United States of America Mathematical Olympiad facts for kids

The United States of America Mathematical Olympiad (USAMO) is a challenging math competition for high school students in the United States. It's held every year and is the final step in the American Mathematics Competitions. Since it started in 1972, it has helped find some of the best young mathematicians. In 2010, the competition grew and split into two parts: the USAMO and the United States of America Junior Mathematical Olympiad (USAJMO).

Students who do well in these nine-hour math proof competitions are invited to join the Mathematical Olympiad Program. This program helps them train and prepare to represent the United States at the International Mathematical Olympiad, which is a worldwide math contest.

Who Can Participate?

To take part in the USAMO, students must be a U.S. citizen or a legal resident of the United States or Canada. Before 2004, only U.S. citizens and permanent residents could be invited. Now, other students living legally in the U.S. can also be invited. However, to join the American IMO team, students must be U.S. permanent residents or citizens.

All participants, no matter where they live, must earn high enough scores in earlier rounds of the AMC contests. You can only take the USAMO if you receive an invitation.

A Look Back at the USAMO

The USAMO began in 1972, thanks to Nura D. Turner and Samuel L. Greitzer. It was first the next step after the AHSME competition. In 1983, another competition called the American Invitational Mathematics Examination (AIME) was added as a bridge between the earlier tests and the USAMO. Then, in 2010, the USAMO was divided into two separate competitions: the USAMO and the USAJMO.

How Students Qualify

About 500 students in total are invited to take the USAMO and USAJMO each year. The way students are chosen has changed over time, but it always involves their scores from earlier math contests.

Current Qualification System

Today, students qualify based on an "index" score. This score combines their results from the AMC 10 or AMC 12 tests with their AIME score.

• For the USAMO, the index is calculated using the AMC 12 score plus 10 times the AIME score. About 260-270 students qualify this way.
• For the USAJMO, the index uses the AMC 10 score plus 10 times the AIME score. About 230-240 students qualify for this competition.

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 were invited because of the COVID-19 pandemic. Also, students who do very well on the United States of America Mathematical Talent Search can qualify for the USAMO or USAJMO if they also score high enough on the AIME.

Past Qualification Changes

Over the years, the exact number of students invited and the specific score cutoffs have changed. For example, in 2006, the number of invited students increased to about 500, partly due to support from the Art of Problem Solving website. The goal has always been to select the top math students from across the country.

The qualification scores (called "indices") needed to get into the USAMO and USAJMO change every year. These scores depend on how difficult the earlier AMC and AIME tests were. For example, in 2020, due to the COVID-19 pandemic, the AIME II and USA(J)MO competitions were held online and were called the AOIME and USO(J)MO.

Test Format and Scoring

The USAMO is a challenging test that requires students to write detailed mathematical proofs.

Modern Test Format (Since 2002)

Since 2002, the USAMO has been a six-question competition that lasts nine hours, spread over two days. This is similar to the IMO. On each day, students have four and a half hours to solve three problems.

Each problem is graded on a scale from 0 to 7 points. A score of 7 means the student provided a perfect, mathematically correct proof. This means a perfect total score for the entire competition is 42 points. The number of students who get a perfect score changes each year, depending on how hard the test is.

Here's what the scores generally mean:

• 0 points: No work or very little progress.
• 1-2 points: Some progress on the problem, but not a complete solution.
• 3-4 points: All the main steps are there, but the explanation might not be very clear. (These scores are not very common.)
• 5-6 points: A nearly complete solution with only small mistakes.
• 7 points: A perfect solution.

Older Test Formats

From 1996 to 2001, the test had two sets of three problems each. Students had three hours for each set, one in the morning and one in the afternoon. Before 1995, the test had five problems to be solved in three and a half hours (or three hours even earlier). Each problem was worth 20 points, making a perfect score 100 points.

How the Test Is Taken

Most years, students take the USAMO at their own high schools. Before 2002, the problems were sent to schools in sealed envelopes and opened only at the official test time. Since 2002, the test problems have been put online on the AMC website 15 minutes before the test starts. After the test, student answers are sent back to the AMC office.

In 2002, the Akamai Foundation, a big supporter of the American Mathematics Competitions, invited all USAMO participants to take the test at a special event at MIT in Cambridge, Massachusetts. All their travel costs were covered. They also invited many younger USAMO participants to an expanded Mathematical Olympiad Program (MOP). This special event was too expensive to do every year, but in 2004 and 2005, funding was found to send some younger students to MOP in a program called "Red MOP."

Today, top students from the USAMO and USAJMO are chosen for MOP based on several factors. This includes members of the IMO team, other high-scoring USAMO students who are not graduating, top USAMO students in 9th or 10th grade, top USAJMO students, and some female contestants. Middle school students can also be invited in special cases.

What Math Topics Are on the USAMO?

The USAMO covers different areas of mathematics. While Calculus is allowed, it's never needed to solve the problems. The types of math problems usually include:

• Algebra
• Geometry
• Number theory
• Combinatorics (which is about counting and arrangements)
• Sometimes Graph theory

These topics are mixed throughout the six problems each year.

What Math Topics Are on the USAJMO?

The USAJMO also covers various math topics, similar to the USAMO, but generally for younger students. Calculus is not required for solutions. The types of math problems usually include:

• Algebra
• Geometry
• Number theory
• Combinatorics
• Sometimes Game theory

USAMO Awards

Students who perform well on the USAMO receive special awards:

• USAMO Gold Award: Given to about 6% of competitors (around 16 students).
• USAMO Silver Award: Given to about 12% of competitors (around 32 students).
• USAMO Bronze Award: Given to about 18% of competitors (around 48 students).
• USAMO Honorable Mention: Given to competitors who score 14 points or more, if they didn't receive a Gold, Silver, or Bronze award. (This award started in 2023.)

Before 2022, the top 12 performers were called "USAMO Winners," and the next 12 or so were "Honorable Mentions."

USAJMO Awards

Students who do well on the USAJMO also receive awards:

• USAJMO Top Honors: Given to the top 9 competitors (as of 2024).
• USAJMO Honors: Given to the next 48 competitors (as of 2024).
• USAJMO Honorable Mention: Given to competitors who score 14 points or more, if they didn't receive a higher award.

The exact number of winners and honorable mentions for the USAJMO has varied over the years. For example, in 2023, there were 13 "Top Winners" and 36 "Winners." Before 2022, the top 12 performers were "USAJMO Winners," and the next 12 or so were "Honorable Mentions."

See also

• American Mathematics Competitions
• List of mathematics competitions