As I’ve told some people, I was in Japan for the past month, working as a member of the IMO 2023 Problem Selection Committee. It was a unique experience, very much different from participating as a contestant, and I’d like to share it a little.
From left to right: Arnaud Maret, Paul Vaderlind, Ivan Guo, Hiroki Kodama, Elisa Lorenzo Garcia, Sam Bealing, Tetsuya Ando, Yuya Matsumoto, Yuta Takaya, Takuma Kitamura, Genki Shimizu, Géza Kós, Tetsushi Ito, me. Not present: Atsuo Yamauchi, Ryu Minegishi.
I am eternally grateful to every one of them for creating such a memorable experience! It was a great pleasure to get to know everybody, and the committee would not have been the same without them.
What it was about#
The actual IMO has only 6 problems each year, and choosing these problems is quite a complicated process. The Problem Selection Committee(PSC) is responsible for one (plus epsilon) step of this process.
- First, every participating country submits no more than 6 problems to the IMO organizers. This is done around April, and the resulting collection is called the Longlist. This year, the Longlist had around 160 problems in total.
- Once the Longlist is compiled, the PSC can get to work. The PSC selects about 30 problems from the Longlist, which make up what is called the Shortlist. Another thing they do is to write up all the solutions to these problems to produce a booklet. A lot of the work was done virtually, but we got to meet up somewhere near Tokyo for 10 days mid-June.
- The Shortlist booklet is then distributed to the jury (which is the collective of all leaders of all participating countries) 5 days before the start of the exam. The jury’s job is to select the 6 final problems from the Shortlist through a painstakingly tedious process. Since the leaders are only given a few days to study the problems, oftentimes they ask for opinions from the PSC.
So that’s how the problems are chosen! If you didn’t like the problems, please note that the PSC is only partly responsible for them.
I volunteered for and was assigned to the Combinatorics team, because, well, even though I do number theory, number theory isn’t really number theory. There were a lot of interesting combinatorics problems, but unfortunately the spots on the Shortlist were limited and we had to reject a lot of nice problems. After a committee-wide discussion, we also decided to filter out all the problems that rely on König’s theorem or Hall’s marriage theorem. The rationale was that students who don’t know these theorem will be at an significant and unfair disadvantage. I’m not sure where the line should be drawn, but I’m hoping that things will clear up once the IMO curriculum is published.
Another bonus of being a PSC member was that we also had to serve as a coordinator, either as a problem captain or as a captain advisor. This position entails
- creating marking schemes,
- getting them approved by the jury before the students finish the exam, and
- overseeing coordination to make sure that the marking scheme is applied consistently.
I was the captain advisor for Problem 3, serving the captain Matsumoto-san. Coordination was a lot more stressful than I expected. Writing the marking schemes was already a pain, since we had to consider all possible configurations of bits of solutions scattered throughout scratch-paper and then construct a reasonable function to $\lbrace 0, 1, \dotsc, 7 \rbrace$. We came up with a scheme that rewards students for a lot of different partial results, so that they could easily get 5 points without actually solving the problem. But this is what the majority of the jury wanted, and I also think it was quite reasonable.
Once we had a solid marking scheme, the actual coordination wasn’t too bad. I think Problem 5 had a lot of trouble, because students were coming up with all sorts of wild solutions that seem to work but couldn’t be easily verified because they’re not written up so clearly. Fortunately, all solutions to Problem 3 followed roughly the same path, which was covered by the marking scheme.
Lessons learned#
Here are some things that I learned while serving as a PSC member. Of course, I’m not speaking on behalf of the committee; these are just my personal thoughts that might or might not be good takes.
If there aren’t good problems, you can’t select good problems. For each subject, we tried to select 2 easy problems, 3 medium problems, and 3 hard problems. But guess what, if there aren’t good easy problems, either you choose bad easy problems or just inflate the overall difficulty. Then it’s not the PSC’s fault that the Shortlist is weird.
Don’t spend too much time carefully wording problems, especially combinatorics problems. Once the problem is chosen to be on the exam, the jury will redo all the work anyways. On the other hand, I think it’s good to write a nice scenario to dress up the problem. I’m very proud that my own coinage “ninja-path” made it to the exam.
Once the number of problems reaches around 11 per subject, you should really try to remember if you’ve seen a similar problem/argument somewhere, or spend some time looking up the problem online. Too many Shortlist problems were gunned down for being “too similar to that other problem” during the jury meeting. Of course, a PSC consisting of 16 people can’t beat the collective knowledge base of 200 leaders plus observers, but I feel like the situation could have been a bit better.
I think it might be good if everybody has a “secondary” subject, so that they evaluate semi-finalized problems on other subjects. This way, about half of the PSC members get to evaluate each problem on the Shortlist. I was too focused on combinatorics problems that I barely looked at other problems until the Shortlist was almost finalized, but every team would have benefited from having more eyes.
When writing a marking scheme, be redundant and put as many examples as possible. When an item is worth $x$ points, I think it’s good to include both a minimal example worth $x$ points and also a maximal example worth $x$ points. This sets up a clear boundary between $x$ points, $x-1$ points, and $x+1$ points.
Nobody can’t predict all possible solutions. At the least, there will always be infinitely many variants of the same solution, and if these variants happen at multiple places, the number of variants get multiplied. Even worse, there might be a completely new solution, and then it’s unclear how many points a student who made partial progress on the new solution should get. Be legally or mentally prepared for all of that.
Coordination can get really confrontational and emotional. Some leaders appeal with reasonable arguments, but some leaders try to bargain their way in. But there is no need to stress too much about it. Just do what you think is the right thing to do, and if they don’t agree, leave it for the chief coordinator/jury to decide.
Moving points seems to be quite popular these days, so be prepared for papers using them. I’ve never heard of it until Ivan told me, but apparently this is some algebro-geometric degree-counting technique for doing Euclidean geometry. Some students seem to love this, and its popularity will only increase over the next few years.
So, did I enjoy being a PSC member overall? Absolutely, it was a childhood dream and I had so much fun participating in the IMO from the organizing side. Will I do it again if I’m invited? Well, I’m not so sure. It was a huge time commitment, basically taking out a whole month from my life. But I also don’t know if I’ll be able to resist the temptation!