Q. #6. What are the tests that lead to the United States of America Mathematics Olympiad (USAMO) and USA Junior Mathematical Olympiad (USAJMO)?

A. The AMC 8 is a standalone contest with benefits of its own (listed in the AMC 8 section of the FAQ). The path to the USAMO and USAJMO begins with either the AMC 10 or AMC 12. Approximately the top 2.5% of AMC 10 students and top 5% of AMC 12 students qualify to take the American Invitation Mathematics Examination (AIME). Students with approximately the top 270 10*AIME + AMC 12 scores are then invited to take the USAMO. The approximately 230 students with the 10*AIME + AMC 10 scores are then invited to take the USAJMO.

From David Shi

Starting with an anecdote. I participated for many years in math contests during my pre-university years, starting in grade 6 (Mathcounts!). I took the AMC 8 in 8th grade and got 21(?) and the AMC 10 that year as well (did rather poorly, <90). Throughout high school I participated in a few contests including CML, CaML, Mandelbrot, SMT (or something like this, stanford math tournament), BAMO, and AMC10/12.

In terms of difficulty:

The AMC 10 is about the easier difficulty for high school contests. The first 10 problems are fairy simple, and the first 5 can be solved by nearly anyone. I hear the logic in this is so people go home and go “nice, I solved the first 10 problems” instead of being like “man, couldn’t solve any problems”. The later problems overlap with some AMC 12 problems. For people that want to pass AIME, this is a much easier path to go. I think I got 23 problems in 10th grade on it (maybe 22, don’t remember now).

The AMC 12 is significantly harder than AMC 10, but I wouldn’t place it at the highest difficulty for entry level contests. It’s fairly difficult though for the last 8 problems or so. Even though all problems can be solved with high school means, sometimes they are tricky. In the AMC 10 there’s not many number theory questions and more complex trig/log/complex numbers problems but on the 12, these are fair game.

The AIME is much much harder than AMC 12. This is like the jump from AMC 8 to AMC 10 problems 20-25. There’s no more multiple choice, and the step up is huge in terms of content that you can be tested on. This is the time you can memorize equations, learn some tricks to solve problems, and work on algebra. I recall coordinate bashing some problems on the AIME. I remember for a few practice problems I manipulated the equation and it went much more easily than if I had tried to solve the problem as it was. While getting 50% of the AMC 12 is pretty good, getting 2 problems is good on the AIME. I don’t remember when I got in my first years of taking it (9 or something when I did it in 2008 but it was an “easy” one) but I recall a 7 in my junior year and I felt pretty satisfied about it.

The USAMO is another monster with a leap of difficulty like AMC 8 to AIME. Not only do you have to be good at math, you have to be able to write rigorous proofs. If you do make it this far though, you’re probably well aware of what goes on.

In terms of being competitive:

The AMC contest is the biggest contest in the USA. Even high schools that don’t do much math may host the AMC contests. It is popular enough that schools like MIT and I believe Caltech ask for your results. Top universities take AIME and USAMO qualifications seriously. About 120k people took the test last year so if you make AIME it’s quite an accomplishment as only the top 1-1.5% do. In my opinion, the AMC is the most competitive out of all contests due to its sheer popularity and how far you can go.