Olympiads are some of the most academically challenging experiences you can test yourself against, and the reward for doing so can be equally substantial. Many universities value students who have participated in prestigious competitions like olympiads because they show dedication, problem-solving skills, and mathematical talent. 

Participating not only enhances your profile but also demonstrates your ability to tackle challenging academic tasks, a quality that top colleges seek. Our blog today is meant to introduce you to the U.S. Math Olympiad and hopefully help you win!

What is the USAMO all about?

The USAMO is one of the most prestigious math competitions in the United States, inviting only the top students in the country to participate. Established in 1972, its purpose is to challenge high school students with rigorous proof-based math problems. Unlike multiple-choice exams, the USAMO focuses on logical reasoning and deep mathematical understanding. 

Over the course of the nine-hour-long exam spread across two days, you will work on complex problems that test your knowledge of the fundamentals of algebra, geometry, number theory, and combinatorics. The USAMO is an invite-only event, and to even reach it, you need to be among the top performers at the AMC 12 and AMC 10 exams hosted by the Mathematical Association of America. 

Top performers at the USAMO get the opportunity to join their Mathematical Olympiad Summer Program (MOP), with the best six representing the U.S. in the International Math Olympiad (IMO), making this competition a gateway to international recognition.

Is the USAMO Prestigious?

Absolutely. The USAMO is highly selective, with around 500 students across the U.S. earning invitations based on their scores from the American Mathematics Competitions (AMC) and the American Invitational Mathematics Examination (AIME). Competing at this level means you are among the best young mathematicians in the country. Simply reaching the stage of the USAMO is unequivocal proof of your mathematical skills and ability to compete in a high-stakes environment. Past winners have contributed to breakthroughs in academic and research fields, further highlighting the prestige and importance of this competition​.

Who is eligible to apply?

To be eligible for the USAMO, you need to meet the following criteria:

• Citizenship: U.S. citizens and permanent residents are eligible.

• AMC Scores: You must first compete in either the AMC 10 or AMC 12. Top scorers in the AMC 12 and the AIME are invited.

• Grade Levels: Both middle and high school students can qualify through AMC performance, though most participants are in high school simply due to the intensity of the exams.

• Invitation: Entry to the USAMO is by invitation only, based on a combination of your scores from the AMC and AIME​.

How does the application process work?

The USAMO is an invitation-only competition, meaning you need to qualify through the AMC and AIME. Here's a step-by-step guide:

1. Participate in the AMC 10 or AMC 12: These exams are held annually in November and January. The registration fee varies from $33 to $116, depending on when you apply. Both are 25-question, 75-minute multiple-choice exams.

2. Advance to the AIME: If you score high enough on the AMC, you’ll be invited to take the AIME in March.

3. Receive your invitation: Around April, top AIME performers will be invited to the USAMO based on their scores.

4. Compete in the USAMO: The competition typically takes place in May.

Participation is generally free, as the contest is invitation-based​.

How is the competition structured?

The USAMO consists of two 4.5-hour sessions, one on each of two consecutive days. Each session presents three problems, for a total of six problems. These problems require proof-based solutions, which means you must explain your answers logically and clearly. The competition does not allow calculators, so you will have to rely on your mental math and reasoning skills. 

Judges grade your solutions from 0 to 7 based on clarity, correctness, and depth of understanding. What you win for this effort is the chance to become a part of the Mathematical Olympiad Program (MOP) and potentially represent your country at the International Math Olympiad! To know more about the IMO, check out our blog here.

10 tips to help you win the USAMO

1. Master the basics: A solid foundation in algebra, geometry, and number theory is absolutely essential. The USAMO is (in)famous for the complexity and depth of its questions, but it still rigidly holds to the basics. Make sure you’ve thoroughly reviewed these topics.

2. Practice proof-based problems: USAMO’s questions all require written proofs, so spend plenty of time practicing explaining your solutions step-by-step rather than just giving the final answer.

3. Study past papers: Familiarize yourself with past USAMO problems. This helps you understand the difficulty and structure of the questions.

4. Focus on logical reasoning: Developing logical reasoning is key. The problems test how well you can think through a complex situation.

5. Do not neglect time management: You have 4.5 hours per session, and while that may seem like a lot, the complexity of the questions is enough to make the time fly by. Evaluate and practice a few different time management strategies to decide on what works best for you. Remember that your submission will be graded on the quality and coherence of your effort, not just your final answer.

6. Join a math club or team: Working with peers can expose you to different problem-solving strategies and foster a collaborative learning environment.

7. Take plenty of mock tests: Simulate the test environment with practice exams to get used to the timing and pressure.

8. Get feedback on solutions: Ask a teacher or a mentor to review your proofs. This will help you refine your problem-solving techniques.

9. Do not skip steps: On the USAMO, even partial credit matters. Always explain your thinking process, even if you’re unsure about the final answer.

10. Stay calm under pressure: The USAMO is challenging, but staying calm and focused will help you think clearly. Trust your preparation, and don’t panic if you hit a tough problem​.

Good luck! Winning the USAMO is no easy task, but with the right preparation, practice, and mindset, you can perform at your best and potentially earn a spot in the prestigious MOP program!

One other option—the Lumiere Research Scholar Program

If you’re interested in pursuing independent research, you could also consider applying to one of the Lumiere Research Scholar Programs, selective online high school programs for students founded with researchers at Harvard and Oxford. Last year, we had over 4,000 students apply for 500 spots in the program! You can find the application form here.

Also check out the Lumiere Research Inclusion Foundation, a non-profit research program for talented, low-income students. Last year, we had 150 students on full need-based financial aid!

Stephen is one of the founders of Lumiere and a Harvard College graduate. He founded Lumiere as a PhD student at Harvard Business School. Lumiere is a selective research program where students work 1-1 with a research mentor to develop an independent research paper.