Musings about the IMO

 (Dylan here.) With the IMO drawing near, I thought I'd stray from maths to do some rambling. 

• What is the IMO?
• What is tested at the IMO? Do I know enough maths?
• Why do people care about the IMO? Should I care about the IMO?
• How do I get to/train for the IMO? How do I know if I'm improving?
• What happens after the IMO?

What is the IMO?

The IMO is a global maths contest for high school students. It's 4.5 hours x 2 days, 3 questions per day. And opening and closing ceremonies, excursions and cultural activities, opportunities to socialise. I'm not sure what the exact mission/vision is, but it's probably to do with maths community, creative and critical thinking, problem-solving, passion for maths, diversity and inclusivity. Wholesome stuff.

What is tested at the IMO?

Good question. There's no bound, but the general soft principle is that the question setup should be easily understood by a child, and a sufficiently bright student can cook up the ideas needed to solve it (so, anything involving a niche area of mathematical theory is much less preferred). The problems are not as open-ended as research problems, but not as closed-ended as, say, sudokus or A-level exam questions. In fact, students that are able to straddle both worlds (i.e. think broad, abstract, explorative, but also zoom in to specific, concrete, special cases) are in a good position to face the IMO gauntlet. 

All this is still pretty vague. To make things more concrete, I thought I'd drag up some data about 21 years of IMO problems that I collected 3 years ago when I was training the Singapore team. [Spoilers galore.]

I kept track of the subtopic (with cryptic symbols and omitted legend), and also the "key ideas" for each problem (it seems that I gave up on geom halfway). These aren't structure-specific (i.e. they tell you nothing about what the problem is about), and they are also inappropriate as hints; instead, these are "spoilers": the distilled ideas that communicate the essence of the solution (upon which one may "naturally" fill in the problem-specific details). 

Taking this birds-eye view seems uncommon to students who have spent years problem-solving. Some might even reject this as an oversimplification of IMO mathematics: "there's no way that so many unique ideas are captured in these vague non-descriptive terms!" and they may be right, depending on how they engage with maths philosophically. Nevertheless, there are some things the data hints at:

1. The flavour of IMO problems is subtly different from other contests such as CMO, CTST, APMO, RMM (though students use problems from all sources to train for the IMO/ISL-sourced selection tests). Simple ideas truly relying on minimal theory, are strongly preferred for the IMO. Problems 1 and 4 often require one idea, while 2 and 5 require two (those who have wrestled with problems of this difficulty before, will know there is inherent difficulty in finding the correct stepping-stone/halfway point/intermediate hypothesis/bridging condition). And (not visible in data above) there is also often a consistent mix between technical and intuitive flavoured problems.
2. The difficulty is not just in understanding a technique, but knowing when it should be "called to use". For instance, I'm sure you're comfortable with the fact that every integer larger than 1 has a smallest prime factor. Yet when a specific diophantine is presented to you, you have to decide (consciously or unconsciously) whether this idea is worth pursuing as a path to a solution.
3. Following up on the previous point, the difficulty is also in integrating techniques. For instance, one may need the soft idea of the discriminant of a quadratic, and then consider the smallest prime factor of that discriminant... If maths skills are ranked by meta-level with pattern-recognition at ground zero, this would probably be done unconsciously on the third floor, and consciously on the fourth. 
4. Combinatorics crosses between different subjects often, which means unlike A, G and N, its a category not of problem type, but of a flavour of ideas! Think discrete, induction, algorithms. (The key ideas listed above also help to "define" clearer what each of A, C, G, and N are.)

Do I know enough maths?

Enough compared to what/who? 

Why do people care about the IMO?

Many reasons, of course. Let me try to list a bunch of perspectives (again, biased by my personal experience and circle). 

• Parents: it's a supportive (healthy?) competitive environment for kids to strive academically at something. Maths leads to the lucrative world of quant finance, and the level of reasoning being developed is also transferrable to other science/tech/medicine jobs. I want to support them in trying to be good at something (which also improves their future prospects). 
• Teachers: it provides motivation (whether internal or external) for students to learn and focus! I want to fuel their desire to learn and improve, and their success also boosts the school reputation.
• Universities: students with IMO medals ease into university maths much faster, and it is positively correlated with them being comfortable with common maths arguments, abstractions, and maths learning in general. They are in a good place to help other maths students in the university community too. Even those that didn't do so well in olympiads, have added drive to work hard to prove themselves in university!
• Companies/recruiters: great, you knew how to think deeply and creatively at a young age! I would like to interview you to see if that makes you an asset to the team. 
• IMO contributors (e.g. coordinators, observers, leaders, PSC): I enjoyed my olympiad journey, and I want to keep up the positive experience for future students!
• Students: math is so much more fun than other school subjects! It's like puzzles, but of pure thought. And it does feel reassuring to understand maths well, when quite a number of peers are struggling (and adults seem to strongly correlate maths ability with general intelligence). It's also nice to be around people who share my interests in maths.

Now to be fair, here are some reasons why people don't care about the IMO:

• The contest is unnecessary pressure. Not just contest pressure, but also added pressure from parents and teachers (who have conflated motives: e.g. they may use contest performance to judge the intelligence of their child, as well as their own performance as a nurturer/educator). It may also indirectly encourage toxicity among some students. A student should pursue maths if they find joy in it, and not for the sake of pleasing others or being top 50 in the world.
• Some topics (such as Euclidean geometry) are not directly useful to undergraduate/research maths. Students spend a disproportionate amount of time learning specific tricks, that they could otherwise have spent learning and developing an appreciation for theory (which is not emphasised, and generally not preferred in the IMO). 
• Because it is also possible to do well at the IMO by grinding through ISLs and TSTs over a year or two, developing problem-level pattern recognition (and not so much idea-level), this is then also encouraged as a subculture as a path to doing well. [E.g. does one learn complex geometry to gain access to a class of problems with otherwise difficult synthetic solutions? Or because imbuing the plane with the multiplicative (and conformal) structure of the complex numbers adds a rich layer of structure with deep consequences to be explored?]
• Discussion is downplayed in both training and the actual contest, although it is everywhere in university maths, research, and other areas of life.
• It's disruptive to an all-rounded high school education. 

I've written both sides pretty polarisingly, and to make things worse, here's an agnostic thought: people outside the world of maths care as much about maths or the IMO as they care about spelling bees or science bowls! I'll leave it to you to find a balanced perspective.

Should I care about the IMO?

Everyone cares about different aspects, and to different intensities, and that's fine; how much you care (about maths/the IMO/the community, say) depends uniquely on your own experiences, so it's entirely up to you.

How do I get to/train for the IMO?

A small digression before the actual answer: this question presupposes that one wants to get to/do well at the IMO. Which is not too bad a goal among all possible goals a child could have, though it hopefully isn't one's only/main goal. The IMO, like other examinations and tests, are not gates or toll booths to your future; they are more like roadside attractions. You can take detours, pit stops or drive as long as you want. 

Now to details. Again, everyone has different learning styles/memory/interests/educational support systems/external pressure/self-expectations/aspects of maths they care about/time and energy; there's no one-size-fits-all; and optimal training will also change over time as you improve/your interests grow. Here are a few general ways to train that I think would be useful (based on my personal experience, and those I know):

1. Spend time. I know people who block out 2 hours every evening, 7 days a week. I can't keep such a schedule, but I did have (usually combi) problems at the back of my mind as a student, that I would think about at recess or in other classes. 
2. Prioritise long-term learning benefits. So don't memorise solutions; take the time to wrestle with key ideas, motivation, soft ideas. Again, the IMO is not the end of any road.
3. Manage pressure. This begins with recognising both external (parents, teachers, peers) and internal (ego, self-worth, competitiveness) pressure. Then processing them together with your passion in maths, as well as your other life priorities; and then finding a sustainable balance.
4. Talk maths. Practise asking about and verbalising maths ideas. Talk to both non-maths people and maths people about maths. Discussing maths promotes clarity of thought and speeds up learning. 
5. Mix up your maths learning. Think about a problem on the train to school. Sit on another problem for a week. Work on one in a 3 hour block. Work on a problem without paper. Think about one for 15 minutes before discussing with someone. Read some notes/blog posts, watch some videos. Try learning the proof of something you used to quote. Return to problems you've solved, or to a solution that you didn't come up with yourself. Try many different ways of engaging with maths.
6. Have a non-maths aspect of your life. Something to decompress at times when maths is draining or stressful; something to put maths into perspective.

Okay, these aren't details like "what sources of problems are the highest quality, or of appropriate level?". But they are the details I think are important. I didn't talk maths often when I was in high school, but looking back, I should have. And at some point in time, I remembered feeling like I needed more time in a day (though I no longer remember the source of this pressure), and started bringing maths problems to sleep. I did a lot of point 2 though, and also my interest stayed strong over the years (feeling tired and overwhelmed in some seasons, but never burnt out; it probably mattered that maths was not my only/dominating interest).

Oh yes, there's also a great deal of luck involved in contest/selection test performance. It isn't inherently good or bad, it's just how it is. 

How do I know if I'm improving?

Great question. I struggled with judging my own maths ability when I was in high school. In hindsight, I should have cared less about quantifying my progress, and instead trust in time and passion. The underlying principle here is really that doing well in contests is not the long-term goal to have, and neither should it be the main metric of progress! (E.g. if you're judging your progress based on mock IMOs every month, you are actually incentivising yourself to prioritise contest-specific problem-solving strategies, which doesn't really include intuition and theory building and other aspects of maths.)

One thing I found helpful was to draw my map of (olympiad) maths, as I currently understood it. I remember my first attempt was pages of bullet points of many disjoint ideas; over the years (and this happened after my final IMO), they fell into broader meta-categories, and by the time I became a trainer, my map had also become clearer and more organised to reflect an improved understanding. 

What happens after the IMO?

Well, it does feel a bit surreal (like in LOTR, the aftermath of the adventure). Then there's catching up on school work, and school exams/A-levels. And also university apps, scholarships, and university maths (if your interest persists). Some self-reflection and consolidation of maths knowledge, then a return to maths. Some will do olympiad training, and will start thinking about set design and problem sourcing. Some will get into research, some into theoretical CS, some into quant finance. Some will get married. Some will eventually write blog posts on a retiree blog. 

Epilogue

I guess I don't feel good writing an entire post without any maths. So here's a trailing food-for-thought: for any finite set $A$ of integers,

$$|A-A| \leq \frac{|A+A|^2}{|A|}.$$

(This is a consequence of the Plünnecke-Ruzsa inequality, the proof of which I've searched long and hard for intuition/motivation, to no success.)

All the best to the current Singapore team, and a figurative toast to those who once walked this road with me; I enjoyed every moment of Saturdays, June trainings, SIMO camps, and international adventures.