### 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?**

**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).- 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.
- 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.
- 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.
- 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?**

**Why do people care about the IMO?**

- 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.

- 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.

**Should I care about the IMO?**

**How do I get to/train for the IMO?**

**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.**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.**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.**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.**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.**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.

**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?**

*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.)

**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?**

**Epilogue**

## Comments

## Post a Comment