

Test format and scoring[edit]

Post-2002[edit]

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[edit]

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[edit]

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[edit]

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[edit]

Here are the subjects on the test in different years by problem number (i.e. what subject each problem was from). Calculus, although allowed, is never required in solutions.

2014:

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

2013:

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

2012:

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

2011:

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

2010:

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

2009:

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

2008:

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

2007:

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

2003:

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

See also[edit]

References[edit]

  1. ^ "American Mathematics Competitions". The Mathematical Association of America. 2014. Retrieved 2014-04-06. 
  2. ^ "United States of America Mathematical Olympiad - USAMO". The Mathematical Association of America. 2006. Retrieved 2006-11-29. 
  3. ^ [1]
  4. ^ [2]

External links[edit]