What I could have done better: an X-man's advice to future generations

(Shi Cheng here.) Hi, it's been a long time since I've been involved in the Maths Olympiad sphere, mostly because of uni, but recently I found out about this blog and I wanted to contribute. This is not a post about Math, but rather, I wanted to share some important mistakes and lessons I learnt from my time as a Math Olympiad student, so that hopefully you will not make them yourself and can maximise your potential as a student. Now, it is possible some of these lessons may already be obvious to you, and perhaps you are judging me for not having realised them earlier. Maybe I should have. But there could be others who may not have realised them, so hopefully this post will stop them from falling into the same traps I did, and for those who are already aware of them, this could serve as a reinforcement.

1. Focus on the big idea(s) of a question. When I was still a Math Olympiad student, I remembered the way I would read solutions is by examining each line, making sure I understood each and every detail and refusing to move on if I didn't. When I moved on to uni, I kept up the same habit when I was revising my courses and going over proofs and statements. This took up a lot of time, and as a result, I was unable to keep up with most of the courses, and had to play catch-up during the holidays. Even worse, I was struggling when it came to solving problems on example sheets. Eventually, I realised the most important thing was the big idea - it's okay to leave a few minor details unverified, what's important is that you had a big picture understanding of what was going on in the course, and that was much more useful towards solving problems on example sheets. I realised the same was true in Math Olympiad. This 'big idea thinking' is what helps you learn from past problems to solve future problems, as you can see which ideas fit in which contexts, and maybe even recognise when you can apply an idea you have seen before, allowing you to make associations between similar problems. I was probably doing this 'big idea thinking' subconsciously, if not I would not have gotten to where I am, but my biggest regret was not paying enough attention to it - I recall others would have conversations about using a certain idea to solve a certain problem, and I would have trouble following such conversations.

   That is not to say that details aren't important, in fact in some questions the big idea is obvious and it's the details that matter (SMO Open 2019 Q3 comes to mind). Writing an airtight solution without holes is also an important skill. But don't lose the forest for the trees.

2. Keep a notebook with you, for you to record all the questions you did, and the 1 or 2 big ideas that each question distils into. When I was a student, I realised I would jump from solving one question to trying another one immediately, without spending adequate time summarising the previous question to make sure I learnt something from it for future questions. This was wrong. Keeping a notebook helps you keep track of what you learn from each question, so that you can see if a similar idea pops up in another question. This also helps to reinforce the 'big idea thinking' from point 1.

   But perhaps a more important reason to have a notebook is if you become a trainer. Trust me, when it becomes your turn to train the juniors, you would most likely have forgotten what you've learnt as a student, especially if you're a guy and have to serve 2 years of NS. I know because this happened to me. When I was asked if I wanted to become a Nat team trainer, I recalled feeling very unconfident about setting training materials. (Eventually I didn't become one, either because of this or because of other reasons, but I faced similar struggles when I was training the HCI internal team, because I was having to dig deep into the trenches of my memory to find questions to use.) Having such a notebook, you can simply refer back to it to see which questions you can collate to form a cohesive training set to give to juniors.

3. Be efficient when solving questions. What do I mean by that? Know when to move on from questions and look at the solution, so that you can maximise the amount you learn in a limited amount of time. This is important because as a student you only have so many years before you graduate and get your last chance at an IMO. For me, I recall spending long hours, sometimes days, on a single, very hard ISL problem before I would finally solve it. (For instance I remember spending the entire duration of a class chalet solving IMO 2015 Q5.) The problem is, the amount of time I would spend on the problem would be well over the amount of time I would realistically get in a competition setting. And in the time I took to solve that problem, I could have looked at the solution, learnt from it, and tried other problems in that timespan. Looking back, I think what kept me from moving on is seeking that dopamine rush, that sense of satisfaction from solving a problem I normally considered out of my reach. And while it is not a bad idea to challenge yourself once in a while, I probably did it to the point where it took up time I should be using to do other questions. There are near-unlimited questions on AoPS and elsewhere, so you will never run out of questions to do.

   With that being said, it is okay to spend more time on a problem if you really think it's worth it. It's not easy to judge when you should continue trying and when it's time to move on from a problem. But spending days on a single problem is probably not a good idea.

4. Don't be afraid to try. I remember when I was solving Q1 during SMO Open 2016, I spent about an hour trying to manipulate things using Sine Rule, before I decided to try spiral similarity, another idea that I had since the start, and immediately solved the question. After that competition, my takeaway was that I had wasted the first hour trying the Sine Rule approach, and should have just tried the spiral similarity approach immediately. Afterwards, whenever I was solving a problem, I would try to reach the correct approach as fast as possible, to minimise the time spent arriving at the solution. The result was that I "tried" fewer things like small cases and approaches that may not make sense initially. Sometimes, it felt like instead of trying to motivate the solution through intuition, I was trying to pluck ideas out of thin air and hoping it would work. I would even look at the solution to a hard problem (such as ISL 2017 N6, which appeared on NTST and no one solved it) and think "I should have just tried this one idea during the paper and it would've worked" without appreciating the amount of time it takes and level of proficiency you need before you are able to arrive at a solution like that. What I want to drive home is, it is okay to spend time to try certain things and fail. Because only by trying things and seeing whether they succeed or fail, you can build your intuition of what works and what does not in different contexts, and that really helps to grow your problem-solving ability. (With that being said, sometimes it is good to make a few trivial observations to reduce the number of things you have to try.)

5. Don't forget to take breaks. I still remember when I was a student, I tried to maximise my ability by forming the mindset that "Any time spent not doing Math Olympiad is time wasted". So I would try to spend all my idle time such as time on the bus/train thinking about problems, or looking at solutions etc.. In uni, when I was trying to catch up with all the coursework (as mentioned in point 1), I experienced burnout and moments where my self-control would totally collapse. It was then that someone explained to me that I needed to take breaks, and how even when you are behind on work, taking breaks can refresh your mind and help you be more efficient when you are actually working. That is why taking breaks is important. On hindsight, I realised my approach to Math Olympiad above was wrong. Rather than maximising time spent, I should have maximised efficiency. And I think the same probably goes for you. It is okay to not feel like doing any Math after a 9-hour intensive training for NTST/IMO. In fact it is probably a good sign if you feel that way, because it means that you have given your all during the training. (Not sure how relevant this is, but I feel like Math Olympiad training is somewhat similar to how athletes train in sports, where after a training session at the gym or elsewhere it is important to give the body time to repair itself and build muscle, and I guess the brain works the same way.)

6. Don't try to predict the difficulty of a question before you've even tried it. This is something I saw in Sheldon's post while browsing the blog, and I thought I'd bring it up because I was guilty of it as well. In his case, he nearly missed out on a spot in the IMO team because he didn't solve a question in NTST he overestimated the difficulty of. I remember something similar happening to me during SMO Senior 2017, where I didn't try Q5 because I was stuck on the earlier questions and thought Q5 would be harder than them and thus not worth trying. After the paper, I tried it and solved it in about 5 minutes. (I bombed that paper btw.)

   Other examples:

• IMO 2016 Q2: I was Sec 3 back then, and had no expectation of myself being able to solve an IMO Q2/5 level problem, but I decided to try it anyway. To my surprise, I was able to solve it in about 2-3 hours at school (timing could be a bit off, it's been a long time).
• IMO 2017 Q4: At that point I had been in Nat team for a while and was able to solve most easy IMO geom (I think), but this particular question stumped me for a long time, I remember I started trying it in the morning and only solved it in the afternoon, and had I been taking the actual paper I might not have solved it at all. I would later find out that 2 of the actual contestants also struggled unexpectedly with this question.

  Basically, if you overestimate the difficulty of a question, you could have blocked yourself from the possibility of solving it even from the start, and that could stop you from trying familiar ideas or small cases because you are operating on the belief that because the question is difficult, those things will not work, when in fact they could, or at least make some progress. On the flip side, underestimating the difficulty of a question could cause you to be unnecessarily flustered when you are stuck on it for longer than you'd like. This is perhaps even more relevant now, because in recent years the medal cutoffs at the IMO have been far more unstable than during my time, when it was always around 2/3/4 questions for B/S/G. So if you go into a question with the wrong mindset about its difficulty, it could easily cost you a medal.