What leads to success at math contests?

I think this is an important question to answer, not the least of reasons being that understanding how to learn is extremely useful both for teaching and learning. [1]

About a year ago [2], I posted my thoughts on what the most important things were in math contest training. Now that I’m done with the IMO I felt I should probably revisit what I had written.

It looks like the main point of my post a year ago was mainly to debunk the idea that specific resources are important. Someone else phrased this pretty well in the replies to the thread

The issue is many people simply ask about how they should prepare for AIME or USAMO without any real question. They simply figure that AOPS has a lot of successful people that excel at both contests, so why not see what they did? Unfortunately, that’s not how it works – that’s what this post is saying. There is no “right” training.

This is so obvious to me now that I’m going to focus more on what I think actually matters. So I now have the following:

  1. Do lots of problems.
  2. Learn some standard tricks.
  3. Do problems which are just above your reach.
  4. Understand the motivation behind solutions to problems you do.
  5. Know when to give up.
  6. Do lots of problems.

Elaboration on the above:

  1. Self-explanatory. I can attest that the Contests section on AoPS suffices.
  2. One should, for example, know what a radical axis is. It may also help to know what harmonic quadrilaterals, Karamata, or Kobayashi is, for example, but increasingly obscure things are increasingly less necessary. This step can be achieved by using books/handouts or doing lots of problems.
  3. Basically, you improve when you do problems that are hard enough to challenge you but reasonable for you to solve. My rule of thumb is that you shouldn’t be confident that you can solve the practice problem, nor confident that you won’t solve it. There should be suspense.

    In my experience, people tend to underestimate themselves — probably my biggest regret was being scared of IMO/USAMO #3’s and #6’s until late in my IMO training, when I finally realized I needed to actually start solving some. I encourage prospective contestants to start earlier.

  4. I think the best phrasing of this is, “how would I train a student to be able to solve this problem?”, something I ask myself a lot. By answering this question you also understand

    a. Which parts of the solution are main ideas and which steps are routine details,
    b. Which parts of the problem are the “hard steps” of the problem,
    c. How one would think of the hard steps of the solution,
    and so on. I usually like to summarize the hard parts of the solution in a few sentences. As an example, “USAMO 2014 #6 is solved by considering the N \times N grid of primes and noting that small primes cannot cover the board adequately”. Or “ELMO 2013 #5 is solved by considering the 1D case, realizing the answer is cn^k, and then generalizing directly to the 3D case”.

    In general, after reading a solution, you should be able to state in a couple sentences all the main ideas of the solution, and basically know how to solve the problem from there.

  5. In 2011, JMO #5 took me two hours. In 2012, the same problem took me 30 seconds and SL 2011 G4 took me two hours. Today, SL 2011 G4 takes me about five minutes and IMO 2011 #6 took me seven hours. It would not have been a good use of my time in 2011 to spend several hundred hours on IMO #6.

    This is in part doing (3) correctly by not doing things way, way over your head and not doing things way below your ability. Regardless you should know when to move on to the next problem. It’s fine to try out really hard problems, just know when more time will not help.

    In the other direction, some students give up too early. You should only give up on a problem after you’ve made no progress for a while, and realize you are unlikely to get any further than you already are. My rule of thumb for olympiads is one or two hours without making progress.

  6. Self-explanatory.

I think the things I mentioned above are at least extremely useful (“necessary” is harder to argue, but I think you could make a case for it). Now is it sufficient? I have no idea.

Footnotes
  1. The least of reasons is that people ask me this all the time and I should properly prepare a single generic response.
  2. It’s only been a year? I could have sworn it was two or three.
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16 thoughts on “What leads to success at math contests?

  1. I would say that the strategy you have described is not anywhere close to being the best strategy for olympiads, but it is quite possibly fairly close to the best FORMALIZABLE strategy. (In particular, this is what I think is commonly referred to as the “Chinese strategy.” On the other hand, in my opinion, the Chinese strategy is actually pretty terrible.)

    Your strategy mostly revolves around the problem-reflection cycle. I find the main issue with this to be that “motivation” is a very cheap form of understanding. The reason why the method works so well anyways is because olympiad math doesn’t have much depth, so this is not actually a huge issue.

    The main issue with “motivation” is that math is all about interconnections. With a “motivation”-based approach, understanding of interconnections is a secondary objective. The main objective a good sense of when each method is applicable. This is not a very deep skill.

    So what would I put as a bullet point? I would put the following as a bullet point: Whenever you see a problem you really like, store it (and the solution) in your mind like a cherished memory. (You really should have a strong emotional attachment to the problem for this to work.) The point of this is that you will see problems which will remind you of that problem despite having no obvious relation. You will not be able to say concretely what the relation is, but think a lot about it and give a name to the common aspect of the two problems. Eventually, you will see new problems for which you feel like could also be described by that name. Do this enough, and you will have a very powerful intuition that cannot be described easily concretely (and in particular, that nobody else will have).

    Here’s an example. A long time ago, I solved some ISL G2 which was actually combinatorial geometry. It had a solution with the extremal principle that I really liked, and I was pretty impressed with myself for coming up with it. I soon solved some other extremal principle combinatorial geometry and felt it was very similar. My “explanation” for it was “Well, you have points, and you have nearby points and faraway points, but you can’t have anything TOO near….” Of course, this is a meaningless explanation. But I basically started calling things along these lines “local-global principles”, which is not really a name anyone should endorse. And pretty quickly I could basically kill almost every combinatorics problem simply by understanding it in terms of my “local-global principles”. (In particular, I can give you an explanation of every combinatorics-themed chapter of Engel in terms of local-global principles.)

    Why does this process not count as formalizable? Because you really have to be genuine in your liking the problem and your wanting to give a name to the relation. Otherwise you might just say “oh they’re related by the extremal principle”, which is really not a relation at all. As I can’t give you a formalization of how to be genuine, I’m not going to count this as formalizable.

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    • That’s a good point, thanks. In my head I never bothered to seperate the notions of intuition and motivation and used the words interchangably (probably a big part of why my (4) above is so muddled and vague), but you are definitely right that if you define “motivation” as “we apply X strategy because of Y situation” then this is indeed only a small part of what constitutes strong intuition. I do feel like you can pick up this from a (deeper) reflection (I think you’re suggesting this as well in your bullet point) but as you said this is not definitely not formalizable.

      (Please correct me if I’m misinterpreting you.)

      And agree 100% that emotions are way more useful than they should be for intuition-based purposes.

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    • The “solving/creativity-based” structure of contests encourages people to focus much more on the “motivation-purely-for-solving” (I believe this is what you’re criticizing) rather than the “natural story/context/interconnections”, but I think this is just a (potentially easily-fixable) problem with current Olympiad culture.

      For example, if authors generally wrote on the story/context/interconnections behind their problems, e.g. how they came up with the problems, related ideas/themes, or even just liberal doses of vaguely-related links/references, I think everyone would gain a lot more out of contests. (This is something I’ve sort of tried to do in the comments sections of solutions to my problems, but still I always subconsciously leave out a lot of background.)

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      • Well in my opinion, basically all problems of olympiads are problems are of olympiad culture :) And I would classify this as a very hard change. Maybe you can create the change at the level of black mop, that doesn’t sound impossible. But when you consider how much time has been spent to explaining extremely basic aspects of the nature of mathematics and then note that the community completely fails to understand these aspects past being able to repeat them….

        I’ve decided that there is too much philosophizing and too little storytelling on these topics. As a mathematical example, take “Look at small examples.” People always say this, but it’s never given a meaning besides “hey if you look at small examples sometimes you can guess a pattern!!!” But there’s a huge amount more to it. Nobody bothers to explain this, maybe because olympiad culture conditions us to believe that anything that we can’t formalize isn’t really real. And on the other side of the coin, people aren’t looking for any real understanding of the nature of mathematics. The culture tells us that the only thing that matters is being good, and so nobody will bother to really understand something which isn’t obviously helpful. And that’s fine, but I’m skeptical of any major short term change happening. If any major short term change were possible, the entire US team should get 42s on the IMO, which clearly didn’t happen.

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  2. Thanks Evan.
    I think people are also looking at time frame (eg how long does it take to go from 3 on AIME to 11+ on AIME). Any insight?

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  4. You say in your post that the Contest section of AoPS suffices for problems. I am currently working on AIME-level problems, and I was wondering if you knew what types of problems in the contest section are similar to AIME level, because I know I will end up running out of AIME problems eventually.

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  6. Hey Evan, I don’t know you personally but somewhat enjoyed the contests you prepared back when I was in high school and am glad to have found your blog now that I’m studying math. I never really found success in math competitions, and while I now recognize a large part of it were personal character flaws (insecurity, cult of genius, thinking that because I learned well in camps that was enough to improve, kidding myself that because I could solve some IMO problems, I should only try to do IMO problems) and naive attitudes towards math, as well as sort of quitting a bit earlier than expected, I think that in a sense the advice of “just solve a ton of problems” and “work hard” was kind of actively harmful to my development.

    With that in mind, I have some comments that are perhaps more relevant if you want to look at contest math as a way of developing good problem solving skills en route to becoming a mathematician and at success at the Olympiads as a consequence of this. I think at least one thing that I clearly noticed from my training and how it was quite lacking in “real mathematics” now that I am at university was the extremely low level of importance placed on actual understanding of mathematics that was somehow encouraged. I say this because now that I’m at university, I tried to apply some of those same methods to my courses, and sometimes failed miserably at really getting anything done. This is because even if I could solve some problems for a course, I didn’t really get what it meant to understand math and I am now only slowly discovering it, while having to fight a bunch of bad habits that I created along the way.
    Don’t get me wrong, I am blissfully happy to have done contest math in high school, because otherwise I probably wouldn’t have thought of math as a viable career path or major, and I met some cool people.

    Anyways, I think that there are some cool parts about contest math that are somewhat hard to learn just by going to university. For instance, an enormous positive is the large amount of solution methods you get to learn. I’m pretty sure that most of my classmates who didn’t do competitions don’t know what strong induction is or have a good feeling for when proof by contradiction can kill a problem instantly. The ones who do certainly learned it on their own. There’s also the sort of independence that it forces you to have. It creates the habit that even if you don’t have to learn something for class, you might want to work it out for yourself. The thing that is problematic about “solve a ton of problems” and “work hard” is that while doing that can certainly get you really high scores at contests, one can probably solve most olympiad problems by just really understanding what’s behind them, in the same way that in college math, solving a problem is the same thing as understanding the theory well enough. Of course, that probably isn’t enough to actually get a super high score, but at that point, one probably doesn’t really feel lost when approaching any of these questions (and training camps probably do suffice!). In that sense, I feel it’s quite a pity that some things are made to look so artificial or magical, because in actual mathematics, there are rarely magical things going on (some look like magic, because someone removed the context in which they were developed).
    I’m curious what your experience with these kinds of things was(as probably some of the people training you had a real math background), and if you remember anything like this from camps.

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  8. I have a grade 4 question that has stumped everyone. The following nos and alphabets are entered in the 4 quadrants of a circle.It is
    5 | 9
    —- -> H 5 –> E H –> 3 E –>?
    H | E E 3 3 7 7 L

    Can anyone tell me the logic?

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  9. Hi
    I have a grade 4 question that has stumped everyone. The following nos and alphabets are entered in the 4 quadrants of a circle.It is
    5 | 9
    —- -> H 5 –> E H –> 3 E –>?
    H | E E 3 3 7 7 L

    Can anyone tell me the logic?

    Like

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