Nico McKinlay: IMO 2024 Report
During July 2024 I had the amazing opportunity to represent New Zealand at the 65th International Mathematical Olympiad in Bath, United Kingdom. It was a great chance to try some cool maths problems, connect with other like-minded high school students, and visit a part of the world I’ve never been to before.
This 65th IMO was particularly special because, as Ross remarked during the closing ceremony, 65 is the smallest number which can be written as a sum of two distinct positive squares in two ways: 82 + 12 = 42 + 72 = 65. (The distinctness restriction is necessary otherwise 72 + 12 = 52 + 52 = 50 would be minimal.)
For New Zealand, this IMO was extra special because we were quite successful at the IMO this year. In particular:
- It is the first time in history that New Zealand beat Australia at the IMO!
- We beat the previous New Zealand ‘total score’ record;
- All six of us won medals – 3 silver medals and 3 bronze medals;
- We managed to keep a ‘clean sheet’ (42/42) on both problems 1 and 4 – New Zealand has only ever done this once before, in 2017.
Travelling to the UK
On the eve of the 7th of July, the NZ team (minus Jerry), Ross, and Kevin met at the Auckland International Airport. Josie and Jerry made their way to the UK separately because Jerry lives in the US. While standing in the bag drop queue, I remember talking to Ross about weird inner products or something, then deciding to spot-check my backpack just to double-check that my passport was in there. I was hit with a sudden surge of panic as I realised I had somehow forgotten to pack my passport. Yikes. Luckily, my older sister was able to drop it off so all was well.
The plane trip was pretty unremarkable. I was pleased to find agreeable audio entertainment (Chopin) on the China Airways flight console, and the plane food was decent, I guess. During the second leg of the flight, Dawn and I become hard-stuck on a geometry problem for quite a while, only to discover that the synthetic solution relies on an annoying radical axis trick involving a circle of radius zero. I tried watching the movie ‘Tenet’ but I gave up halfway through because the plot was too difficult to follow.
Pre-IMO training
Prior to the actual IMO we sat some mock IMO exams created from the 2023 shortlist. We marked each other’s mock scripts, which had two benefits: firstly it helped us get a feel for what sorts of things might earn partial marks, and secondly it forced us to empathise with Ross and Kevin who have the job of deciphering our messy IMO scripts. My first two mock exams went okay: 7-1-0 and 7-0-0. However things started getting a bit dicey in the last two mock exams. For some reason during the third mock, I decided to work on the geometry P2 and ignore the very paragraphy P1. After constructing some random points to help deal with arbitrary angle conditions, I stumbled across a useful synthetic observation which allowed me to complex bash the problem. Unfortunately I then proceeded to get stuck on P1, leaving me with 0-7-0. On the final mock, I couldn’t make any progress on P1 or P2. It turned out that my anguish on P1 was due to a failure to do the small cases comprehensively enough (lesson learned: do your small cases properly). I was able to bash an important synthetic observation on the P3 geometry but naturally I couldn’t find the final step of the problem because it’s a P3. This left me with a very unusual score of 0-0-3 for the final mock.
Pre-IMO fun
During the pre-IMO period we did lots of fun stuff in Bath and around the campus. Visiting the eponymous Roman baths was particularly fascinating. While the architecture surrounding the baths is largely Victorian-era reconstruction rather than Roman original, it was interesting to get a glimpse into the lives of the ancient Romans.
We also hung out with the Netherlands team quite a bit. I fondly recall playing frisbee piggy-in-themiddle with them while walking around the Bath skyline track and also on the University of Bath campus.
The IMO exam
The first day of the exam led to some people (Jerry) jokingly labelling this year’s IMO as the ‘international sequences olympiad’. I had a bit of a tumultuous journey towards my solution to P1, because I firstly fakesolved, then ‘fixed’ my fakesolve, then realised that the fix to my fakesolve was actually a fakesolve itself, then finally wrote down a real solution. By the time I had stopped hallucinating about how the floor function works and found a legit solution to P1, nearly four hours had gone by. So I spent the last half hour scribbling down random ideas and steps for P2, which somehow got me 1 point. This left me with 7-1-0 for day 1.
The second day also wasn’t smooth. After staring at the geometry diagram for 1 hour, I was out of synthetic ideas, so I decided to look for ways to bash the problem. Luckily a convenient complex bash was possible. Not so luckily, I made a mistake in my bashing calculations and could not find the mistake. Another hour went by and after reading through my work line by line I still couldn’t find the mistake. I realised I needed to take a more efficacious approach in my corrective search. I had the idea to think wishfully about what number or value needed to change to stuff to cancel, and with this in mind, I noticed that inserting a ‘2’ in one of my fractions would help cancel stuff out. Tracing this all the way back through my calculations, sure enough, I’d dropped a factor of 2 somewhere. But that wasn’t the end of my woes. When using the method of distributivity of complex conjugation, one can only check that two directed angles are equal, but not distinguish between supplementary and equal angles. So I spent even more time coming up with a rigorous argument to move from equal directed angles to supplementary angles required in the question. It turns out that this was not an efficient use of time because the mark scheme didn’t penalise ‘directed angle handwaving’ so I could have just handwaved it. With less than an hour to go, I started playing around with small cases on P5, and then with only about 10 minutes left in the exam, I suddenly realised the answer. I scribbled down the construction as fast as I could, but didn’t have enough time to prove that my construction is optimal so I ended up taking a 1 mark penalty for that. All in all, I was very pleased to end up with 7-6-0 for the second day.
Why does everyone hate Turbo?
So yeah remember that problem which I scribbled down a solution in the last 10 minutes? Turns out that it trolled a lot of people, especially stronger students (update: Evan Chen gave it a MOHS rating of 35). It seems that many contestants became fixated on proving the optimality of non-optimal strategies, failing to realise the strategy that was hiding in plain sight. Here is a humorous photo of the results of an audience-poll question during Grant Sanderson’s lecture (‘Which problem from the 2024 IMO was your least favourite?’):
I think the major practical difficulty with P5 is the fact that it is a P5. Students expected some kind of serious and non-trivial argument for a P2/P5 combi, not some simple trick. I don’t think P5 would be out of place on NZMO1, but it is all too easy to be intimidated by the placement of the problem as P5. Perhaps not an ideal question for the IMO when its biggest difficulty is the mere conception that it is difficult. But, I’m not complaining, because P5 allowed me to snag a bronze medal (and hilariously, nearly even a silver medal).
Post-IMO fun
With IMO exams out of the way, it was time to party, go on excursions, and like actually socialise with the other contestants (spoiler: I’m not great at socialising). On Thursday the NZ team went on the London walking tour, which was a great opportunity to see some of the major tourist attractions in London, such as the London bridge, Big Ben, Westminster Abbey, and other things. We had a lovely chaperone who demonstrated a particular unremitting diligence in counting the six of us at every possible junction to ensure that no one got lost. Our tour guide provided quite interesting commentary although he did seem to walk rather fast for the group (but he carried around a flag so we could spot him in a packed crowd so that was fine, I guess). Here are some photos:
During the train rides on the London walking tour, Haotian and I played lots of ‘colour wars’, an interesting (and somewhat strategically complex) game part of an app called ‘Two Player Games’. I jokingly referred to the game as “the only interesting combi problem of this trip” (I have a personal vendetta against combinatorics for some reason). Eventually we decided that Player 2 seems to have the advantage, but that plenty of tricks are available to Player 1 to try and stay in the game.
Some other highlights of the post-IMO fun:
Myself (NZL3) and an Estonian (EST4) joined forces with four Italians to participate in the 6-a-side football tournament.
Jane Street (generous sponsor of NZMOC) hosted a maths olympiad celebration which was heaps of fun and filled with nice food and refreshments, chess games, table tennis. We got to stay at a fancy hotel (London Hyatt), listen to Matt Parker speak, attend the video games section of the science museum, and go on a nice boat ride.
Miscellaneous photos
More Turbo the snail drama
Selfie with Evan Chen (thank you for the bashing handouts ❤)
Selfie with Po-Shen Loh (he sat next to me on the bus!)
Playing mini-golf with the Netherlands team
Chris working on a puzzle – arrange five 3-4-5 triangles (no overlap) such that the overall shape has an axis of symmetry:
Playing in the badminton doubles tournament with Dawn. We made it to the semi-final then got crushed by the Australian duo William (AUS2) + Laura (AUS1) (William is super pro at badminton)
The final of the badminton doubles tournament. It was tied at one set each then the Australians clutched up winning the final set 22-20.
Jerry handing out some rubber ducks to other contestants (“it’s a traditional NZ souvenir, trust”)
IMO closing party:
Questions
- How has attending this event demonstrated greater knowledge of available career paths in science and technology?
Although I did not learn directly about career paths in science and technology, by attending this event I found a greater appreciation for the most important connection between the IMO and careers in science and technology: the ‘problem-solving mindset’. The discourse surrounding the decision between pure maths and more applied fields such as science, technology and finance was certainly present in several conversations amongst students and leaders at this event.
- How has attending this event enthused or inspired you to pursue science and technology careers?
During the Jane Street celebration I was inspired by the possibilities of using general mathematical thinking and problem-solving skills in a more applied setting.
- Has attending this event changed about how you feel about science or technology?
I would say attending Terence Tao’s lecture on artificial intelligence in research mathematics helped me understand the role of such modern technology in mathematics and general interconnectedness between mathematics, science, and technology.
- How has attending this event and participating with like-minded students been of benefit to you?
I think interacting with the other students has been both humbling and inspiring, but also just heaps of fun.
- Now that you have had time to reflect about your experience what have you learnt about yourself?
I suppose I’ve learned to approach things with a positive mindset because good things can unexpectedly happen (P5), and also that I need to get better at socialising.
- What did you enjoy about your experience?
I believe this entire report answers this question sufficiently.
Conclusion
I would like to thank the IMO organisers, Jane Street, Ross, Kevin and Josie, for their roles in making this trip possible and making it an amazing experience. Thanks everyone, it was a blast.