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How to Compete In the USA Math Olympiad (USAMO)

By Eric Eng

By Eric Eng

a trophy and a certificate

One of the best ways to set yourself apart academically and prove that you deserve a spot at one of the country’s most prestigious universities is to compete in national scholastic competitions. One such competition is the USA Math Olympiad (USAMO), the nation’s premier math competition for the highest achieving math students in the country.

What is the USA Math Olympiad (USAMO)?

The USA Mathematical Olympiad (USAMO) is a premier national mathematics competition that identifies and challenges the brightest high school students in mathematical problem-solving and proofs. Administered by the American Mathematics Competitions (AMC) and sponsored by the Art of Problem Solving (AoPS), the USAMO has expanded its participants from approximately 250 to around 500.

The competition is designed for students with a grasp of algebra, geometry, number theory, combinatorics, and probability, along with advanced problem-solving skills such as induction and proof by contradiction. It is a proof-type, invitation-only exam that selects its competitors based on their performance in the AMC 10, AMC 12, and AIME. 

The test itself spans two days, comprising two 4.5-hour sessions without the use of calculators. Each of the six questions focuses on precalculus concepts, demanding a deep understanding and innovative thinking for success.

Scoring for the USAMO is tough, with each response graded on correctness, completeness, and clarity. Each question can earn a participant a maximum of 7 points, leading to a total potential score of 42. Given the complexity and the proof-oriented nature of the questions, scores often range in the single digits, especially for first-time participants.

Students gathering for the International Math Olympiad.

How do you qualify for the USA Math Olympiad (USAMO)?

Students can start participating in the AMC program at any age, as long as their parents, teachers, or mentors think they are ready for the AMC 8, AMC 10, or AMC 12. Some students as young as 8 years old have joined the American Mathematics Competitions.

We’ll break down everything you’ll want to know about joining the USA Math Olympiad (USAMO). 

Take the AMC 10 or AMC 12

To compete at the USA Math Olympiad (USAMO), you’ll have to first take the AMC 10 or AMC 12 tests, which are national tests administered by the Mathematical Association of America. Only those students who achieve the top scores will be invited to the next stage of the arduous process. When it comes to the AMC tests, the MAA recommends that students in the two lower high school grades take the AMC 10, while juniors and seniors take the AMC 12.

Each test features 25 questions, with each tester getting 75 minutes total to complete the exam. Each correct answer is worth 6 points and each unanswered question is worth 1.5 points.

One thing of note is that test takers do not face any deductions for wrong answers, and test takers don’t technically need to get every question right to reach a qualifying score. In fact, you simply need your score to be in the top 5% of scorers in the AMC 12 or the top 2.5% of scorers in the AMC 10 to qualify. If you do qualify, you then move on to the next round of testing.

Take the AIME

From there, you will then take the American Invitational Mathematics Exam, otherwise known as the AIME. The AIME is a highly difficult 15-question exam in which testers have 3 hours to complete. Every answer in the AIME is an integer between 999, inclusive.

One thing to keep in mind is that whether you took the AMC 10 or AMC 12, you will be taking the same exact AIME. To make the task even more challenging, while students have multiple chances to qualify while taking the AMC 10 or AMC 12, you’ll only have one chance at the AIME. It goes without saying that you are going to want to prepare for it as best you can.

Once your score has been recorded, your AIME score is multiplied by ten and added to your AMC score. That final score determines whether or not you qualify for the USA Math Olympiad (USAMO).

Typically, an index of 210 or higher is necessary to qualify, which combines your AMC Score + 10*AIME score. So if you have a 120 on the AMC and 10 on the AIME, that is enough to qualify for the USAMO.

Students posing for the International Math Olympiad.

Qualify for the Math Olympiad

If you do well enough on the AIME, chances are very good that you will qualify for the USA Math Olympiad (USAMO). The very top scorers will have the opportunity to represent their school and country in the International Math Olympiad. This is an incredibly difficult process, and it is important to remember that very few students actually make it this far. 

What to Do if Your School Doesn’t Offer the AMC Exams?

If your school doesn’t offer the AMC exams, you can take several steps to facilitate participation. Start by encouraging your school principal, math teacher, or gifted education coordinator to register the school for the competition. If necessary, you could offer to cover the registration and material costs to make it easier for your school to agree. 

Consider organizing a math club or leveraging an existing one to sponsor and fund the competition’s registration. Arrange for the school to register directly since competition materials need to be sent to the school. Alternatively, some colleges and universities host the AMC exams, especially the AMC 10/12 on the “B” date, which could be another option for participation.

What is the best way for beginners to prepare?

Typically, math through pre-calculus will cover the majority of topics that are tested on the AMC 10, AMC 12 and AIME. With that being said, math competition problems are going to be quite a bit harder than the questions you have to answer on your nightly homework or even exams.

If you are not yet studying pre-calc at school, your first step to prepare for this process will be learning the problems and how to solve them for your AMC 10 or AMC 12 test day. If you want to start working ahead of your class, considering asking a teacher at your school if they are willing to meet with you outside of class so you can begin learning.

There are numerous excellent books available for studying. A good starting point is to explore the Problem Books Section at the MAA Bookstore. Particularly useful are the Contest Problem Books Volumes I – IX, which contain all the American Mathematics Competition problems from 1950 to 2007. These books are highly recommended for anyone looking to improve their AMC scores.

You may join a Mock USAMO. A Mock USAMO is a contest that simulates the actual USAMO. When organized on platforms like AoPS/MathLinks, participants are typically given a broad time window to complete the mock USAMO.

AoPS also offers an online school that includes the Worldwide Online Olympiad Training (WOOT) program. This program is designed to help students prepare for the USAMO, national Mathematics Olympiads, and the IMO. 

But wait, there’s more! AoPS hosts numerous free Math Jams, with some specifically focused on discussing USAMO problems. Check their Math Jam Schedule for upcoming sessions.

To win the International Math Olympiad, the US team trained with its rivals

How can you learn problem-solving skills to qualify for the USA Math Olympiad (USAMO)?

One of the best ways to prepare for and succeed at taking the AMC is being able to do rote problems and to downright master the concept of problem-solving and using them to solve difficult problems.

Another great way to excel during this process is to utilize the website and books known as The Art of Problem Solving. This is a well-known and oft-used resource for those who took the AMC and AIME. 

The Art of Problem Solving offers opportunities to learn how to solve certain types of problems that you are likely to see on the exams. The site also advertises a list of the best prep books to purchase as well as online forums where you can speak with other AIME hopefuls about questions you may have, studying, and strategies to succeed.

The Art of Problem Solving (AoPS) textbooks teach the fundamentals of algebra, geometry, number theory, and combinatorics in order to master the material on the AMC. Learning the fundamentals through the Art of Problem Solving textbooks could put you way ahead of your peers.

Another great option is to utilize online resources from the MAA, as well as doing lots of practice problems on the practice exams. There’s really no substitute when it comes to doing practice problems, and by applying the methods that you learn from AoPS, you’ll be on track toward qualifying for the USAMO.

While it might feel good to only focus on what you are answering and correcting, it is important to figure out what is going on in your computations when you answer questions incorrectly, especially if you consistently answer the same kinds of questions.

There are lots of recourses out there meant to help students prepare for the AMC exams and to improve their math problem solving skills. The great thing about this kind of work is that even if you unfortunately fall short of your goal to advance to the USA Math Olympiad (USAMO), the lessons you learn while pursuing that goal will help you throughout your high school career, college career and professional career.

Tips on how to prepare for the USA Math Olympiad (USAMO)

There are some more awesome ways to prepare for success at the USA Math Olympiad (USAMO). In order to make sure that you are using the best studying practices, consider the following tips to help you study like a pro.

  • Find great practice problems.

When you are looking for problems to try and master to prepare for the AMC exams, make sure that you are working with problems that will actually be similar to those that you will find on the official test. You want to prepare using questions that fit both format and type. Of course, any form of math-related problem-solving questions will be helpful, but if your concern is doing well on the AIME, you will want to be preparing with questions that you will find on the AMC.

  • Ask for help if you need it.

If you are struggling with a specific question or the same kind of question over and over, don’t hesitate to reach out for help, either from your teacher at school or through a useful message board that you find online.

  • Make great use of your time.

Practicing is important, but make sure that you are making the very most of your practice time as well. Make sure to time yourself so that you are simulating a real test-taking environment.

Find a quiet room, do not use outside help, and sit at a great desk or table that is quiet and private. As you review your answers, there is nothing wrong with bringing in outside resources to figure out what you’ve done right and what you’ve done wrong.

  • Make sure to focus on areas of weakness.

When you are studying, make sure to spend the most amount of time focusing on the areas that you find yourself struggling with most. As you work through your answers, make sure to keep track of the problems that you did not solve as well as concepts that you are not confident in. Log your mistakes in your workbook or a separate journal to keep track of trends that might help you improve overall.

It is also important to not just log your mistakes and then move on. Make sure that you understand why you made the mistakes that you did, what you didn’t know, what you thought you knew, and make a plan for how you will solve similar problems in the future.

  • Keep an eye out for the small mistakes

Something that you should pay very close attention to while you study is making sure that you avoid tiny mistakes that can have a major impact on your final answer. Obviously, something as small as an incorrect decimal point will result in your coming up with a drastically different answer than the correct one. It is also important to be taking these sample exams while using a timer.

If you feel rushed, you will be more likely to make small mistakes. By making sure that you can pay close attention to detail while dealing with the pressure of taking a test on the clock, you will be greatly prepared for the actual test day.

  • Set a regularly scheduled time to study.

It is best for your brain to set aside a dedicated block of time each week for studying. By practicing at least once per week, you will be able to retain the skills you need to build your ability and confidence. In fact, the best way to do it is to treat your studying like another class like you have in school.

It is also important to remember that the earlier you start for the exam, the better your chances are of succeeding. Even if you don’t plan on trying to take the AMC 10 until your sophomore year, there is absolutely nothing wrong with starting to study in your freshman year. In fact, it’s encouraged!

  • Join other math competitions

If you are lucky enough to have a math team or club at your school, there is nothing wrong with joining in for extra practice. Doing regional, state, or even local competitions will only help you learn how to manage your nerves and gain confidence. It will also give you lots of chances to practice.

Don’t forget, the earlier you start preparing, the better you will likely do when the big AIME test day finally comes. For parents, remember that some middle schools and even elementary schools have math clubs that can get your children thinking about numbers in a productive and challenging way from a very young age.

What should you do when test day actually arrives?

How do you make sure that you’re best prepared the night before the test? Here are a few little tips to keep in mind.

  • Don’t do lots of studying the night before you take the AMC. At that point, it’s up to you to trust in your knowledge and experience and give your mind the rest it needs to be performing at its peak the next day.
  • Make sure you get a healthy meal and a great night’s sleep. That doesn’t mean you have to eat a huge meal and sleep for 10 hours. Simply give your body and mind the fuel and rest it needs.
  • Finally, before you go to bed, sit down with your parents (or whoever will be taking you to the exam) and review the schedule and directions you will need to make sure that you arrive with ample time. After all, the last thing you want to worry about on test day is whether or not you’re going to show up late.

Performing well in the USA Math Olympiad is as much an intellectual game as it is a mental game. Hopefully, by implementing the tactics above, you can qualify for USAMO be considered one of the top 500 students in the country for mathematics, and stand out from your peers in the college admissions process.

Final Thoughts

By competing in the AMC10 and 12, possibly advancing to the American Invitational Mathematics Exam (AIME), and reaching the holy grail of the USA Math Olympiad (USAMO), you will significantly increase your odds of Ivy League universities such as MIT or Caltech

If you’re looking to get into the top colleges in the country, you’re going to need to make a strong case to the admissions officers that you’re going to be successful academically, and performing well in the USA Math Olympiad is one way to separate yourself from the rest of the pack. 

Frequently Asked Questions

1. Who can take the AMC?

The American Mathematics Competitions (AMC) are open to students from any registered school who meet the age and grade requirements for each specific competition: AMC 8, AMC 10, and AMC 12. Following the AMC 10 and AMC 12, students may qualify for the AIME by ranking in the top percentages of scorers. The prestigious USAMO extends invitations to top-performing students in the U.S. or Canada, and to U.S. citizens studying abroad. The primary requirement to participate in these contests is that the school must be registered with the AMC program.

2. Where can I get past AMC papers and solutions?

Past AMC problem papers and solutions are available directly from the AMC office under various publications. For a comprehensive collection, interested individuals can purchase flash drives that contain historical contest problems and solutions from specific decades, ensuring access to original, unedited content. The Math Club Package offers a compilation of contests from the 2000s, including AMC 8, 10, 12, AIME, and USAMO, all on one flash drive.

3. How do we get our organization involved in the AMC contests?

The American Mathematics Competitions initially cater to public schools and select accredited private schools, including colleges and universities. Organizations such as home schools, learning centers, and testing centers interested in participating must provide comprehensive details about their academic structure to the AMC office. The review process takes approximately 4 to 6 weeks, after which a decision will be communicated via email.

4. I don’t know about the American Mathematics Competitions (AMC). What are the different contests?

The American Mathematics Competitions, organized by the Mathematical Association of America (MAA), aim to foster mathematical interest and enhance problem-solving skills through engaging competitions. These contests provide teachers and students a platform to engage with challenging mathematical problems that complement the educational curriculum. The competitions range from the AMC 8, aimed at younger students, to the more advanced USAMO, providing various levels of challenge and recognition.

5. What books should I buy to study from or to improve my scores?

To aid in preparation for the AMC, the MAA Bookstore offers a range of problem books that catalogue historical competition problems. The Contest Problem Books Volumes I – IX, for instance, cover problems from 1950 to 2007 and are recommended for those looking to deepen their understanding and improve their performance in future competitions.

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