Background: I had the privilege of ‘interviewing’ Warren Li OE, in which I asked him some questions about general math, the IMO (International Mathematical Olympiad), and Cambridge. Just in case you don’t know, Warren Li is a 4-time IMO participant (with 1 Gold medal and 3 Silver medals), who was at Eton from 2014-2016. He has just finished 4 years at Trinity College, Cambridge (studying math), and is heading to Princeton in the US to pursue a PhD. We wish him best of luck with this (especially given the current pandemic)!

General questions:

What are your earliest memories of math?

When I was very young, my parents enrolled me in KUMON, an education programme where you do daily worksheets in maths and English. KUMON taught me maths up to about an introductory A Level standard, and is where my route to ‘interesting’ maths began.

What do you enjoy most about math?

What I love about maths is how you may use it to explain concepts in so many different areas of interest. Maths is the main constituent of theoretical physics, statistics, engineering and so on. For me, maths really helps me to understand certain aspects of these complicated systems at a deeper, more conceptual, level.

What is/are your favorite area(s) of math?

In my Olympiad days, I really enjoyed number theory and geometry, as these heavily involved the ‘pattern spotting’ that makes maths beautiful. Nowadays, I’m more interested in analysis and differential equations. Analysis is the study of ‘limits’, and is important in understanding differential equations as it explains rigorously what it means for something to change infinitesimally. Differential equations are ubiquitous in science, and understanding how solutions of differential equations behave is crucial to understanding how some scientific model works. I find this beautiful because it’s still about ‘pattern spotting’ – the questions you ask are often about spotting patterns in the solutions, or understanding when such patterns are broken.

When did you first think about making IMO?

I probably started seriously thinking about IMO when I went to my first Olympiad training camp, which was held in Oxford in August 2012. That’s when I first started getting real exposure to an environment full of like-minded problem solvers.

When did you first do BMO1/2?

I first did BMO1 when I was 11, but I only got 7 marks that year. My first BMO2 was 2 years later, when I was in year 9 (still at my old school so not F block!)

What do you think was the tipping point from the year before you made IMO to the one after?

A lot changed that year – I went to my first training camp as I said above, and started to get really into Olympiad problem solving. I think the biggest tipping point, though, was that I did well enough in BMO2 to make the team for another international competition, the Romanian Master of Mathematics. I think it’s here that I built the confidence to see that if I continued working hard I could really compete on the world stage.

How many hours per day were you training in your first year? Was this the same across all years?

I’d guess around three hours a day. That’s probably roughly the same across most years, and if anything I worked the hardest during my third year of IMO prep when I was really pushing for a gold medal.

What does ‘training’ look like for you (e.g. do a problem, check the solution)? Do you count thinking about problems during the day?

Certainly most time ‘training’ is spent doing problems. An ideal situation would be that I would either solve the problem, or spend around a day thinking about it in one way or another. If I couldn’t solve it after a day, I’d probably look for hints / motivating ideas from other people before reading a solution. Obviously some time is needed to build theory as well, but the emphasis for IMO problems is definitely on problem solving ideas rather than theory.

Do you have any textbook recommendations? Does getting the right textbook even matter that much?

As I said, theory isn’t supposed to be a huge part of preparing for Olympiads, so there isn’t really a ‘right textbook’. But if I were to recommend one, it would be Arthur Engel’s ‘Problem Solving Strategies’, particularly as it focuses on the subtle but ingenious tricks-of-the-trade in IMO problems, rather than any obscure high powered theorems.

Is there one thing (if that’s possible) you think helped you make and succeed at IMO the most?

In hindsight, one of the things that really helped was having a network of people I could communicate with and bounce ideas off (both adults and other students). As well as having someone to turn to when I was stuck, it also meant I could see a variety of different insights into any given problem. Also, they could act as yet another source of interesting problems.

I haven’t really thought about this before! Perhaps I would choose the 6th problem from the 1988 IMO. This was a number theory problem that, before being submitted to the contest, was sent to the four most prominent Austrian number theorists of the time, and none of these experts could solve it in the six hours they were given. Despite this, there were 11 IMO contestants who managed to solve it. That’s because the main solution technique was something called Vièta jumping, something which anyone who knows how to solve quadratic equations can do in theory, but something which wouldn’t come to the mind of the four experts.

Do you still do IMO math problems from time to time?

Regrettably, these days I don’t spend a lot of time doing actual IMO problems, especially those at the harder end of the spectrum. Though I do still have some involvement in mentoring current students, so I do still spend some time thinking about problems of that style.

What would you say to someone aspiring to make IMO?

I’d just say keep solving problems, especially ones which you yourself think are challenging. If you keep experiencing the euphoria of solving problems that you initially thought would be impassable, you’ll have more and more motivation to find even more new ideas. The other thing, as I’ve previously mentioned, would be to find like-minded people to develop these ideas with.

Is there anything you regret about IMO?

One of the best things about being at an IMO is to meet people all around the world who are just as interested in maths as you are. The thing I regret is that I didn’t spend enough time during the actual competition talking to people from other countries, particularly the non-English speaking ones. I’m just about to start a PhD in Princeton, and it amazes me how many people there are, even at that one school, that I could’ve met at some IMO beforehand.

Have you been doing any research in Cambridge?

Unfortunately, I haven’t had the opportunity to do any proper mathematical research in Cambridge. Interestingly, the research experience I have actually comes from my internships in the quantitative finance industry – although the problems are different, the research mindset I developed should hopefully be really useful for my research endeavours that lie ahead.

How do you recommend preparing for the Trinity interview?

The Trinity interview is a bit special in that you have to do an hour long exam beforehand, then talk about some of the questions you looked at with your interviewers. But the advice I’d give probably doesn’t depend on the format – for any problem you are given, you want to stop and think a bit to think about sensible approaches, and make sure that you can carefully explain each step of any potential approach you take, even if they might not work. In maths, there’s no benefit from approaching something with ‘fuzzy’ ideas which you yourself don’t understand fully, and your interviewer will be able to tell if you’re doing this.