Math contest platitudes, v3

I think it would be nice if every few years I updated my generic answer to “how do I get better at math contests?”. So here is the 2019 version. Unlike previous instances, I’m going to be a little less olympiad-focused than I usually am, since these days I get a lot of people asking for help on the AMC and AIME too.

(Historical notes: you can see the version from right after I graduated and the version from when I was still in high school. I admit both of them make me cringe slightly when I read them today. I still think everything written there is right, but the style and focus seems off to me now.)

0. Stop looking for the “right” training (or: be yourself)

These days many of the questions I get are clearly most focused on trying to find a perfect plan — questions like “what did YOU do to get to X” or “how EXACTLY do I practice for Y”. (Often these words are in all-caps in the email, too!) When I see these I always feel very hesitant to answer. The reason is that I always feel like there’s some implicit hope that I can give you some recipe that, if you follow it, will guarantee reaching your goals.

I’m sorry, math contests don’t work that way (and can’t work that way). I actually think that if I gave you a list of which chapters of which books I read in 2009-2010 over which weeks, and which problems I did on each day, and you followed it to the letter, it would go horribly.

Why? It’s not just a talent thing, I think. Solving math problems is actually a deeply personal art: despite being what might appear to be a cold and logical discipline, learning math and getting better at it actually requires being human. Different people find different things natural or unnatural, easy or hard, et cetera. If you try to squeeze yourself into some mold or timeline then the results will probably be counterproductive.

On the flip side, this means that you can worry a lot less. I actually think that surprisingly often, you can get a first-order approximation of what’s the “best” thing to do by simply doing whatever feels the most engaging or rewarding (assuming you like math, of course). Of course there are some places where this is not correct (e.g., you might hate geometry, but cannot just ignore it). But the first-order approximation is actually quite decent.

That’s why in the introduction to my geometry book, I explicitly have the line:

Readers are encouraged to not be bureaucratic in their learning and move around as they see fit, e.g., skipping complicated sections and returning to them later, or moving quickly through familiar material.

Put another way: as learning math is quite personal, the advice “be yourself” is well-taken.

1. Some brief recommendations (anyways)

With all that said, probably no serious harm will come from me listing a little bit of references I think are reasonable — so that you have somewhere to start, and can oscillate from there.

For learning theory and fundamentals:

For sources of additional practice problems (other than the particular test you’re preparing for):

  • The collegiate contests HMMT November, PUMaC, CMIMC will typically have decent short-answer problems.
  • HMMT February is by far the hardest short-answer contest I know of.
  • At the olympiad level, there are so many national olympiads and team selection tests that you will never finish. (My website has an archive of USA problems and solutions if you’re interested in those in particular.)
    The IMO Shortlist is also good place to work as it contains proposals of varying difficulty from many countries — and thus is the most culturally diverse. As for other nations, as a rule of thumb, any country that often finishes in the top 20 at the IMO (say) will probably have a good questions on their national olympiad or TST.

For every subject that’s not olympiad geometry, there are actually surprisingly few named theorems.

2. Premature optimization is the root of all evil (so just get your hands dirty)

For some people, the easiest first step to getting better is to double the amount of time you spend practicing. (Unless that amount is zero, in which case, you should just start.)

There is a time and place for spending time thinking about how to practice — one example is if you’ve been working a while and feel like nothing has changed, or you’ve been working on some book and it just doesn’t feel fun, etc. Another common example is if you notice you keep missing all the functional equations on the USAMO: then, maybe it’s time to search up some handouts on functional equations. Put another way, if you feel stuck, then you can start thinking about whether you’re not doing something right.

On the other extreme, if you’re wondering whether you are ready to read book X or do problems from Y contest, my advice is to just try it and see if you like it. There is no commitment: just read Chapter 1, see how you feel. If it works, keep doing it, if not, try something else.

(I can draw an analogy from my own life. Whenever I am learning a new board game or card game, like Catan or Splendor or whatever, I always overthink it. I spend all this time thinking and theorizing and trying to come up with this brilliant strategy — which never works, because it’s my first game, for crying out loud. It turns out that until you start grappling at close range and getting your hands dirty, your internal model of something you’ve never done is probably not that good.)

3. Doing problems just above your level (and a bit on reflecting on them)

There is one pitfall that I do see sometimes, common enough I will point it out. If you mostly (only?) do old practice tests or past problems, then you’re liable to be spending too much time on easy problems. That was the topic of another old post of mine, but the short story is that if you find yourself constantly getting 130ish on AMC10 practice tests, then maybe you should spend most of your time working on problems 21-25 rather than repeatedly grinding 1-20 over and over. (See 28:30-29:00 here to hear Zuming make fun of them.)

The common wisdom is that you should consistently do problems just above your level so that you gradually increase the difficulty of problems you are able to solve. The situation is a little more nuanced at the AMC/AIME level, since for those short-answer contests it’s also important to be able to do routine problems quickly and accurately. However, I think for most people, you really should be spending at least 70% of your time getting smarter, rather than just faster.

I think in this case, I want to give concrete descriptions. Here’s some examples of what can happen after a problem.

  • You looked at the problem and immediately (already?) knew how to do it. Then you probably didn’t learn much from it. (But at least you’ll get faster, if not smarter.)
  • You looked at the problem and didn’t know right away how to start, but after a little while figured it out. That’s a little better.
  • You struggled with the problem and eventually figured out a solution, but maybe not the most elegant one. I think that’s a great situation to be in. You came up with some solution to the problem, so you understand it fairly well, but there’s still more for you to update your instincts on. What can you do in the future to get solutions more like the elegant one?
  • You struggled with the problem and eventually gave up, then when you read the solution you realize quickly what you were missing. I think that’s a great situation to be in, too. You now want to update your instincts by a little bit — how could you make sure you don’t miss something like that again in the future?
  • The official solution quoted some theorem you don’t know. If this was among a batch of problems where the other problems felt about the right level to you, then I think often this is a pretty good time to see if you can learn the statement (better, proof) of the theorem. You have just spent some time working on a situation in which the theorem was useful, so that data is fresh in your mind. And pleasantly often, you will find that ideas you came up with during your attempt on the problem correspond to ideas in the statement or proof of the theorem, which is great!
  • You didn’t solve the problem, and the solution makes sense, but you don’t see how you would have come up with it. It’s possible that this is the fault of the solutions author (many people are actually quite bad at making solutions read naturally). If you have a teacher, this is the right time to ask them about it. But it’s also possible that the problem was too hard. In general, I think it’s better to miss problems “by a little”, whatever that means, so that you can update your intuition correctly.
  • You can’t even understand the solution. Okay, too hard.

You’ll notice how much emphasis I place on the post-problem reflection process. This is actually important — after all the time you spent working on the problem itself, you want to update your instincts as much as possible to get the payoff. In particular, I think it’s usually worth it to read the solutions to problems you worked on, whether or not you solve them. In general, after reading a solution, I think 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.

For the olympiad level, I have a whole different post dedicated to reading solutions, and interested readers can read more there. (One point from that post I do want to emphasize since it wasn’t covered explicitly in any of the above examples: by USA(J)MO level it becomes important to begin building intuition that you can’t explicitly formalize. You may start having vague feelings and notions that you can’t quite put your finger on, but you can feel it. These non-formalizable feelings are valuable, take note of them.)

4. Leave your ego out (e.g. be willing to give up on problems)

This is easy advice to give, but it’s hard advice to follow. For concreteness, here are examples of things I think can be explained this way.

Sometimes people will ask me whether they need to solve every problem in each chapter of EGMO, or do every past practice test, or so on. The answer is: of course not, and why would you even think that? There’s nothing magical about doing 80% of the problems versus 100% of them. (If there was, then EGMO is secretly a terrible book, because I commented out some problems, and so OH NO YOU SKIPPED SOME AAAHHHHH.) And so it’s okay to start Chapter 5 even though you didn’t finish that last challenge problem at the end. Otherwise you let one problem prevent you from working on the next several.

Or, sometimes I learn about people who, if they do not solve an olympiad problem, will refuse to look at the solution; instead they will mark it in a spreadsheet and to come back to later. In short, they never give up on a problem: which I think is a bad idea, since reflecting on missed problems is so important. (It is not as if you can realistically run out of olympiad problems to do.) And while this is still better than giving up too early, I mean, all things in moderation, right?

I think if somehow people were able to completely leave your ego out, and not worry at all about how good you are and rather just maximize learning, then mistakes like these two would be a lot rarer. Of course, this is impossible to do in practice (we’re all human), but it’s good to keep in mind at least that this is an ideal we can strive for.

5. Enjoy it

Which leads me to the one bit that everyone already knows, but that no platitude-filled post would be complete without: to do well at math contests (or anything hard) you probably have to enjoy the process of getting better. Not just the end result. You have to enjoy the work itself.

Which is not to say you have to do it all the time or for hours a day. Doing math is hard, so you get tired eventually, and beyond that forcing yourself to work is not productive. Thus when I see people talk about how they plan to do every shortlist problem, or they will work N hours per day over M time, I always feel a little uneasy, because it always seems too results-oriented.

In particular, I actually think it’s quite hard to spend more than two or three good hours per day on a regular basis. I certainly never did — back in high school (and even now), if I solved one problem that took me more than an hour, that was considered a good day. (But I should also note that the work ethic of my best students consistently amazes me; it far surpasses mine.) In that sense, the learning process can’t be forced or rushed.

There is one sense in which you can get more hours a day, that I am on record saying quite often: if you think about math in the shower, then you know you’re doing it right.

 

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17 thoughts on “Math contest platitudes, v3

  1. I’m still surprised that people come to me for advice expecting me to say something other than a rehash of the points here, or expecting me to give a quick way to get good results. It shouldn’t be surprising that it takes a time investment to get better; if there was quick way then everyone would be doing it (EMH).

    I’m waiting for the day someone approaches me and asks me advice because they think they’ve been practicing more than their proportional improvement. That would be great.

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    • I do get questions of the form “I haven’t improved for the last T time, can you advise?”, but most of the time it’s not accompanied by a description of what their past practice has been. I’ve started replying to these by simply asking “what have you tried?” since it’s much easier to diff that way.

      On the other hand, I do think there are people who don’t score as well as they hope, despite having really worked hard and not doing anything obviously wrong. They are somewhat rare, but they do exist, and I often feel bad because I don’t really know what to tell them. Typically they succeed later on in non-contest settings though, because of various non-math skills (or just sheer work ethic) that they pick up in the process.

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  2. I agree strongly with most of this post, except for the AMC/AIME bit where you say “it is not necessary to continually drill problems 1-20 over and over if you are already getting 130-ish. I agree this does absolutely nothing for your problem solving abilities, but I would still argue this helps your results(which unfortunately are very important if you wish to take the USAMO), regardless of how boring it is.

    Two years ago, I was 1.5 points off the JMO cutoff, and consequently spent most of the next year practicing for JMO/AMO. When practicing for AMC 10/12 and AIME, I did only a few practice tests and just worked on the hard problems, and I was satisfied with my ability to solve at least 3/5 of the last five on AIME within 30-45 minutes or so. However, when doing this, I completely underestimated the power of the first few problems on the AIME/AMC; on the AMC 12, I didn’t even reach the last 5 on the 12 because of a few timesinks (which were very easy problems, just me being somewhat dumb) and made lots of “silly mistakes”(I generally don’t like this term because whenever I use it it causes me to gloss over them instead of analyzing them, but whatever). As a result, I got a 106.5/124.5 on the AMCs and a 8 on AIME, missing the cutoff by 7.5 points this time and my chance at MOP for that year, even though I had gotten problems 12,13, and 15 right on the AIME. In short, I spent most of my time agonizing about how hard the last five were, thinking I had solved 1-10 easily when in reality I had not(missing 5 of them in the end).

    (as another anecdote, this approach led to my downfall at PUMaC; I had placed 15th in combo the year before, and then worked on the last few, and then missed 1,3,5 on the real test).

    The point is not about how unfortunate I was or anything(it was 100% my fault), but rather that drilling problems 1-20 allows you to get used to the time conditions and management, and also be used to “timesinks” (on the 12 there seems to be at least one “timesink” between problems 8-14, from what I have seen). By only working on the last five, you(or at least me) subconsciously undermine the importance of the first ten(or twenty for AMC) and risk missing those. You are essentially saying “my score is 120+6k where k is the number of problems I solve in the last five”, which is clearly not true.

    I think this does not apply for USAMO(i.e., there is little necessity to solve 1/4s after you are good enough at them) , because once you figure out the main idea(s) of a USAMO problem(excluding pure bash solutions, but even then you can tell when you mess up on these because the end conclusion is something like 1=1, done, instead of arbitrary answer on AMC) you are basically done (and given you write it up properly should get full credit), whereas on the AMC/AIME the main ideas are typically easily identified but then a whole lot of number crunching-which you must do accurately-separates you from getting those 6 points. Therefore, when practicing for the AMC/AIME, I would recommend drilling those easy problems, as by doing these you practice the skills needed to cope with “silly mistakes” and “timesinks”. If you are unable to reach the last five on a practice test, I would recommend analyzing the first twenty and see where time could be cut(in particular if you wasted a bunch of time on a problem because you misread it or something) and then attempt the other five, timing yourself for each problem to do so before looking at the solutions/checking answers. This might not help you grow that much as a problem-solver, but it will help your AMC score, which will enable you to try the tests which actually stretch your problem solving muscles.

    The last paragraph is the main reason why I have extremely different mindsets to AMC practice and USAMO practice: AMC is a contest at which I must do my best at(so I can actually make the USAMO), one which I do not enjoy particularly. AMC practice feels more like a job, a mandatory task, a school assignment. On the other hand, I can do USAMO problems whenever I want, regardless of what mental state I am in, and USAMO practice feels more like a hobby for me. For AMC, my preparation consists of doing every recent past AMC and reflecting, while for USAMO my preparation consists of doing whatever problem(s) I want whenever I want. And from what I have heard from other top students (at the risk of sounding overconfident of my own abilities), they see AMC as a hurdle that must be overcome rather than as a source of fun, hard problems. Most of the time, the truly great problems on the AMC are harmed(in my opinion) by the number crunching they require and/or their late placement on the test (so very few people will even see them, let alone attempt them).

    sorry if my ideas here are very loosely organized(or if I repeated things multiple times). Also this post was intended to be about the importance of doing practice AMCs to their full but seems to have turned into a post about why I don’t like the AMCs. Oops.

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    • also, if I seem too critical of short- answer contests, I would like to say that I enjoy PUMaC, HMMT, and CMIMC problems, because they are hard, fun, and typically don’t require a load of number crunching (and the time limit feels more generous than on AMC/AIME, probably because of the lack of number crunching).

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      • I realized that my post could be misconstrued as me advising people to practice all of the problems when practicing for AIME/AMC; this is not the case. My only message from that terribly long post was that before the AMC, do full AMC tests, and don’t think of your score as dependent on the number you get correct in the last, say , 5 problems. If you are practicing for the AMC at, say, the beginning of the year(and really anytime until the week(s) before the exam), you should be focusing only on the hard problems for sure and there is no need to waste time with the easier ones; just don’t assume that you will easily get all of the easier ones correct as this will cause you to underestimate your own ability to make “silly mistakes” or fall into “time sinks”.

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    • This is a fair point. I guess in my post I was focusing on how to get better at problem-solving, which is not quite the same as how to maximize scores, particular with easier/faster contests (as you’ve noted).

      Though, I expect someone in your situation (in which you *can* solve all the problems but just aren’t fast/accurate enough) probably will notice this. I expect the situation of “never really getting the last five problems” is more common.

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  3. Couldn’t agree more. There’s no perfect plan when it comes to preparing for math olympiads. However, there are clearly bad strategies.

    What’s your take on doing an “IMO style” test a day instead of doing problems in order to improve in a specific subject. My country’s deputy leader had us, the team for IMO, take such a test each day in our training program (for about 40 days), but we genuinely felt no improvement by training in such a way.

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    • Gut feelings:

      (1) The biggest comment I have is that if “genuinely felt no improvement” you are probably right. This applies to anything else. I think most IMO-level contestants have the self-awareness to realize when they are improving or not. (Don’t take my advice over your own freedom of though. You are the world expert in you.)

      (2) To respond to the specific question, I think IMO style tests are important but to a certain degree. One per day for 40 days does seem excessive to me. (But to push the opposite direction, at USA MOP we have 9 tests and I think this is too *few*. I would push to 12 tests if I was dictator.)

      (3) Obviously the situation is worse if you do practice tests to the exclusion of everything else.

      (4) My feeling is that it also depends on where you are in terms of theory/fundamentals. Especially in geometry — I think if you don’t know the standard material you should be working through my book or something similar rather than just doing tests. Conversely, the black MOP kids (top ~12 in the USA, most of who might gold-medal on a good day, many who are certainly better than me) know way more than stuff they need to, and for them I find it hard to think of things that would be significantly better than more practice tests.

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  4. So, I completely agree with your advice.

    For someone who got a 105 on the AMC 10A 2019, what contests would be above their level?

    I know this goes against not looking for the right training, however, I just want some examples.

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    • Later half of AMC10/12’s, maybe COMC (Canada’s contest) come to mind as examples of things that could be right. Older AIME’s (say pre-1995) may also be suitable. Note there is a gradient of difficulty within any individual contest, too.

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    • @wow to reply to your second comment, anything is possible. Personally, I went from solving about 4-6 questions on AIME to solving about 10-11 in about three weeks (before the AIME…), so depending on your diligence I would say that anything is possible.

      by the way, I am pretty sure Evan said explicitly somewhere in this post that questions regarding “maximum improvement in x time” are very difficult to answer. just do your best, and let the cards fall after that.

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  5. On #2: I feel that this type of thinking stems from exposure to AMC/AIME-style questions which oftentimes require you to think and make a drastic simplification before diving in. Now there’s a tradeoff between highbrow theorizing and lowbrow plug/chug, with the former being better if you’ve seen a technique before (or if you’re sure of an upper bound on how much time you need to theorize), and the latter if not. When it comes to, say, reading model theory for the first time, the latter regime seems more optimal. To conclude: math contests prepare you pretty badly for research!

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