Olympiads: The Infinitely Overdue Retrospective

I put this question in my FAQ, because at least two people have asked me this question, and that’s how frequent a question needs to be to be on my FAQ: I got an IMO¹ gold medal in 2012, as a ninth grader, and an IOI gold medal in 2014, as an eleventh grader. I could have kept going to either, or even decided to try taking the IPhO or something, but I didn’t. Why not?

The short answer: It was a rough utilitarian calculation. By continuing, I would probably displace somebody else who would gain more from being on an IMO/IOI team than I would. Besides, I wanted to do other things in high school, so I wasn’t losing much.

I think the short answer actually captures most of my thinking when I made the decision back then, and it’s not really new; I said as much at the end of 2013. But behind it was a lot of complex thoughts and feelings that I’ve been ruminating over and trying to put into words for the better part of a decade. Hence, this post.

There is a natural question that precedes the frequently asked one that I have never been asked, something I am now realizing I never honestly asked myself and never tried to answer deeply: Why did I participate in the IMO and the IOI in the first place?

There are a lot of obvious answers. I got the opportunity and the motivation to learn a lot of interesting math and algorithms, and meet a lot of smart people. There are less obvious answers, such as Evan Chen’s list of lessons from math olympiads. (The post is specifically about math, but I think what it says generally applies to programming competitions and probably most other olympiads too. I am also going to somewhat interchangeably discuss math olympiads and all olympiads for the rest of this post, just because I know the most people talking about math olympiad philosophy; it should be possible to generalize similarly.) Practically, if materialistically, the medals helped my college and internship applications and netted me some scholarship money. Besides, it was fun.

But looking back at all of this, I’m afraid the number one reason I participated in the olympiads is that I thought that was what people like me — people who discover in middle or high school that they’re good at math — were “supposed” to do.

I don’t regret participating in these competitions. What I do regret is not thinking sooner, and more carefully, about why I was participating in them. What were my underlying long-term goals and values? How would competitions help me achieve them? In what ways would competitions not help me achieve them?

I listed above that competitions enabled me to meet a lot of smart people. I’m sure I realized this and enjoyed it, but in hindsight I was pretty bad at making the most of it by maintaining relationships with these people. I do wonder if, to some extent, the way competitions work led me away from this. Olympiad contestants may prepare for contests by solving problems collaboratively that they can’t solve by themselves, but only with the implicit recognition that this is unrealistic for the competition that matters at the end, so you have to turn everything into lessons you can apply yourself or it doesn’t count. However, solving (liberally defined) problems collaboratively is realistic pretty much everywhere else, and is a better foundation for longer-term friendships. On the other hand, this isn’t as inherent a limitation of olympiads as I just made it seem, since there are plenty of meta-olympiad tasks that contestants could work together on without needing to worry about keeping things realistic. How do you best teach a concept to somebody else? How do you prepare useful study materials for others? How do you decide which problems to practice next? How do you stay motivated while practicing? I wish I had tried harder to do and think about these things with others.

Another thing I did less of was coping with risk and the possibility of failure,² especially in light of the last part of Evan’s essay, which I hope you read.

I really want you to work hard, but I really think if you don’t do well, if you fail, it’s better to you.

I had a hard time relating to this when I first heard it, but it makes sense if you think about it. What I’ve tried to argue is that the benefit of math contests is not that the contestant can now solve N problems on USAMO in late April, but what you gain from the entire year of practice. And so if you hold the other 363 days fixed, and then vary only the final outcome of the USAMO, which of success and failure is going to help a contestant develop more as a person?

I think I met my goal each of the three times I participated in an international olympiad, and this probably caused me to develop less as a person. Looking back, I don’t think I really thought about the possibility of failure, and what my contingency plans would be if I failed. But I started down the olympiad track so early, against such a small pool of contestants from my country, that I think I could have been justifiably confident about making the IMO team eventually. I am not totally sure what I wish I had done differently here; I think all three of my goals were reasonable balances between realistic and ambitious, and I got pretty lucky. But I wish that after olympiads, I had kept this in mind and pushed myself harder towards loftier goals — in fact, that I had pushed myself harder to simply find or formulate these loftier goals…³ which brings me to my last point.

The biggest thing I think I failed to do while participating in olympiads was the fairly meta task of thinking about, refining, and working towards less well-defined goals, such as (but not only) figuring out what I wanted to do with my own life. The IMO is a very regular event with a clear objective function that was handed to me on a silver platter as something to optimize for. There are huge unknowns, i.e. what the problems on contest way will be, but they are known unknowns. The path to success on the competition can be phrased very simply, albeit uselessly: become able to solve more problems. You get a math problem, and you need to figure out the right key idea or ideas that will solve it. Then you have to write up a solution rigorously, without making any logical mistakes (which I think I was relatively good at). The objective function was so detailed and so multifaceted that I was able to put years of effort into optimizing it without thinking very hard about my underlying objective function. What did I really care about? Was I satisfied with “getting a gold medal on the IMO” as some kind of fundamental inherent goal — the Mount Everest that people climb “because it’s there”⁴ — and should I be doing everything in my power to get there?

To be clear, I don’t think there is anything wrong with deciding to do the IMO “because it’s there”. Everybody has their own life to live and their own meaning to find. Besides, as I’ve already said, trying to do well at math olympiads gives you plenty of auxiliary benefits. I can’t even say whether my past self would have been satisfied with such a justification for doing the IMO. But present me is definitely not.

What are my life goals? The most salient feature is that I care a lot about creation — making new things exist, things that would not exist otherwise.⁵ Ideally, I’m creating things that many other people can use and enjoy. As educational and as fun as taking the IMO was, it and similar contests do not involve creation to me, because they consist of producing solutions to problems specifically chosen to have accessible “olympiad-size” solutions, as hard as they may be to find. Again, however, this isn’t an inherent limitation of olympiads, since there are countless olympiad-adjacent tasks that do involve creation: writing new olympiad training material, composing new problems, running a successful math club or camp,⁶ hell, just maintaining an interesting blog. Upon reflection, I always found other olympiad competitors more impressive when they were successful at one of these things rather than just when they achieved any high scores or prestigious awards. And this unspoken feeling that I hadn’t done the same is why the pride and joy at winning all those awards faded so quickly. I don’t know why it took me five years to write it down. I regret not trying harder to do any of these things, precisely because these problems are less well-defined and there’s so much less guidance out there on how to solve them. I think these factors would have made the process of trying to solve these problems that much more educational — in particular, that they would have taught me some skills, such as those around managing poorly-defined goals, that I was not learning from just doing olympiads — and I would have valued the results that much more.

And while we’re on the topic: it’s weird to say it, but I don’t think I actually enjoy “competitions”, in the broadest possible sense as a type of interpersonal interaction, much at all. The approximate reason is that I dislike losing much more than I like winning, so the expected value is rather low. It’s not so much that I strongly hate losing as it is that winning, for me, gets tempered by feeling bad for the people who lost.⁷ I liked everything I got to do and everybody I got to meet while preparing for these competitions, which I’ve already talked about enough. I enjoyed solving olympiad problems, and I’m sure that I’d find a math competition infinitely more enjoyable than a competition on some random topic I care nothing about, but the inherent nature of competition itself? Meh.

Overall these considerations are what shaped my (lack of) participation in the Putnam competition in college. I took the class informally known as the Putnam seminar freshman fall, got to hang out with a few more math nerds, and took the Putnam because that was what I was supposed to do, but in the contexts of future years it became apparent that the Putnam was a math competition stripped down to the parts I enjoyed the least (and there was no way I would do well enough to make up for the experience with money).⁸ So it made sense to stop.

You know what would be cool? Same setup as the IMO, people from all across the world solving olympiad problems in a room, except we’re all working together to get a high enough total score to beat the final boss in an alternate reality game or something. I haven’t thought this out and I’m pretty sure it makes no sense. But it would be cool.

In any case, had I thought more carefully about all of these things, I might have gotten more value and satisfaction out of math competitions and decided to continue doing them. Or I might just as well have decided at a similar point that I was better off doing other things. But since I hadn’t done this thinking, the olympiads I did attend just left me feeling vaguely like I was missing something. After the novel experiences of being with like-minded peers who were as good as math as I was and the struggle and exhilaration of practicing, competing, and finally succeeding at my own goals, doing more olympiads felt like it would only be a (time-consuming, unreliable) way to acquire scholarships and impress colleges even more, and I didn’t see how I’d get more out of more competitions. Why not let other people have the peer group and go through the experience?

So far I’ve tried to phrase everything in this post in terms of my personal experience with and changing attitudes towards competitions, since I don’t have much confidence about other people’s experiences with competitions, which is exactly because I drifted away from the contest scene. But I have spent a lot of time wondering about whether I have any critique about olympiads in general.

Probably the most commonly heard critique is the “mantra” that math competitions are different from math research: how research problems take much longer to solve, are less “cut-and-dried” (as Terence Tao put it), and so on. Instead of going over these arguments in detail, I will just link to Evan Chen’s counterargument, which links to plenty of them. Evan argues that, although the “mantra” may be literally true, it’s too often used to criticize olympiads by implying that skill at math competitions is somehow not valuable or “virtuous”, while skill at math research is. The “mantra” is used to recruit math competitors into math research by suggesting that math research is somehow the one true test of skill and devaluing their accomplishments, which is harmful because the skills and accomplishments of math competition have inherent value and are too important to be confined to careers in math research. I am somewhat torn on the post, as you can see from my comments there, but now I fully agree with the core I summarized here; we’ll come back to it and some other ideas mentioned there later.

There are other critiques, like Cathy O’Neil’s first point that math competitions are unnecessarily discouraging to most competitors, which makes a lot of sense and is probably one of the more important effects of math competitions to be concerned about (since, by design, most participants in math competitions don’t win). However, since I lack the personal experience to argue convincingly about this, I won’t dwell on this; this short discussion is not meant to imply that I think this problem is not important. But I think about the inverse from time to time. Do math competitions provide the right kind of encouragement to the winners? O’Neil says no, but pretty much only in the context of winners who go on to study math in grad school:

[…] I also know quite a few people who were absolutely dominant in math contests in their youth who really seemed to suffer later on from that, especially in grad school. From my armchair psychologist’s perspective, I think it’s because they got addicted to the rush of doing math really fast and really well […] and when they get to grad school and realize how hard math really is, they can’t stand it.

But if you let go of the assumption that every successful math olympiad participant can or should go to grad school and continue in math research, this concern seems to be casting the net very narrowly. Put more starkly, and perhaps provocatively: what if the failing of math contests was not getting the aforementioned winners “addicted to the rush of doing math really fast”, but of misleading them into “continuing” to study math in grad school, even though it didn’t suit them?

I see some corroboration from another critique that was made as a comment to Evan’s post, even though it’s posed as a disagreeing opinion:

Several harmful ‘mantras’ that the Olympiad tribe use are: […]

1) Only Olympiad people are ‘legit’. The truest measure of intelligence is performance on Olympiads.

People with research experience […] are viewed with suspicion. The general idea is that, even if they know a lot of stuff in some narrow subfield, they must think slower than the Olympiad gods, and hence they’re less legit. […]

2) The only thing that matters is ‘being legit’. Smart people (i.e. Olympiad people according to (1)) are the only people worth talking to or working with.

Olympiad people wall themselves off, in places like SPARC (>80% Olympiad when I went), Random, and 3W. This causes some serious myopia and destroys the horizon-expanding benefits of college. How can you speak about economic policy, for example, if you don’t know anybody who had to work during high school? If you don’t know anybody that worries about being able to find a job? Not everybody can or will work in Silicon Valley and Wall Street.

I also ran into this several times when checking out finance companies. The key benefit, repeated over and over again, was that you got to work with Smart People. That your job was to rake in the money you deserved, from your ability to outsmart the market. Needless to say, mantra (2) is an extremely useful recruiting tool for finance, which sucks in Olympiad people like crazy. And though I agree with you on the questionable impact of pure math research, finance is definitely worse.

3) The only useful application of mental energy is solving hard technical problems.

Communication, teaching, and all creative and physical activities are severely undervalued. [This mantra] pushes Olympiad people out of many fields where they could make substantial impact. Pretty much any job that involves directly helping human beings is ruled out under (3). Public service is out. Law, medicine, and engineering is out. Administrative roles are out, despite their enormous possible impact. Software engineering is almost ruled out, unless you’re at some fancy startup with only Smart People (as per mantra (2)).

When I first read it, the comment seemed alarmingly accurate in reference to myself. 3W, aka Floorpi, is the hall I’m living on as an MIT undergraduate, and has had disproportionately many IMO medals per capita for a while.⁹ I applied to SPARC too, even though I didn’t end up attending.¹⁰ The career options I was considering and the ones I had mentally rejected were mostly all laid out, and seemed to reflect the career options of a lot of people I knew too, with a few exceptions. Although I had decided not to enter finance fairly early on for exactly the reasons described, there was a period of time after I read somebody’s post on Hacker News about their cool projects at some finance company during which I wondered what I was missing out and whether I had made the wrong decision. It was rereading exactly this comment that reminded me why I had made that promise to myself.¹¹

At the same time, the more I thought about it, the less I believed that the number of Olympiad people who literally believed these mantras was large enough to explain the effects. These mantras are actually quite extreme beliefs, and even if you relax the “only”s I cannot match them to the behavior of any Olympiad person I know; I certainly had never believed any of them. What gives?

I think more Olympiad people (including me) actually believed qualitatively weaker, more plausible-sounding versions of these mantras, which could very well still have similar impacts — statements along the lines of:

(1′) As an Olympiad person, I am the most similar to other Olympiad people.

(2′) Olympiad people are the happiest and most productive when working with and surrounded by similar people (i.e. other Olympiad people according to (1′)).

(3′) An Olympiad person’s mental energy is most effectively used to solve hard technical problems.

These statements sound much more plausible, partly because they were probably true for most Olympiad people when they were doing Olympiads. I know I am not alone among math Olympiad participants in finding a peer group at olympiad training camp that I had never had before, and having one of the best times of my life with them. And math olympiad problems were surely a better use of mental energy in high school than whatever the standard school curriculum had to offer. These beliefs also reinforce each other past high school. For instance, if you are the most similar to other Olympiad people, and if you trust they made career decisions wisely, then it makes sense to pick the same careers they’ve picked. Likewise, if you want to surround yourself with other Olympiad people, it makes sense to go into the same careers, even the same companies, that they do. In fact, each statement is probably still true for some Olympiad people after Olympiads, so I think it’s worth going into how they might become false.

Statement (1′) might break down because the universe of discourse becomes much larger after high school olympiads: not only do you get the opportunity to meet many more people, there are also many more axes along which you can be similar to somebody else. There are many interests that may be shared but aren’t accessible before college or similar levels of independence; more shared experiences to be forged; and more opportunities to join forces for a cause that you might never have considered in high school.

Statement (2′) perhaps stays the most likely to be true, but its implications change substantially if you reject (1′), and even then it still might not be totally true. I think the first part, that people are happiest when working with similar people, is quite plausible, but whether they are the most productive really depends on what kind of productivity you care about. There probably still are problems that are best solved with a high concentration of olympiad people who have similar skills, but there are also a lot of complicated problems with technical and nontechnical components that you probably want to solve with a combination of cross-disciplinary approaches from people with different perspectives, and if I had to guess I’d guess that there are more important problems today of the latter type than of the former.

And similarly, statement (3′) might break down because your metric of effectiveness changes when you actually want to achieve something different. A lot of things people do in high school are driven by growth and trying to distinguish yourself. In addition, you’re always racing the clock of eligibility for high school things, so it’s easier to justify focusing more on things you’re absolutely the best at, where you have the most chance of accomplishing something before you graduate. But choosing a career past high school is about far more than choosing what kind of problems you’re the best at and want to spend most of your time solving. Other things you might take into account include salary, work-life balance, future career flexibility, ease of travel, who you’ll be working with, who you’ll be working for, and perhaps the most meaningful,¹² the kind of impact — social, political, economic, cultural, you name it — you’ll have on the world and the people around you. There are many more reasons to do or learn something you’re not the best at.

And so collectively, I think these beliefs, combined with the characterization of math research as the “virtuous” follow-up career to math competitions, may push math olympiad contestants into more closed-off social groups and a narrower range of conventionally “mathy” careers and fields of study than ideal. To be maximally clear, I think there are some Olympiad people for which these beliefs will stay true — they will actually be best suited for research, or will still be most productive if they are near other Olympiad people in the future, or will care most about purely technical problems and be the right fit to continue working on them — but there are also many for which they won’t. And at least for myself, although I might have had an inkling of doubt about them that drove me to stop participating in olympiads when I did, I think these beliefs influenced where I am today more than they should have. I felt unquestioningly for a long time that I was suited, maybe even somewhat obligated, to continue studying math (doubly so because math is an absurdly flexible major at MIT) and to pursue a “math career”: if not pure math research, then research in an adjacent field like computer science or economics, or some comparably logicky or quantitative endeavor like finance or software engineering. I felt I should be looking for other people who did well in math contests and doing things similar to what they did, which in most cases happened to be all of the above.¹³ I felt guilty for taking the bare minimum of math classes needed to graduate, even though I was just more interested in the other classes. I think this was a mistake because of all the reasons I outlined that (1′) through (3′) break down past high school, plus exactly what the first Mantra literally says — olympiads are very different from research — but along an aspect that people don’t talk about enough: they’re different enough that being interested in one does not mean you will be interested in the other, and there’s nothing wrong with this!

I have absolutely read about these sentiments before, like in the original Mantra post and its quote from Richard Rusczyk in particular:

When people ask me, am I disappointed when my students don’t go off and be mathematicians, my answer is I’d be very disappointed if they all did. We need people who can think about these complex problems and solve really hard problems they haven’t seen before everywhere. It’s not just in math, it’s not just in the sciences, it’s not just in medicine — I mean, what we’d give to get some of them in Congress!

But I don’t think I understood it until now. Take running for Congress as an example — I have an instant reaction that it wouldn’t be the right choice for me, and can back it up with plenty of reasons to object: I don’t have the social or rhetorical skills, the ability or fortitude to handle public relations, the background knowledge in politics or a sufficiently close field; I have different strengths that I’d be wasting; I wouldn’t enjoy the job; I don’t believe that me being in Congress is an effective way to change the world in a way I care about. But if the last objection were sufficiently false, I think running for Congress would deserve serious consideration, even despite all the other objections; they’re surmountable, in some cases with meta-skills one could well have cultivated in math competitions: handling challenges, staying motivated, collaborating effectively, explaining things clearly, figuring out what things you need to learn and then learning them. And I don’t think I could possibly have had enough information to rule out that last objection before college. I still love math, but I care at least as deeply about a lot of other things, and just because I was never exposed to a competition about them, just because I dedicated the core of my high school career to math competitions before I could explore other possible interests and causes to pursue, doesn’t make these other passions less legitimate or less worth caring about.

The decision of whether to do math olympiads or some other academic competition or anything else altogether in high school is fairly different from the decision to keep studying mathematics in college, and is radically different from the decision to keep doing math for the rest of one’s career. These life decisions are too important and too different to be informed too strongly by a sense of tribal identity and attachment to the word “math” formed in high school.

And so although I’ve probably belabored the question that this post is nominally about, maybe I can try to answer a more interesting question, because after thinking through everything in this post, I would probably have chosen to do the same number of olympiads: why should I have stopped? I should have stopped to try doing radically different things to gather more information, both about what I did and didn’t enjoy, and about what I did and didn’t care about. I should have figured out that I wanted to make something big, learned why this is hard, tried to do it anyway, and realized how much more I valued the result.

I make no claims at all about how widely applicable that specific advice to my past self is, but I think the big question behind it worth thinking about earlier and more often, at every stage of your life, instead of ever just continuing to do things because it’s what you’ve done before or what society thinks you “should” do after what you’ve done before:

What do you really want?

thanks to Evan for reading a draft of this post