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United States of America Mathematical Olympiad
United States of America Mathematical Olympiad
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 American Mathematics Competitions. In 2010, it split into the USAMO and the United States of America Junior Mathematical Olympiad (USAJMO).

Top scorers on both six-question, nine-hour mathematical proof competitions are invited to join the Mathematical Olympiad Program to compete and train to 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.[1] Only US citizens and permanent residents could be invited to the USAMO until 2003,[2] other students legally residing in the US can be invited since 2004.[3] Starting from IMO 2022, only U.S. permanent residents and citizens may join the American IMO team.[4] 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.

History

[edit]

The USAMO was created in 1972 at the initiative of Nura D. Turner and Samuel L. Greitzer,[5][6][7] and served as the next round to the AHSME until 1982. In 1983, the American Invitational Mathematics Examination was introduced as a bridge between the AHSME and USAMO. In 2010, the USAMO split into the USAMO and USAJMO.[8]

Historical 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 competition's history.

Present

[edit]

AMC 12 based indices are determined by taking AMC 12 Score + 20*(AIME Score). AMC 10 based indices are determined by taking AMC 10 Score + 20*(AIME Score). Cutoffs, based on AMC 12 indices, are determined so that approximately 260-270 students qualify for the USAMO. Cutoffs, based on AMC 10 indices, are determined so that approximately 230-240 students qualify for the USAJMO. If a student took the AMC 10 and 12 (i.e. AMC 10A and 12B or AMC 12A and 10B) and qualified for both the USAMO and USAJMO, the student must take the USAMO. In 2020, due to grading constraints caused by the COVID-19 pandemic, lower numbers of students were admitted (223 USAMO qualifiers and 158 USAJMO qualifiers). In addition, students who qualify for the AIME through scoring at least 68/75 on the United States of America Mathematical Talent Search can qualify for the USAMO by scoring at least 11 on the AIME or the USAJMO by scoring 9-10 on the AIME, provided the student is eligible.[9]

2011

[edit]

Since 2011, the goal has been to select approximately 500 students total for the two Olympiads where 270 students qualify for the USA Mathematical Olympiad (USAMO) and 230 students qualify for the 2011 USA Junior Mathematical Olympiad (USAJMO). Selection for the USAMO and USAJMO are made according to the following rules:

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.[10]

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: [1]

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.
  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]

From 1998 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
2020* 12A + (10*AIME I): 233.5 and above

12A + (10*AOIME): 234 and above

12B + (10*AIME I): 235 and above

12B + (10*AOIME): 234.5 and above

10A + (10*AIME I): 229.5 and above

10A + (10*AIME II): 233.5 and above

10B + (10*AIME I): 230 and above

10B + (10*AIME II): 229.5 and above

223 USAMO; 158 USAJMO
2019 12A + (10*AIME I): 220 and above

12A + (10*AIME II): 230.5 and above

12B + (10*AIME I): 230.5 and above

12B + (10*AIME II): 236 and above

10A + (10*AIME I): 209.5 and above

10A + (10*AIME II): 216.5 and above

10B + (10*AIME I): 216 and above

10B + (10*AIME II): 220.5 and above

275 USAMO; 235 USAJMO

2018 12A + (10*AIME I): 215 and above

12A + (10*AIME II): 216 and above

12B + (10*AIME I): 235 and above

12B + (10*AIME II): 230.5 and above

10A + (10*AIME I): 222 and above

10A + (10*AIME II): 222 and above

10B + (10*AIME I): 212 and above

10B + (10*AIME II): 212 and above

242 USAMO; 156 USAJMO

2017 12A + (10*AIME I): 225.5 and above

12A + (10*AIME II): 221 and above

12B + (10*AIME I): 235 and above

12B + (10*AIME II): 230.5 and above

10A + (10*AIME I): 224.5 and above

10A + (10*AIME II): 219 and above

10B + (10*AIME I): 233 and above

10B + (10*AIME II): 225 and above

280 USAMO; 208 USAJMO

2016 220.0 for USAMO with AIME I, 205.0 for USAMO with AIME II 210.5 for USAJMO with AIME I, 200.0 for USAJMO with AIME II 311 USAMO; 198 USAJMO
2015 219.0 for USAMO with AIME I, 229.0 for USAMO with AIME II 213.0 for USAJMO with AIME I, 223.5 for USAJMO with AIME II
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

*In 2020, the AIME I took place as normal on March 11, 2020. However, the escalating COVID-19 pandemic which had just shut down most U.S. Schools forced the postponement of the AIME II, which was scheduled for March 19, and the USA(J)MO which was scheduled for mid-April. Both competitions were eventually rescheduled in June as online competitions which students participated in at home and were renamed as the AOIME (American Online Invitational Mathematics Examination) and the USO(J)MO (United States Online (Junior) Mathematical Olympiad) respectively. They were sponsored by Art of Problem Solving (AoPS).

Test format and scoring

[edit]

Post-2002

[edit]

Since 2002, the USAMO has been a six-question, nine-hour mathematical proof competition 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. The top scorers are published by MAA and the recognition changes over the years. Please refer to Award section for more details.

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 sophomores and juniors to MOP as well, in a program popularly called "Red MOP."

Top USAMO and USAJMO participants are selected to MOP through multiple criteria of entry. As of 2016, the IMO team members, the next approximately 18 non-graduating USAMO students, the next approximately 12 USAMO students in 9th or 10th grade, the top approximately 12 students on the USAJMO, as well as some varying number of female contestants from the USAMO or USAJMO are invited to MOP, with middle school students invited on a case-by-case basis.

Exam content for USAMO

[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.

2023:
  1. Geometry
  2. Algebra
  3. Combinatorics
  4. Number theory
  5. Combinatorics
  6. Geometry

2022:

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

2021:

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

2020:

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

2019:

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

2018:

  1. Algebra
  2. Algebra
  3. Number theory
  4. Number theory
  5. Geometry
  6. Algebra

2017:

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

2016:

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

2015:

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

2014:

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

2013:

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

2012:

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

2011:

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

2010:

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

2009:

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

2008:

  1. Number theory
  2. Geometry
  3. Combinatorics
  4. Combinatorics
  5. Combinatorics
  6. Combinatorics

2007:

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

2006:

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

2005:

  1. Combinatorics
  2. Number theory
  3. Geometry
  4. Combinatorics
  5. Combinatorics
  6. Number theory

2004:

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

2003:

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

2002:

  1. Combinatorics
  2. Algebra
  3. Algebra
  4. Algebra
  5. Combinatorics
  6. Combinatorics

2001:

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

2000:

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

Exam Content for USAJMO

[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.

2017:

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

2016:

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

2015:

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

2014:

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

2013:

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

2012:

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

2011:

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

2010:

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

USAMO Awards

[edit]
  • USAMO: 2022 -
  1. USAMO Gold Award: at least approximately 6% of competitors (approximately 16)
  2. USAMO Silver Award: at least approximately 12% of competitors (approximately 32)
  3. USAMO Bronze Award: at least approximately 18% of competitors (approximately 48)
  4. USAMO Honorable Mention: competitors who score 14 points or more, provided they did not receive a different award. (year 2023 - present) [11]
  • USAMO: 2021 and before
  1. USAMO Winners: the top 12 performers
  2. USAMO Honorable Mentions: the next approximately 12 performers

USAJMO Awards

[edit]
  • USAJMO: 2024
  1. USAJMO Top Honors: Top 9 competitors
  2. USAJMO Honors: the next 48 competitors
  3. USAJMO Honorable Mention: competitors who score 14 points or more, provided they did not receive a different award.
  • USAJMO: 2023
  1. USAJMO Top Winners: at least 12 highest scoring competitors (actual number: 13)
  2. USAJMO Winners: the next approximately 20% of competitors (actual number: 36)
  3. USAJMO Honorable Mention: competitors who score 14 points or more, provided they did not receive a different award.
  • USAJMO: 2022
  1. USAJMO Winners: approximately top 70 competitors
  2. USAJMO Honorable Mentions: the next approximately 70 competitors
  • USAJMO: 2021 and before
  1. USAJMO Winners: the top 12 performers
  2. USAJMO Honorable Mentions: the next approximately 12 performers

See also

[edit]

References

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

United States of America Mathematical Olympiad

View on Grokipedia
from Grokipedia
The United States of America Mathematical Olympiad (USAMO) is an annual, invitational proof-based mathematics competition for high school students in the , consisting of a six-question, two-day, nine-hour examination that challenges participants on advanced topics in , , , and . It selects approximately 250 top-performing students based on combined scores from the (AMC) 12 and (AIME), with a parallel USA Junior Mathematical Olympiad (USAJMO) for about 250 top scorers from the AMC 10 and AIME since 2010. Established in 1972 at the initiative of mathematicians Nura D. Turner and Samuel L. Greitzer to identify and nurture exceptional mathematical talent, the USAMO was endorsed by the (MAA) and first administered on May 9 of that year as a five-problem test, evolving into its current format to better prepare participants for international competition. The competition has since become a cornerstone of the (AMC) series, which begins with broader exams like the AMC 10 and AMC 12 to funnel elite students into the USAMO and USAJMO. High scorers on the USAMO and USAJMO advance to the Mathematical Olympiad Program (), a summer where further testing determines the six-member U.S. team for the (IMO), the world's premier high school mathematics contest held annually since 1959. The U.S. team, selected through this rigorous process, has achieved notable success at the IMO, including first-place finishes in 2015, 2016, 2018, 2019, and 2024, reflecting the competition's role in fostering world-class mathematical problem-solving skills.

Overview

Eligibility and Qualification

The Mathematical Olympiad (USAMO) and the United States of America Junior Mathematical Olympiad (USAJMO) are invitational proof-based competitions administered by the (MAA) for high-achieving students in the United States and . Eligibility requires full-time enrollment in an accredited school or homeschool program in these countries, with participants selected based on performance in prerequisite exams. For the USAMO, eligible students must be in grade 12 or below and under 19.5 years of age on the day of the (AMC) 12 exam, typically held in November. The USAJMO targets younger participants, requiring students to be in grade 10 or below and under 17.5 years of age on the day of the AMC 10 exam. These age and grade restrictions align with the respective AMC eligibility, ensuring participants are pre-collegiate high school students as of the qualification date. If a student qualifies for both competitions based on their scores, they receive an invitation to the USAMO. Qualification begins with the AMC 10 or AMC 12, multiple-choice exams taken annually in November. The top 2.5% of AMC 10 scorers and the top 5% of AMC 12 scorers advance to the (AIME), a 15-question exam held in . From there, approximately 250 students are selected for each based on a composite index: the USAJMO Index (AMC 10 score + 10 × AIME score) for AMC 10 participants, and the USAMO Index (AMC 12 score + 10 × AIME score) for AMC 12 participants. The MAA determines cutoffs annually after AIME results are finalized, adjusting for exam difficulty and score distributions to select the top performers. The USAJMO was introduced in 2011 to provide a dedicated pathway for younger high achievers, distinguishing it from the USAMO by focusing on students from the AMC 10 pool while maintaining similar proof-oriented format and selection rigor. This separation allows the MAA to better identify talent at earlier grade levels for advanced training and international team consideration.

Purpose and Significance

The Mathematical (USAMO) and the USA Junior Mathematical (USAJMO) primarily aim to identify and nurture exceptional mathematical talent among high school students for advanced training and international representation. These competitions select top performers for invitation to the Mathematical Program (MOP), a intensive summer training initiative designed to develop skills necessary for competing at the (IMO). By focusing on proof-based problem-solving, the olympiads emphasize deep conceptual understanding and creativity, serving as a critical pipeline to the U.S. IMO team. As the apex of the U.S. high school competition landscape, the USAMO and USAJMO distinguish themselves through their rigor and selectivity, drawing only the nation's most promising young mathematicians. High achievers from these events form the core of the U.S. delegation to the IMO, where the team has demonstrated sustained excellence, securing first-place team finishes in 2015, 2016, 2018, 2019 (tied with ), and 2024, along with second-place finishes in 2023 and 2025, and a legacy of multiple gold medals that underscore the program's effectiveness in talent development. For participants, success in the USAMO and USAJMO fosters proficiency in advanced mathematical reasoning, equipping students for future academic pursuits and professional opportunities in fields like , applied sciences, and . The experience cultivates resilience and innovative thinking, with many advancing to prominent roles in academia and research; for instance, Reid Barton, a three-time IMO gold medalist from the U.S. team, later earned a PhD from and contributed to . Administered by the (MAA), these olympiads contribute to broader national efforts in by encouraging problem-solving abilities and inspiring widespread engagement with mathematics among youth. Through this sponsorship, the MAA promotes an environment where creative exploration of mathematical ideas can flourish, ultimately strengthening the U.S. mathematical community.

History

Establishment and Early Development

The of America Mathematical Olympiad (USAMO) was established in 1972 by the Mathematical Association of America (MAA) to identify and nurture mathematically talented high school students and to prepare a national team for the (IMO). The initiative stemmed from advocacy by Nura D. Turner, who published an article in 1971 calling for a national olympiad to elevate U.S. participation in international competitions. Samuel L. Greitzer served as the first chairman of the USAMO committee, while Murray S. Klamkin contributed significantly as a problem setter and committee member. The inaugural USAMO took place on May 9, 1972, as an invitation-only proof-based examination modeled after the IMO's emphasis on rather than multiple-choice formats. Approximately 100 top-scoring students from the American High School Mathematics Examination (AHSME) were invited to participate, with 106 ultimately contacted by the MAA. The exam consisted of five problems requiring rigorous proofs, designed to select the most promising candidates for advanced training. The top eight performers from this event formed the basis for the U.S. IMO team, marking the beginning of structured preparation for international competition. In its early years through the , the USAMO solidified its role as the culminating stage of the MAA's , building directly on the AHSME to create a pipeline for elite talent. Greitzer and Klamkin coached the emerging U.S. IMO teams, with Turner handling organizational logistics, enabling the first official U.S. participation at the 1974 IMO held in . Participation remained selective, limited to around 100-150 invitees annually, focusing on depth over breadth to foster profound mathematical insight. By the mid-1980s, the competition's prestige grew, supported by grants from the Corporation for awards ceremonies and MAA resources for administration, while integration with the newly introduced American Junior High School Mathematics Examination (AJHSME) in 1985 began broadening the talent pool at earlier grades.

Key Milestones and Format Changes

In the , the Mathematical (USAMO) saw important adjustments to its structure to support deeper problem-solving and handle rising interest from high school students. In , the format shifted from a single-session exam with five problems to six problems spread across two three-hour sessions separated by a one-hour break, allowing more time for proof-based work and better mirroring the intensity of international competitions. By 2000, participant numbers had grown to approximately 250 qualifiers, drawn from top performers on the (AIME), signaling the competition's evolution into a broader platform for elite talent identification. The early 2000s brought further refinements to align the USAMO more closely with the International Mathematical Olympiad (IMO). In 2002, the format was updated to its modern structure: two consecutive days, each featuring three problems over 4.5 hours, for a total of nine hours of examination time, emphasizing endurance and comprehensive proofs in a style akin to the IMO. This change reduced the previous same-day sessions and enhanced focus on rigorous mathematical reasoning without multiple-choice elements. A major milestone occurred in 2010 with the introduction of the of America Junior Mathematical Olympiad (USAJMO), creating a separate track for younger participants in grades 8 through 10 who excelled on the AMC 10 and AIME. This split allowed more high school students (grades 11-12) to qualify for the USAMO, expanding access while reserving the junior event for earlier-grade talent and fostering a more inclusive pipeline to advanced competitions. Post-2010 updates emphasized equity through the established index-based qualification system, combining AMC and AIME scores to select diverse high achievers. In the 2020s, the prompted temporary adaptations, including virtual proctoring for the 2020 USAMO and USAJMO (renamed USO(J)MO for that year), with reduced qualifier numbers (223 for USAMO and 158 for USAJMO) due to logistical challenges. Starting in 2022, a medal system was added for USAMO (, silver, ) and recognition levels for USAJMO, providing clearer honors for top performers. These developments reflect an overall trend from an elite-only focus in its early decades to a broader talent development pipeline, supporting more students in progressing toward IMO representation and mathematical excellence.

Qualification Process

Current Selection Mechanism

The selection process for the United States of America Mathematical Olympiad (USAMO) and the USA Junior Mathematical Olympiad (USAJMO) commences with participation in the (AMC) 10 or AMC 12, which are held annually in November. The AMC 10 is intended for students in grade 10 or below and under 17.5 years of age on the test date, while the AMC 12 is open to those in grade 12 or below and under 19.5 years of age. Top performers—typically the top 2.5% on the AMC 10 and top 5% on the AMC 12—advance to the (AIME), a 15-question test administered in February. Qualification for the olympiads is determined by composite indices combining AMC and AIME scores, with cutoffs established annually based on overall performance to select the top approximately 250-300 students in total. In late 2025, the MAA updated the multiplier to 20 for selections beginning with the 2026 USAMO/USAJMO, to further emphasize AIME performance. Students qualifying via the AMC 10 index are assigned to the USAJMO, which targets younger participants in grade 10 or below, while those via the AMC 12 index are assigned to the USAMO; dual qualifiers are invited to the USAMO. Invitations are announced shortly after AIME scores are released, typically in late February or early March, with the USAMO and USAJMO exams held over two consecutive days in late March. Special provisions exist for homeschooled students and U.S. citizens residing abroad, who may participate by arranging approved proctoring through a manager or , provided they meet eligibility criteria and register appropriately. An appeals process is available for disqualifications or score disputes, directed to the MAA's on Competitions via for review.

Historical Evolution of Selection

The United States of America Mathematical Olympiad (USAMO) selection process underwent several key changes from its founding in through , transitioning from subjective invitations to more standardized, score-based mechanisms to better identify high school students with exceptional mathematical ability for advanced proof-based competition and (IMO) preparation. From 1972 to 1995, selection was direct and relatively informal, with approximately 100 participants invited based on top rankings from the American High School Mathematics Examination (AHSME) Honor Roll, supplemented by recommendations from teachers and math organizations; no standardized scoring index was employed, allowing the committee flexibility to include promising students beyond raw exam performance. The introduction of the in 1983 added an intermediate step, where top AIME scorers (roughly the top 2.5-5% of AHSME participants) advanced to USAMO, increasing the pool to about 150 invitees while maintaining a focus on AHSME and recommendation-based filtering until the mid-1990s. Between 1996 and 2001, the process formalized further with the continued use of AIME as the primary qualifier following the AHSME (renamed AMC 12 in 2000), inviting approximately 100-150 top scorers based on combined performance to ensure a consistent cohort size for the proof exam. This period emphasized objective score thresholds to reduce subjectivity, though the number remained modest compared to later expansions. The 2002-2005 era saw the debut of an explicit qualification index formula, defined as the AMC score plus 10 times the AIME score, used to rank and select approximately 250 participants; this weighted approach prioritized AIME proficiency while accounting for AMC breadth, marking a shift toward algorithmic precision in selection. From 2006 to 2010, the index formula continued as AMC + 10×AIME, with continued inclusion of top younger students from AMC 10 in the USAMO pool, expanding access to approximately 250 participants overall and broadening the talent base. The 2011 transition split the competition into the senior-level USAMO and junior-level USAJMO, with dynamic cutoffs applied to invite roughly 100 students to each based on index scores, allowing for grade-appropriate challenges and larger cohorts to foster deeper IMO preparation. These evolutions were driven by the need to enhance diversity in participant backgrounds, reduce reliance on regional biases in recommendations, and better align selection with the rigorous demands of IMO training by incorporating intermediate assessments and inclusive pathways for emerging talent.

Qualification Scoring Indices

The qualification process for the (USAMO) and the (USAJMO) relies on a composite scoring index that combines performance on the (AMC) 10 or 12 and the (AIME) to select approximately 250-300 top participants each year. This index normalizes the differing scales of the AMC (out of 150 for AMC 12, out of 120 for AMC 10) and AIME (out of 15) into a single metric, with cutoffs determined annually based on the overall score distribution to ensure selectivity. As of the 2026 cycle, the formula is AMC score + 20 × AIME score (previously +10 × AIME since 2002). Mathematically, this is expressed as: Index=AMC score+20×AIME score\text{Index} = \text{AMC score} + 20 \times \text{AIME score} For USAMO, the maximum possible index is 450 (150 + 20 × 15), while for USAJMO it is 420 (120 + 20 × 15). Cutoffs fluctuate yearly depending on exam difficulty and participant performance; prior years (using +10) typically ranged from 200 to 250 for USAMO and 180 to 230 for USAJMO, with separate thresholds for combinations of AMC 10A/10B/12A/12B and AIME I/II. Future cutoffs with +20 are expected to be higher (e.g., 400-500), calibrated annually. In cases of tied indices, ties are broken by the higher AMC score, followed by other factors if necessary. The following table illustrates representative cutoff indices for recent years (using +10 formula), highlighting annual variation; for example, in 2024, USAMO cutoffs reached as high as 248 due to competitive scoring on AIME I with .
YearUSAMO Cutoff Examples (AMC 12 + 10 × AIME)USAJMO Cutoff Examples (AMC 10 + 10 × AIME)
2024245 (12A + AIME I), 248 (12B + AIME I), 220 (12A + AIME II), 228 (12B + AIME II)236 (10A + AIME I), 232 (10B + AIME I), 207 (10A + AIME II), 205 (10B + AIME II)
2023223 (12A + AIME I), 227 (12B + AIME I)194 (10A + AIME I), 190.5 (10B + AIME I)
2022222 (12A + AIME I), 227.5 (12B + AIME I)203.5 (10A + AIME I), 190.5 (10B + AIME I)
2010208.5 (general)188.5 (general)
From 2002 to 2010, the USAMO used the index formula AMC 12 score + 10 × AIME score, selecting around 250-275 high school participants without a separate junior division; cutoffs during this period generally fell between 190 and 220, reflecting similar selectivity goals. Prior to 2002, qualification was determined solely by ranking the top approximately 250 highest AIME scores, with AMC scores serving only as tiebreakers when AIME results were equal, emphasizing raw problem-solving performance on the 15-question AIME without a weighted composite. This pre-index approach aimed to identify elite talent from the roughly 5,000-10,000 AIME participants annually but was replaced to better integrate AMC performance and account for broader accessibility.

Exam Format

USAMO Structure and Scoring

The Mathematical Olympiad (USAMO) is administered as a two-day proof-based examination, consisting of six problems divided equally between the sessions. Each day features three problems to be solved within 4.5 hours, for a total testing time of nine hours, typically held in March at designated official sites across the . This format emphasizes rigorous mathematical proofs, requiring participants to provide complete, original solutions that demonstrate deep understanding and creativity. Scoring for the USAMO is conducted by panels of expert mathematicians, who award partial credit based on the correctness, completeness, clarity, and originality of solutions. Each of the six problems is worth up to 7 points, yielding a maximum total score of ; partial credit is given for incomplete or partially correct proofs, with no points for entirely incorrect attempts. The (MAA) recognizes high performers through awards: gold medals to approximately the top 6% of participants, silver to the next 12%, to the following 18%, and honorable mentions to those scoring 14 or more points overall. Since 2002, the USAMO has adopted its current structure to better align with the International Mathematical Olympiad (IMO) format, shifting from earlier single-day or shorter-session models to the two-day, 4.5-hour-per-day schedule with an emphasis on extended proof-writing and original approaches. This change allows participants more time to develop sophisticated arguments without rushing, fostering deeper problem-solving skills. Approximately 250 students qualify annually for the USAMO, selected from top performers on the AMC 12 and AIME based on a composite index score. In contrast, the USAJMO employs a parallel structure but targets younger high school students qualifying via the AMC 10 or AMC 8 pathway.

USAJMO Structure and Scoring

The of America Junior Mathematical Olympiad (USAJMO), established in 2010 to create a dedicated pathway for younger high school students, mirrors the structure of the senior-level USAMO but with adjustments for age-appropriate rigor. Approximately 250 top-scoring participants, selected based on their performance in the AMC 10 or AMC 8 and AIME, are invited to compete in this proof-based examination. The format consists of six problems administered over two consecutive days, with three problems per day. Each daily session lasts 4.5 hours, resulting in a total examination time of nine hours, typically held in late . Scoring in the USAJMO utilizes a granular partial credit approach, where each of the six problems is graded out of 7 points, yielding a maximum total score of 42. Evaluators assess submissions for mathematical accuracy, logical completeness, and expository clarity, awarding points incrementally for valid partial solutions or insightful attempts even if the full proof is not achieved. This system promotes thorough reasoning over rote answers and allows recognition of progress; for instance, contestants scoring 14 or more points receive Honorable Mention, while the top approximately 20% are honored as winners. Key distinctions from the USAMO include a focus on slightly more accessible problems to broaden engagement among students in grades 10 and below, while preserving olympiad-level depth in , , , and . This design encourages participation from emerging talents without diluting the competition's emphasis on creative problem-solving.

Pre-2002 Formats

The United States of America Mathematical Olympiad (USAMO) was established in 1972 as a proof-based competition to select top high school students for potential inclusion on the U.S. team for the (IMO). From its inception through 1995, the exam followed a consistent single-session format consisting of five problems to be completed in 3.5 hours, with scoring out of a maximum of 100 points (20 points per problem). This structure emphasized depth in problem-solving, allowing participants to develop detailed proofs under time constraints that tested endurance and creativity akin to international standards, though shorter than the IMO's two-day format. In 1995, the final year of this original format, the USAMO retained the five-problem, 3.5-hour single-session structure, serving as the capstone before integration with the (AIME) qualification process became more formalized. The problems covered advanced topics in , , , and , requiring rigorous justifications, and the total score remained out of 100 points. This exam marked the end of the pre-multi-session era, with approximately 250 qualifiers from prior rounds participating annually. Beginning in and continuing through , the format evolved to a two-session structure on a single day, featuring six problems divided equally into two sets of three, with each session lasting 3 hours and a 1-hour break between them. Scoring shifted to 7 points per problem, yielding a maximum of 42 points overall, which allowed for finer gradations in evaluating proof quality and completeness. This adjustment increased the total examination time to 6 hours of active work, aiming to assess a broader range of skills while maintaining the emphasis on proof-writing depth; it represented a step toward greater alignment with the IMO's multi-session approach, though still condensed into one day for logistical efficiency. The change facilitated more problems without excessively prolonging the event, streamlining qualification for the Mathematical Olympiad Program () training .

Test Procedures

Administration and Logistics

The Mathematical Olympiad (USAMO) and the USA Junior Mathematical Olympiad (USAJMO) are administered annually by the (MAA), typically over two consecutive days in late March. For the 2025 competition cycle, the exams were held on March 19 and 20, with each session running for 4.5 hours, typically from 1:00 p.m. to 5:30 p.m. Eastern Time. Registration occurs exclusively through the MAA's online AMC platform following qualification determination after the (AIME), with invitations sent to eligible participants based on their composite scores. This process ensures that approximately 250-270 students qualify for the USAMO and 200-250 for the USAJMO each year, varying by cutoff scores, streamlining administrative efficiency. Exams are proctored in person at designated official testing sites nationwide, including accredited schools, universities, and other approved venues in the United States and . Participants must register for a nearby site upon receiving their invitation, and remote or virtual proctoring is not permitted under current policies to maintain exam integrity. During the in 2020 and 2021, temporary virtual options were implemented for some MAA competitions like AIME to accommodate disruptions, but USAMO and USAJMO maintained adaptations ensuring integrity, with post-pandemic administration reverting to mandatory in-person settings. U.S. citizens and permanent residents enrolled full-time in accredited schools or homeschools in the U.S. or are eligible, with logistics coordinated to facilitate participation. Test materials are strictly regulated to promote fair proof-based problem-solving: participants receive blank scratch paper (non-graph), rulers, compasses, and erasers from the , while personal items such as calculators, electronic devices, or reference materials are prohibited. No questions on the exam require computational aids beyond manual calculation. Strict no-communication protocols are enforced, with proctors required to collect all phones and smartwatches prior to the start and to monitor the room continuously to prevent any interaction among participants or with external sources. The MAA oversees all , site assignments, and overall coordination, ensuring compliance with these guidelines across hundreds of locations.

Rules and Proctoring Guidelines

The United States of America Mathematical Olympiad (USAMO) and USA Junior Mathematical Olympiad (USAJMO) are administered under strict proctoring guidelines to ensure fairness and integrity. Exams are conducted exclusively at designated official testing sites across the country, supervised by trained proctors such as teachers, school administrators, or university faculty who are unrelated to any participants. Proctoring involves continuous monitoring of students to prevent communication or external assistance, with identity verification required upon arrival to confirm eligibility and prevent impersonation. No remote or unsupervised proctoring is permitted under any circumstances. Students must adhere to rigorous conduct rules during the two-day, 9-hour proof-based exams (4.5 hours per day, covering three problems each day). Prohibited items include all electronic devices (such as phones, smartwatches, and calculators), which are collected by s before the exam begins; any use results in immediate disqualification. Permitted materials are limited to blank scratch paper, pencils, erasers, rulers, and compasses for geometric constructions. No collaboration or discussion of problems is allowed until after 8:00 a.m. ET the day following the final session, and bathroom breaks require leaving all materials with the proctor under supervision. For the digital format implemented since 2021, students type their solutions directly into a secure MAA online platform using a single-tab browser on a provided or tablet, with submissions automatically uploaded at the end of each day; paper-based scanning and upload were used in transitional years but are no longer standard. Post-exam, solutions undergo manual review for by comparing proofs for unusual similarities, though automated tools are not typically employed due to the creative nature of proofs. Grading follows an anonymous process to eliminate bias, with students instructed not to include names, schools, or identifying information on submissions. Answer sheets are evaluated by committees of experienced graders, including former participants and professional mathematicians, who assign scores from 0 to 7 points per problem based on completeness, correctness, and clarity of proofs, yielding a maximum total of 42 points. The process typically takes about 3-4 weeks, after which scores are posted on the secure AMC platform. Appeals are limited to requests for rescoring of specific problems, involving a recheck for clerical errors or overlooked partial credit; additional student explanations submitted with appeals are disregarded to maintain and fairness. Violations of rules, including or use of unauthorized aids, result in immediate score cancellation and disqualification from the cycle. Egregious or repeated offenses may lead to a lifetime ban from all MAA competitions, with proctors required to report suspected irregularities directly to MAA integrity officials. Such incidents have been rare throughout the USAMO's , particularly in pre-2000 paper-based eras when participation was smaller and oversight more localized.

Mathematical Content

Topics in USAMO

The United States of America Mathematical Olympiad (USAMO) primarily tests advanced topics in four core areas of pre-calculus mathematics: , , , and . In , problems often involve sophisticated inequalities, such as those requiring AM-GM or Cauchy-Schwarz applications, and polynomial manipulations, including , , and symmetric functions. problems emphasize Euclidean techniques like similarity, congruence, and area ratios, alongside concepts such as cross-ratios and harmonic divisions. focuses on Diophantine equations, , and prime factorizations, frequently demanding properties of integers and divisibility. covers , including connectivity and colorings, as well as , which explores unavoidable structures in large sets. USAMO problems are proof-based, necessitating creative and rigorous arguments rather than mere computation, with common techniques including invariant methods to track unchanging properties across transformations and generating functions to enumerate combinatorial objects. These proofs highlight originality, such as devising novel substitutions or case analyses. Unlike introductory contests, USAMO eschews entirely, relying instead on tools. The difficulty aligns with (IMO) standards, featuring problems that challenge even top high school students through intricate setups, like functional equations where one must determine forms satisfying recursive relations or geometric transformations preserving specific incidences. Scores typically reflect this rigor, with problems rated on a 0-7 scale per question, where full credit demands complete, elegant proofs. Over the years, USAMO maintains a balanced distribution across the core topics, with each exam usually including one or two problems from each area to ensure comprehensive testing. For instance, past exams have featured problems like proving that certain polynomials have no real roots beyond a given degree (algebra), determining the minimal number of lines intersecting a configuration of points without three collinear (), solving for integer solutions to equations primes (), and counting independent sets in a graph under symmetry constraints (). This equilibrium avoids overemphasis on any single domain while progressively increasing complexity across the six problems.

Topics in USAJMO

The of America Junior Mathematical Olympiad (USAJMO) emphasizes foundational mathematical topics tailored for younger participants, typically top performers from the AMC 10 and AIME who are in grades 10 or below. The core areas include , , , and , with problems designed to develop proof-writing skills without requiring advanced techniques. In , problems often involve solving equations, manipulating polynomials, and analyzing sequences, such as arithmetic or geometric progressions, to establish properties or find patterns. Elementary geometry focuses on classical configurations like triangles, circles, and polygons, requiring constructions, congruence, or similarity arguments to prove relationships. Introductory covers primes, divisibility, and , with tasks centered on Diophantine equations or basic properties of integers. Simple introduces counting techniques, permutations, and the to solve problems involving selections or distributions. The problem style prioritizes rigorous proofs using guided methods, such as for recursive structures or basic inequalities like AM-GM for optimization, fostering logical deduction over computational routines. Difficulty remains challenging but accessible, avoiding tools like complex numbers or , to build confidence in foundational reasoning. For instance, a 2018 USAJMO problem required proving a property of sequences using induction, highlighting without abstraction. Trends in USAJMO problems show a progression toward USAMO-level sophistication, with recent exams increasingly integrating topics, such as combining and in a single proof. Past problems consistently emphasize conceptual understanding and creative application, preparing participants for advanced olympiads by reinforcing core skills through varied, logic-driven challenges.

Awards and Outcomes

USAMO Honors and Prizes

The of America Mathematical Olympiad (USAMO) recognizes outstanding performance through a tiered system of honors and prizes, emphasizing prestige and rather than monetary rewards. Top performers receive medals in , silver, and categories, awarded based on score rankings to ensure a minimum distribution among participants. These medals are presented by the (MAA), the organizing body, and serve as a hallmark of excellence in advanced problem-solving for high school students. Gold medals are awarded to at least approximately the top 6% of USAMO participants, recognizing the highest achievers who demonstrate exceptional mathematical insight across the competition's six proofs-based problems. Silver medals go to the next tier, encompassing at least approximately 12% cumulatively (thus the 6% to 12% range), while medals are given to at least approximately 18% cumulatively, highlighting strong but slightly less dominant performances. For instance, in the 2024 USAMO, 15 students received gold medals, 38 silver, and 54 , out of a participant pool determined by prior qualifying scores. These thresholds can vary annually depending on the exam's difficulty and overall score distribution, but the percentage guidelines maintain consistency in rewarding excellence. In addition to medals, contestants scoring 14 or more points out of a maximum of 42 receive Honorable Mention recognition, acknowledging solid proficiency in olympiad-level . All USAMO participants, regardless of level, are awarded certificates of participation by the MAA, which document their involvement in this prestigious national competition. High-scoring medalists, particularly those in the gold category, are invited to attend the Mathematical Olympiad Program (), an intensive summer training camp that further develops their skills; this invitation underscores the competition's role in nurturing future mathematical talent. Unlike some international contests, USAMO offers no cash prizes, prioritizing the intrinsic value of intellectual accomplishment and opportunities for advanced study.

USAJMO Honors and Prizes

The of America Junior Mathematical Olympiad (USAJMO) recognizes outstanding performance among its approximately 250-300 invitees, who are typically ninth- and tenth-grade students, through a tiered system of honors designed to motivate early development in mathematical problem-solving. Approximately 20% of participants receive Top Honors or Honors, with Honorable Mentions awarded additionally to those scoring 14 or more points out of 42. Top Honors are awarded to the highest-scoring contestants, generally the top 20-25 students achieving scores of 30 or more points, while Honors go to the next tier of high achievers, often those scoring around 25 points. These distinctions highlight exceptional proof-based reasoning at the junior level, with thresholds adjusted annually based on overall performance to ensure fairness. For example, in the 2025 USAJMO, 27 students received Top Honors, 50 Honors, and 85 Honorable Mentions. Unlike the USAMO, which features gold, silver, and bronze medals for its top 36% of participants, the USAJMO emphasizes broader recognition without metallic prizes, focusing instead on certificates of achievement issued by the (MAA). The primary purpose of these honors is to encourage mathematical talent among younger students, providing validation and resources for further growth. High-scoring USAJMO participants, particularly those earning Top Honors, are invited to the (MOP), a summer training initiative that integrates junior qualifiers into advanced coursework and serves as a stepping stone to the USAMO and eventual international competitions. This pathway ensures seamless progression for promising juniors, with lower score thresholds compared to the USAMO—such as 14 points for mentions versus higher bars for senior medals—tailored to the developmental stage of participants.

Pathway to International Competitions

The top approximately 70 high scorers from the USAMO and USAJMO are invited to the Mathematical Olympiad Program (MOP), a three-week summer designed to foster advanced problem-solving skills and prepare participants for international competition. For instance, 76 students attended the 2025 MOP. Held annually since at various university locations, MOP brings together talented high school students for intensive mathematical instruction under expert mentors. During , participants engage in rigorous training and take a series of exams, including the Team Selection Tests (TSTs), which consist of multiple proof-based assessments such as the December TST, January TST, and others weighted into an overall IMO Index score. From these results, six students are chosen to represent the at the (IMO), the premier global competition for high school mathematicians. The has fielded teams at the IMO continuously since its first participation in 1974, achieving consistent excellence with multiple top rankings over the decades. In recent years, the U.S. team secured first place at the 2024 IMO held in the , earning a total score of 192 points and multiple individual gold medals. In 2025, the U.S. team placed second at the IMO in with 216 points and five gold medals. Organized by the (MAA) and supported by the (NSA), the pathway from USAMO/USAJMO through MOP to the IMO has profoundly influenced mathematics by nurturing exceptional talent, with alumni including Fields Medalist , who earned an IMO gold medal in 1988 before receiving the prestigious award in 2006 for contributions to partial differential equations, , and .

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  26. https://artofproblemsolving.com/wiki/index.php/American_Mathematics_Competitions
  27. https://ivyleaguecenter.org/2016/02/17/recent-usamo-and-usajmo-qualification-indices/
  28. https://www.maa.org/news/2025-usamo-and-usajmo-thresholds-now-available/
  29. https://areteem.org/blog/usamo-and-usajmo-qualification-cutoffs/
  30. https://ivyleaguecenter.org/2024/03/01/2024-usamo-and-usajmo-qualification-cutoffs/
  31. https://artofproblemsolving.com/wiki/index.php/American_Invitational_Mathematics_Examination
  32. https://maa.org/wp-content/uploads/2025/08/2025-USAMO-Awardee-List.pdf
  33. https://artofproblemsolving.com/wiki/index.php/United_States_of_America_Junior_Mathematical_Olympiad
  34. https://artofproblemsolving.com/wiki/index.php/USAMO_historical_results
  35. https://www.jstor.org/stable/27963084
  36. https://artofproblemsolving.com/wiki/index.php/1995_USAMO
  37. https://prase.cz/kalva/usa.html
  38. https://maa.org/wp-content/uploads/2025/08/2025-2026-Hosting-MAA-Competitions-Guide.pdf
  39. https://mop.evanchen.cc/tests/selection/
  40. https://web.evanchen.cc/handouts/Syllabus/Syllabus.pdf
  41. https://www.maa.org/maa-invitational-competitions/
  42. https://artofproblemsolving.com/wiki/index.php/USAJMO_Problems_and_Solutions
  43. https://maa.org/wp-content/uploads/2024/11/2024-USAMO-Awardees-FINAL.pdf
  44. https://maa.org/wp-content/uploads/2025/06/2025-USAJMO-Award-Winners-List.pdf
  45. https://maa.org/student-programs/youth-program-awards-winners/
  46. https://maa.org/wp-content/uploads/2025/10/2025-Mathematical-Olympiad-Program-Participants.pdf
  47. https://gonit.app/blog/how-to-qualify-for-the-imo-in-the-usa/
  48. https://www.imo-official.org/country_info.aspx?code=USA
  49. https://maa.org/math-values/imo-medalists-and-their-contributions/
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