United States of America Mathematical Olympiad facts for kids

Kids Encyclopedia Facts

The United States of America Mathematical Olympiad (USAMO) is a special math contest for talented high school students in the United States. It started in 1972 and is the final part of the American Mathematics Competitions. In 2010, it split into two contests: the USAMO and the United States of America Junior Mathematical Olympiad (USAJMO).

Doing well in the USAMO or USAJMO is a big achievement for high school students. Top scorers in these contests are invited to join the Mathematical Olympiad Program. Here, they train to represent the United States at the International Mathematical Olympiad, a worldwide math competition.

Who Can Participate?

To take the USAMO, you must be a U.S. citizen or live legally in the United States or Canada. However, only U.S. citizens and permanent residents can join the American team for the IMO.

Everyone who takes part must also get high enough scores in earlier rounds of the AMC contests. You can only take the USAMO if you are invited.

History of the Contest

The USAMO began in 1972. It was started by Nura D. Turner and Samuel L. Greitzer. For many years, it was the next step after the AHSME contest. In 1983, the American Invitational Mathematics Examination (AIME) was added as a bridge between the earlier contests and the USAMO. In 2010, the USAMO was divided into two separate contests: the USAMO and the USAJMO.

How Students Are Chosen

About 500 students in total are chosen each year for the USAMO and USAJMO. Before 2006, only about 250 students were chosen. The AMC Committee has changed the selection process many times over the years to keep this number consistent.

How Students Are Chosen Now

To qualify for the USAMO, students usually take the AMC 12 test. Their score from the AMC 12 is added to 10 times their AIME score. This creates a "USAMO index." About 260-270 students with the highest USAMO indices qualify for the USAMO.

For the USAJMO, students usually take the AMC 10 test. Their score from the AMC 10 is added to 10 times their AIME score. This creates a "USAJMO index." About 230-240 students with the highest USAJMO indices qualify for the USAJMO.

If a student takes both the AMC 10 and AMC 12 and qualifies for both the USAMO and USAJMO, they must take the USAMO. In 2020, fewer students qualified due to the COVID-19 pandemic. Also, a special rule was made for female students to help more of them qualify for team selection.

Students who qualify for the AIME through the United States of America Mathematical Talent Search (USAMTS) can also qualify for the USAMO or USAJMO based on their AIME scores, even if they didn't take the AMC 10 or 12.

Changes in How Students Were Chosen (2011)

Since 2011, the goal has been to select about 500 students in total. This includes about 270 for the USAMO and 230 for the USAJMO.

Students who are U.S. citizens or live in the U.S. and Canada can take these tests if they have high enough scores. The USAMO uses scores from the AMC 12 and AIME. The USAJMO uses scores from the AMC 10 and AIME.

Only students who took the AMC 12 are usually eligible for the USAMO. Only students who took the AMC 10 are usually eligible for the USAJMO. This means the USAJMO is mostly for students in 10th grade or below. If a younger student qualifies for the USAMO through their AMC 12 score, they must take the USAMO.

The top students are chosen based on their index scores. Also, one student with the highest score from each U.S. state that isn't already represented gets invited.

Changes in How Students Were Chosen (2010)

In 2010, the USAMO split into two parts. About 270 students, mostly from the top AMC 12 participants, took the USAMO. Students who only took the AMC 10 participated in the USA Junior Mathematical Olympiad (JMO).

The selection was based on an "Olympiad index" (AMC score + 10 * AIME score). Only AMC 12 takers were generally eligible for the USAMO. Only AMC 10 takers were eligible for the JMO, which limited it to 10th graders and younger.

A small number of AMC 10 takers who scored very high on the AIME were also invited to the USAMO. The top student from each state not already represented was also invited.

Changes in How Students Were Chosen (2008)

In 2008, the goal was to select about 500 top scorers from the AIME and AMC contests for the USAMO. Selection was based on a "USAMO index" (10 times AIME score + AMC score).

The first group chosen were the top 330 students who took the AMC 12. Then, about 160 younger students (10th grade and below) with high AIME scores were selected. The student with the highest index from each state or territory not yet represented was also invited.

Changes in How Students Were Chosen (2006-2007)

Starting in 2006, the USAMO expanded to include about 500 students. The selection process was similar to 2008, focusing on the USAMO index. The goal was to select about 240 top AMC 12 students first. Then, about 120 younger students (10th grade and below) with high AIME scores were chosen. Again, the top student from each state not yet represented was invited.

Changes in How Students Were Chosen (2002-2005)

From 2002 to 2005, about 250 top scorers were chosen for the USAMO. The selection used the USAMO index. The first group was about 160 students with the highest scores from the AMC 12. Then, younger students (grades 7-10) with high AIME scores were selected. The top student from each state not yet represented was also invited.

Changes in How Students Were Chosen (2001 and Earlier)

From 1998 to 2001, students were chosen in groups:

First Group: The top 120 students.
Second Group: The next 20 students in 11th grade or below.
Third Group: The next 20 students in 10th grade or below.
Fourth Group: The next 20 students in 9th grade or below.
Fifth Group: One student from each state, U.S. territory, and APO/FPO schools, if not already chosen.

Recent Qualification Scores

Here are the scores needed to qualify for the USAMO and USAJMO in recent years. The "index" is usually your AMC score plus 10 times your AIME score.

Year	AMC 12	AMC 10	Total number of qualifiers
2020*	12A + (10AIME I): 233.5 and above 12A + (10AOIME): 234 and above 12B + (10AIME I): 235 and above 12B + (10AOIME): 234.5 and above	10A + (10AIME I): 229.5 and above 10A + (10AIME II): 233.5 and above 10B + (10AIME I): 230 and above 10B + (10AIME II): 229.5 and above	223 USAMO; 158 USAJMO
2019	12A + (10AIME I): 220 and above 12A + (10AIME II): 230.5 and above 12B + (10AIME I): 230.5 and above 12B + (10AIME II): 236 and above	10A + (10AIME I): 209.5 and above 10A + (10AIME II): 216.5 and above 10B + (10AIME I): 216 and above 10B + (10AIME II): 220.5 and above	275 USAMO; 235 USAJMO
2018	12A + (10AIME I): 215 and above 12A + (10AIME II): 216 and above 12B + (10AIME I): 235 and above 12B + (10AIME II): 230.5 and above	10A + (10AIME I): 222 and above 10A + (10AIME II): 222 and above 10B + (10AIME I): 212 and above 10B + (10AIME II): 212 and above	242 USAMO; 156 USAJMO
2017	12A + (10AIME I): 225.5 and above 12A + (10AIME II): 221 and above 12B + (10AIME I): 235 and above 12B + (10AIME II): 230.5 and above	10A + (10AIME I): 224.5 and above 10A + (10AIME II): 219 and above 10B + (10AIME I): 233 and above 10B + (10AIME 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 happened as planned. However, due to the COVID-19 Pandemic, the AIME II and USA(J)MO were delayed. They were later held online in June and renamed AOIME and USO(J)MO.

Test Format and Scoring

After 2002

Since 2002, the USAMO has been a math contest with six problems. Students get nine hours to solve them, spread over two days. This is the same format as the IMO. On each day, students have four and a half hours to solve three problems.

Each problem is scored from 0 to 7 points. A score of 7 means the solution is perfect and mathematically correct. The highest possible score is 42 points. The number of perfect scores changes each year depending on how hard the test is. Top scorers are recognized by the MAA (Mathematical Association of America).

Here's what the scores mean:

0 - No work, or very little effort.
1-2 - Some progress was made, but the problem wasn't fully solved.
3-4 - All steps are there, but they might not be very clear. These scores are rare.
5-6 - A complete solution with only small mistakes.
7 - A perfect solution.

From 1996 to 2001

The test had two sets of three problems. Students had three hours for each set. One set was given in the morning, and the other in the afternoon.

Before 1995

The test had five problems to solve in three and a half hours (or three hours in earlier years). Each problem was worth 20 points, so a perfect score was 100 points.

How the Test Is Given

Most years, students take the USAMO at their own high schools. Before 2002, the problems were sent to schools in sealed envelopes. These could only be opened at the exact test time. Since 2002, the problems have been put on the AMC website 15 minutes before the test starts. Students then fax their answers back to the AMC office when the test is over.

In 2002, the Akamai Foundation, a big supporter of the contests, invited all USAMO participants to take the test at MIT in Cambridge, Massachusetts. They paid for all the travel costs. Akamai also invited about 160 younger students (not high school seniors) to a bigger Mathematical Olympiad Program (MOP). This special event was too expensive to do every year, so it stopped. However, in 2004 and 2005, funding was found to send 30 incoming freshmen to MOP in a program called "Red MOP."

Top students from the USAMO and USAJMO are chosen for MOP based on several things. As of 2016, this includes the IMO team members, about 18 top USAMO students who are not graduating, about 12 top USAMO students in 9th or 10th grade, and about 12 top USAJMO students. Some female contestants from the USAMO or USAJMO are also invited. Middle school students are invited sometimes, depending on their performance.

USAMO Test Topics

Here are the types of math problems on the USAMO test each year, by problem number. You can use Calculus to solve problems, but it's never required.

2023:

2022:

2021:

2020:

2019:

2018:

2017:

2016:

2015:

2014:

2013:

2012:

2011:

2010:

2009:

2008:

2007:

2006:

2005:

2004:

2003:

2002:

2001:

2000:

USAJMO Test Topics

Here are the types of math problems on the USAJMO test each year, by problem number. Like the USAMO, Calculus is allowed but never required.