AMC, AIME, USAMO Contradiction

To get to the national swimming championship, you need to win the state running championships.

What? Is that a joke? Perhaps you’re having the same reaction. Because this is exactly what is happening with math competitions. The official USA math competition has three rounds: AMC, AIME and USAMO.

AMC is a multiple-choice competition with 25 problems in 75 minutes. To be good at it, you need speed, accuracy and the ability to guess well.

AIME is 3 hours long and has 15 problems. The problems are a different level of difficulty and guessing will not help you. Though AIME is also multiple-choice, unlike AMC where you choose out of 5, in AIME you choose out of 1,000. But you still need speed and accuracy. A small arithmetic mistake will cost you the whole problem.

USAMO is a competition that runs for 9 hours and has 6 problems. The problems are much harder and you have to write proofs. Proofs? What proofs? Where are the proofs coming from? It is like you got to the national swimming championship because you are a great runner, but you do not know how to swim.

As the result of this system of selection, the USA team at the International Math Olympiad has diverse skills: these kids are good at all three types of the math competitions. It is like taking an Olympic triathlon team to the Olympic swimming event.

However, the US loses by selecting in this way. There are many kids who are great mathematicians: they may be good at difficult problems but not that good at speed-racing problems. An arithmetic mistake costs you only one point at IMO, but a whole problem at AIME. There are kids who are deep mathematicians prone to small arithmetic mistakes. They could get a gold medal at IMO, but they can’t pass AMC or AIME.

Not only that. As many AMC problems are boring and do not require ideas, AMC might discourage kids from all math competitions at an early stage.

I will write later with my ideas about how to change AMC. Right now I would like to offer a solution to a smaller problem. I am sure that the US math team organizers know many cases in which a non-senior kid does great at USAMO and is potentially a team member for the next year’s US IMO team, but, oops, next year he can’t pass AMC.

I suggest the following: USAMO participants are allowed to go to next year’s AIME no matter what their AMC scores are. USAMO winners are allowed to go to the next year’s USAMO no matter what their AIME results are. This way kids who have proved that they are great swimmers do not need to compete in running again.



  1. misha:

    I have another suggestion: AMC should be scrapped or renamed AAC, American accounting competition.

  2. Woo Chang:

    Interesting post. I once compared AMC/AIME and preliminary rounds of KMO (Korea Mathematics Olympiad) and noticed how different they are. Indeed, it takes speed to do well in AMC, while you need deep thinking and good proofs even at the preliminary rounds of KMO.
    I felt KMO early rounds were more difficult because of that.

  3. Ted Alper:

    found your site via your post on AOPS re: linguistics puzzles [unfortunately it showed up a little late for this year’s NACLO rounds]. I don’t have time to read everything now, but I very much like everything I’ve read so far.

    One point. While I very much agree with your point about the different set of skills, the threshold for making the AIME from the AMC is pretty low for USAMO-caliber students. 100 points on the AMC 12 is not much at all. There is also the way into the AIME by doing well on the USAMTS, which is at least closer in conception to olympiads.

    I think there’s more truth to your point as far as qualifying for the USAMO from the AIME, though, just as a logistical problem, finding a way to efficiently and accurately find the *most* IMO-capable out of 250,000+ students at the start is tough, even if the quality was not such a fuzzy one. Allowing previous USAMO winners to take the USAMO again is fine, but that will only be a drop in the bucket [it’s a good empirical question, though: how many USAMO winners 11th grade and younger failed to qualify for the next year’s USAMO?]

  4. Mary O'Keeffe:

    It is also important to note that the purpose of the AMC/AIME/USAMO is not just to select the most promising IMO team–I don’t think one could justify all the resources devoted to it, if that were the only purpose, but also to engage and inspire a large number of young students to develop mathematically. The reality is that the standard US high school mathematical curriculum gives students little opportunity to develop their skills as proof-writers, let alone as olympiad problem-solvers.

    One also needs to take a long-term view. Some students selected by the current AMC/AIME process may not do well on the USAMO the first time they take it, BUT I would still argue that the experience of taking it may inspire them to work on developing their skills for future years, and that is a very valuable and important accomplishment of the USAMO as well.

    To partially address Ted Alper’s question, I don’t have numbers, but I am aware of a number of students who have qualified for USAMO prior to 11th grade who failed to qualify in subsequent years. Trivial mistakes on the AIME can be notoriously costly, and there’s a tradeoff between spending time meticulously checking over more accessible problems and spending time trying to crack the most elusive problems.

    In fact, over the years, I understand that several USAMO honorable mentions and so-called “blue moppers” failed to qualify for USAMO in a subsequent year. In the time when Titu Andreescu ran MOP, I understand that Moppers were told that they could petition to take the USAMO if they failed to make USAMO in a subsequent year. It was apparently an “unwritten rule” known to moppers, but I don’t know of anyone in that position who actually invoked the provision. I have heard that some USAMO honorable mention/blue moppers who could have petitioned Titu to take the USAMO in a subsequent year, declined to do so, feeling that there was a matter of honor involved in adhering to the written rules.

    I don’t know if the rule about former moppers being able to petition to take still applies (and given that the number of USAMO qualifiers have increased so greatly in recent years, it is probably much rarer for USAMO honorable mentions and blue moppers to fail to qualify for USAMO in subsequent years.) In the late 90s, about a quarter of a million students took the AHSME (precursor to AMC12–there was no AMC10 then) and about 160 or 170 were invited to write USAMO each year. In recent years, the number of students taking the AMC exams is harder to determine (because many students take the contests on both the A and B dates), but it is clearly less than 200,000 given any reasonable estimate of multiplicity, yet the number of USAMO qualifiers has roughly tripled to over 500.

    There are no perfect systems, but I think that the US system has worked reasonably well in producing IMO teams that have performed very credibly over the years. True, they haven’t managed to defeat China in recent years, but given China’s very large population and their highly systematic and rigorous training from an early age, I think it’s hard to argue that tinkering with the American selection prcoess would do much to address it.

    For me, a bigger priority than reforming the USAMO selection process is drawing many more American students into discovering the joy of challenging mathematics and developing their problem-solving skills. Given the reality of the standard high school mathematics curriculum as well as the practical realities of grading large numbers of olympiad-style problems, I think the AMC exams serve an important function in drawing many students in to attempting more challenging problems than they would otherwise see during their high school years.

    The problem I see is how to get more students to engage even in that initial entry level. It’s sad to see that, despite all the AMC efforts (designing four AMC entry level exams–10A,10B,12A,12B instead of the single AHSME that existed a decade ago) and despite the huge increase in demographics (the total number of students in high school is much larger now than it was a decade ago), the total number of students taking the AMC10/12 exams has actually dropped. I think that more encouragement and outreach is the answer.

  5. Tanya Khovanova’s Math Blog » Blog Archive » Multiple Choice Proofs:

    […] limit, this encourages students to stop thinking and go for guessing. I recently wrote an essay AMC, AIME, USAMO Contradiction, in which I complained about the lack of proofs in the first two rounds of math […]

  6. John:

    Oh, come on. If you can’t do easy math, then you shouldn’t be doing hard math. The AMC and AIME are designed so that even if you are a great, but slow and inaccurate, olympiad solver, there is still such a huge safety net. If you can’t get past AMC and AIME, chances are you can’t get past USAMO.

  7. Shah:

    to the USAMO qualifiers, AIME and AMC is pretty trivial. the main reason for AIME and AMC is to get students interested. for some reason, people feel that proofs are too hard. I agree with the fact that in AIME one computation mistake can cost you the entire problem, its pretty rough. HOWEVER, the main reason on why USA doesnt do as well as it could do is not because of the team selection test. its because in general, there is a LACK OF INTEREST IN MATH. this boils down to the teaching. i would say only about 15 (1%) of students understand the difference between real math and school math. the selection process is actually not bad because there are 2 rounds of proof tests and a summer camp. but the main problem is the competitiveness.

  8. J:

    Your argument is weak, Shah.
    Her point is there might be students who could excel at IMO but never even make it to USAMO because they do not pass AMC.
    You’re just reiterating the obvious fact that USAMO qualifiers do well at AIME and AMC.

  9. Edward:

    The AMC is not so much speed as it is if you can do the problems accurately. If you have taken MATHCOUNTS before, those problems are much more fast paced. I made it to the AIME this year in 8th grade, but the AMC 10 was fairly easy. I finished it with around 20 minutes to spare. A couple problems are fairly difficult as to my level, but the others are not easy to make mistakes on. MATHCOUNTS is verrry easy to mess up on, I missed national last year becaue I screwed up 4 questions I could do.

  10. T.:

    It would be very, very unusual for a serious IMO candidate to be filtered out by the AIME. It is essentially impossible for such a person to be stopped at the first round, the AMC.

    The problem that the AIME does create for the US IMO team is that students who are potential IMO material but aren’t naturally good at the AIME have to spend extra time in their training for the sole purpose of getting past this one strange exam. This reduces the time they can spend preparing for the proof-based olympiads.

    A more serious problem, for selecting an IMO team, is that the USAMO itself can be flaky. It is not unusual for IMO-worthy students (especially 12th graders) to be stopped at the USAMO because they had a bad day, or misread one problem.

  11. I agree:

    Two words: Tim Chu.

    Game over.

  12. Tanya Khovanova’s Math Blog » Blog Archive » Why Americans Should Study the Moscow Math Olympiads:

    […] have already written about how American math competition are illogically structured, for the early rounds do not prepare students for the later rounds. The first time mathletes […]

  13. Jimmy:

    This is a very good post. I myself know someone who was very bad at AMC, though good at AIME. One year he made USAMO barely because he aced the AIME. The next year he could only take AMC once because he had to go to a funeral the other date. He did not even pass to AIME that year. He was quite disappointed but after receiving his acceptance to MIT he did not care any more.

  14. MathyMom:

    I coundn’t agree more with you!

  15. David:

    “It would be very, very unusual for a serious IMO candidate to be filtered out by the AIME. It is essentially impossible for such a person to be stopped at the first round, the AMC.”

    This. The number of people with the brilliance and hundreds and thousands of hours of dedication necessary to be IMO high performers who can’t get past an easy math IQ test is zero basically. The fact that you only have a handful of examples (of people who didn’t even qualify for the IMO, much less place) “over the years” shows your point is off-base.

    Americans care less about the IMO compared to other countries for the same reason Soviets cared more about chess than all but one American named Bobby: opportunity. There are a lot more profitable and interesting ways to ply that math G and being a USAMO qualifier repeatedly or especially placing is usually enough to get one into MIT etc. and open the doors to these.

    The easiest way to solve the concerns about occasional slippage or time spent studying for the AIME is give one “AIME point” for every USAMO problem solved the prior year. That handles most of the concerns about budding IMO stars missing future USAMOs.

    “It is also important to note that the purpose of the AMC/AIME/USAMO is not just to select the most promising IMO team–I don’t think one could justify all the resources devoted to it, if that were the only purpose, but also to engage and inspire a large number of young students to develop mathematically.”

    Also this. These tests are among the only reliable ways for students to stand out and get into Caltech and MIT to pursue lots of non-math subjects that require high math ability.

    Calculus BC and the SAT are a joke and have way too low a ceiling, and the AMC/AIME are performing a crucial sorting function for all the future physicists, computer scientists, and engineers out there. This is more valuable to the US and its GDP than slightly better sorting for the one or two people who have the math IQ necessary to prove anything new these days. Those people are truly a different breed, and will find their way to a math Ph.D. with or without a cute little competition along the way.

  16. Ron:

    David, you are so wrong about that first statement, and your ignorance on the subject is quite apparent. You seem to not have taken AMC or AIME in the past few years or know much about qualification for USAMO or the cutoffs in recent years.

    There are numerous people who have scored highly on the USA(J)MO one year, getting USAJMO HM or even qualifying for Red MOP, but they fail to make USAJMO the next year or USAMO in any of their later years (11th/12th grade) in high school. Many were definitely at the level of blue MOP or TST group contention. I personally know several of them.

    One of these people who I know pointed out that the skillset for solving olympiad problems is often quite different from the skillset for computational problems, especially the AMC where competitors are at a much bigger time crunch. For example, solving olympiad geometry problems really does not help very much with solving AMC or AIME type geometry problems. In olympiad geometry, there is an emphasis on much more advanced and varied techniques (such as barycentric coordinates and projective geometry, inversion, etc.) while AMC/AIME geometry is much more “bashy” and uses things like dropping uninteresting perpendiculars, assigning variables, doing trig, etc. On the other hand, the synthetic geometry in olympiads is often very different. As one of my MOP friends noted, doing a lot of olympiad geometry can actually hurt your ability to solve AMC/AIME type geometry problems.

    The other problem is that with AMC/AIME type problems (especially AMC), it is always easy to make simple mistakes under the time pressure. No amount of olympiad experience can really help you with that. You have to prepare specifically for this. Being good at olympiads does not necessarily mean that you are good at solving these types of computational problems very quickly and accurately. These days, the cutoff for USAMO qualification is usually in the high 230s, and one needs a good AMC 12 score 126+ and a high 11+ on AIME to comfortably qualify for USAMO, which is no easy task, even if you’re very good at solving olympiad problems.

  17. old hand:


    you give examples of moderately elite outcomes (USAJMO HM or mid level MOP) so it’s not surprising some of the marginal (i.e. lucky) cases of this would not qualify the next year. The topic of discussion was mainly about realistic IMO candidates being filtered out by AMC and AIME, and as people said above the second is rare and the first is pretty hard to imagine as AMC12 is trivial for them. Maybe there are some deep but really slow thinkers, but the IMO and Putnam are also speed contests to a substantial extent (based on more advanced material of course).

    Of course olympiad level geometry helps on the AIME. To assert otherwise is ridiculous. All the little tricks that help with proofs come up as time savers in numerical problems of the AIME type. Deeper knowledge and problem solving experience accelerate work on simpler problems.

  18. Lazar Ilic:

    Fascinating old post to revive. I am hopeful that the MAA Mathematical Association Of America will continue to support the USAMTS United States Of America Mathematical Talent Search route with an AIME American Invitational Mathematics Examination score of 11.

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