Jay Zhao: IMO 2024 Report

The Exam

With about ten pages of cluttered and incoherent diagrams in front of me and absolutely no progress made on problem 4 (Geometry), I put on my glasses to look at the time, 1 hour and 40 minutes left… s%#(!@t.

Fun Fact: I scored 0/0/0 on my first mock in the UK. This was because I misread the problem one (Geometry).

I quickly reread the statement of the problem and I had actually got the problem right this time. This means my inability to solve the question was simply an issue of skill, at this point I was certain I was not going to solve the problem.

On day one I had only managed to solve only one problem: the simple Algebra problem and then failed to solve the Number Theory problem, even though Number Theory was supposed to be my favourite topic. Then I was going to end day two without solving a single problem and come home with a total of 7 points + some partials and an honourable mention at best.

To add to my sadness, minutes later the Mongolia team member sitting diagonally from me would very noticeably pull out a purple folder and transfer the papers from his desk into the folder. Purple was the folder designated for problem 4… f%#$!%!&!.

“Wishful thinking is the formation of beliefs based on what might be pleasing to imagine, rather than on evidence, rationality, or reality. It is a product of resolving conflicts between belief and desire…” I decided to think wishfully.

No cyclic quadrilaterals had appeared yet, If (BXZP) was cyclic then (CYZP) is probably also cyclic? Oh… that solves the problem. Are they cyclic?... Oh yeah. they are.

I was able to solve problem 4 and moved onto problem 5 after scribbling down an angle chase to finish off my solution.

Fun Fact: In my first at home mock it took me about 3 hours to solve problem 1 (Geometry), then about 45 minutes to solve problem 2.

Problem 5 I found to be pretty simple and after finishing off problem 5 with about an hour left, I would make no substantial progress on P6 so I decided to rewrite my solutions for problems 4 and 5.

Problem 5

Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters in 2022 of the cells. Initially, Turbo does not know where any of themonsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.

Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.

Determine the minimum value of n for which Turbo has a strategy that guarantees reaching the last row on the nth attempt or earlier, regardless of the locations of the monsters.

The numbers 2024, 2023 and 2022 are completely arbitrary and in fact this problem can be generalised to a (N+2) x (N+1) grid containing N monsters.

We can easily reach the end in at least N+1 attempts by choosing the first column, trying to go all the way to the bottom, and if we fail, moving onto the next column until we eventually find a column with no monsters in it. This would be an O(N) solution meaning on the order of N.

Typically O(N) or O(N²) combi problems have a very obvious construction and the difficulty of these combi questions is in proving you can’t do any better than your construction. Many contestants, including myself at some point, thought the solution would be O(log₂N), meaning that the number of attempts would increase by 1 every time N doubled. Typically O(log₂N) solutions would involve some sort of algorithm that involves “repeatedly halving”, like 2023 IMO P5 which can be solved using a binary tree (binary meaning 2), perhaps many contests considered log₂ because of last year's combi.

But actually, the solution was just 3, meaning an O(1) or constant solution. Similar to 2022 ISL C7 (where the answer was also 3), however unlike this year's P5 the difficulty in solving 2022 C7 doesn’t immediately vanish once you know what the answer is, it requires a really clever construction, whereas the construction in this year's P5 was relatively simple.

Conjecture: If a combi question has an O(1) solution, the answer is 3. This is because 3 is the smallest number that is not completely trivial. Consider the following variations of P5:

1. There are no monsters
   The solution to this would just be for turbo to do whatever he wanted and then eventually walk to the end (very exciting!) and finish in 1 attempt.
2. The monsters disappear once you encounter them
   Turbo could just walk straight down a column and if he was caught by a monster he could walk straight down the same column where there is now no monster and finish in 2 attempts.

You can see that these two variations are way too simple to be put on a contest and after finding the construction you would have to show that anything smaller didn't work, so if the answer was k you would usually have to manually show that 1,2,...k-1 all didn’t work, and for k>3 this is too tedious. Meaning 3 is the only option left.

Moments after the IMO

The clock hit zero and the day two exam ended, and it seemed that everyone around me had gotten 3 for P5, initially I thought this was really good as it confirms I had gotten the right answer, however I soon began to think that if everyone solved P5, then that must mean it was really easy and I conjecture that the bronze cut-offs were going to be at least 7+7+7+1=22 (3 problems + 1 partial).

The night after the exam I spoke to members of the Hong-Kong team who were shocked I did not solve problem 2 and told me that it was not only routine but that they all solved it instantly besides one person who they were all making fun of…

Then they were all super surprised I solved problem 5 which none of them solved bar one (not the one they made fun of unfortunately, that would’ve been pretty funny). I had given each of the team members a rubber duck and they had begun violently assaulting my ducks while imagining it was Cheuk Hei Chu from Hong Kong, the one who proposed the problem.

The Hong Kongers asked me to play Texas Holdem with them which was a little too complicated for me. They also asked me if I spoke Cantonese (I do not) and began swearing in Cantonese to each other. Unfortunately I had to leave when I was summoned by Ross Atkins to meet outside.

That same day I was asked by a british organiser if the chinese translations of the exam were good, I believe his chinese is probably better than mine and I told him I did the exam in english.

While doing problem 5, I named my oracle function Hao who would place the monsters in a way to oppose Turbo. Life would imitate fiction a few days later when NZL5 (Haotian Wang) placed his foot on an innocent snail during a joint New Zealand and Netherlands excursion to Bristol. A short moment of silence soon followed as we mourned the death of Turbo the snail.

During lunch, someone from the Netherlands discovered a web page where the IMO results were being released in real time. I had gotten 7/7 on P1, and 7/7 on P5 and my P4 remained hidden. It turns out that each person had one of their scores hidden (for some reason).

Kevin had told us at dinner that New Zealand had gotten a total of 120 points, he also told us that we had achieved perfect scores on P1 and P4 (beating both USA and China on P4), using this information I was able to deduce that I had gotten a perfect score on P4 and deduce the scores of everyone in New Zealand. This would give me a total score of 7+2+0 + 7+7+0 = 23 (although I still believed bronze cutoffs were in the 20s). We were also told that we would beat Australia for the first time in history which was extremely exciting.

That night I did some statistics in my dorm room. I obtained all the scores from the web page and filled in the hidden scores by giving everyone 7 on P1 and P4 and then replacing all other missing scores with someone else's score from the participants' team. I had obtained medal cutoffs of 17/23/29 which would mean I’d get a silver medal.

The real cutoffs were announced to be 16/22/29 (I was a bit bummed to be off). Meaning that NZL2, NZL4 and NZL6 (me) would get silver medals, and that NZL1 NZL3 and NZL5 would get bronze medals with NZL3 getting 21 points (1 off silver).

Closing Ceremony

During the closing ceremony I sat next to a Chinese team member who I was too scared to speak to. I also sat behind 2 Canadians who had very cool Canadian flag headbands.

The Australian team graciously gifted me a koala by unknowingly sticking it on my back, which is perhaps symbolic of how Australia is behind New Zealand.

Ross had been gloating to all the other team leaders about New Zealand’s victory over Australia and throughout the day we had been receiving congratulations from other countries including from the Australians on our success.

However Australia did at least have one gold medalist which New Zealand could unfortunately not match this year.

We then went outside where a carnival was set up. I went around pretty much every food stall and munched up on food. We took pictures with the Australian team and then I spent most of the time waiting in line for rolled ice cream.

Fact: If you don’t eat all the food you lose money

I then found NZL3 and NZL5, played connect 4 and then attempted one of Ross Atkins chess puzzles, he did not like my solution and insisted I find another one.

Fun and Games at the IMO

Before the IMO, New Zealand and the Netherlands partook in joint pre-IMO training, everyday we would eat breakfast and pull up to an exam room to sit a 4.5 hour long mock IMO. We would then eat lunch and play Frisbee which I was very good at besides the part that involves throwing the frisbee. However, all good things eventually come to an end and we would eventually be forced back to the exam room to get the solutions spoiled and then mark our scripts. Fortunately, after all of this we were free to have fun and enjoy ourselves.

We ventured frequently to the student union which was one of my favourite spots pre-imo, where the New Zealand team and later also the South African team would partake in competitive foosball, card games and some pool, all of which I was not very good at. The student union also had a sports area: I played badminton with NZL1, NZLD and NZLC where NZL1 beat us so badly that she was recruited to go play with a group of professionals and was replaced by NZL3. Later I would come back and decide to play table tennis with NZL5, where we were challenged by the Turkish team.

My favourite spot during the IMO was the XTX hub, where I got lots of free pink lemonade, coffee and boba. There were also lots of games at the XTX hub and I was challenged by an Australian to some of the games at XTX hub and eventually connect 4, a game which I began to enjoy after figuring out how to count properly. NZL1 would very rudely criticise my connect 4 skills during this game and I would then go on to beat them multiple times in a row. I also played NZLC in connect 4 a few times which was quite fun because I didn’t win all of them.

My favourite room in the XTX hub was the chill room, which is kinda a funny name. It had a bunch of bean bags, some flexagons and a big whiteboard in the middle which of course was filled up with maths problems and theorems that some students decided they wanted to share. NZL5 decided to write up a proof we learnt from Ross Atkins to the statement: All angles are 90 degrees. The Dutch Deputy would give this proof a 7/7 which was quite impressive for NZL5 who tended to lose marks for skipping details. A pair of Canadianswould later pull up and after looking at the board for a few seconds one of them said “Oh that's easy, just invert at the point at infinity.”

I attended a few talks at the IMO: A talk by Daniel Tammet on Autism, One by Grant Sanderson (3blue1brown), a talent show hosted by Matt Parker, and a talk by a Google Researcher on AlphaGeometry.

Evan Chen, who wrote my favourite maths olympiad book Euclidean Geometry in Maths Olympiad, also showed up and later the famous New Zealand gold medalist Ishan Nath would also also show up to meet us at the IMO.

It’s really cool being around lots of very smart people at the IMO. I encountered people from Hong Kong, Costa Rica, South Africa, Turkmenistan, Netherlands, Canada, Turkiye, Germany, Australia, China, Japan and more.

Fun Fact: A member of the Costa Rican team came up to me one day and told me that: FIrstly, that New Zealand was the best country and secondly, that Eric, Brian and Grady (Past IMO Contestants) were the goats.

I cannot give an exact recount of what happened on each day, but I really enjoyed playing card games with the team, with the exception of Mao which I hated. My favourite games were coup and jungle speed.

Exploring the UK

The United Kingdom is known for many things: it’s world class food (Lime Tree), its brilliant weather, its lack of crime, high quality of life and the abundance of bins. But most famously of all are the numerous extremely famous attractions that we were about to visit.

Bath

The first thing we did when we arrived in Bath was visit the Roman bath houses and explore Bath. We then visited a fudge shop in Bath where I got free chocolate fudge. The shopkeeper seemed really excited when we told him we were here for a competition, however his smile seemed to fade when NZLC alerted him that it was actually a maths competition.

We also decided to embark on a Bath Skyline walking tour with the Dutch. Which involved walking around in the trees for a couple hours (not worth it) and not getting a single glimpse at the Bath Skyline besides at the very start of the walk. However I did enjoy playing catchwith the Dutch team even though I had to hold Ramsey and carry my bag because I did not want my bag to be covered in ants.

During the walk the Dutch team of course asked me to try liquorice which was definitely one of the things I’ve ever eaten.

London

After Day 2,

The London Walking Tour was quite fun, besides the fact that the weather was a little too nice and ended up being a bit too hot. I had been under the (obviously false) assumption that London was extremely rainy. We saw Buckingham Palace, Big Ben, the London Eye and a few other spots before heading back to Bath. I also learnt quite a few fun facts like the origin of the phrase “Laughing my head off”, and that buildings in London were named stuff like “Stack of Pizzas”, “Walkie Talkie” and “The Shard”.

Unfortunately, I actually lied when I said London was known for its abundance of bins and it’s in fact quite the opposite. I felt like the city of London wanted me to just litter after I spent what felt like an hour walking around with rubbish in my hands. It turns out London’s scarcity of bins actually comes from lingering fears of terrorist attacks, as it had been plagued by terrorist attacks for years of people dropping bombs in public bins. I also learnt that my school has had its bins removed because people were setting them on fire (how?)

Bristol

The next day, New Zealand and the Netherlands decided to go on an improvised excursion to Bristol after our trips to Bletchley Park were cancelled.

We decided to visit the Clifton Suspension Bridge, which involved a walk through the trees and which was beginning to feel like a repeat of our Bath Skyline walk. Adding to this, the weather was almost equally as good as the previous day.

Jane Street

After the IMO the New Zealand team was invited to go to London and attend an after party hosted by Jane Street. I don’t think Australia was invited, another victory for New Zealand.

The Hotel we stayed at was extremely fancy

We were invited to a video game museum, where I spent most of the time walking around grabbing drinks and fancy food and then spending the last part of the day playing tetris.

The next day we were invited to the Jane Street headquarters. We were meant to meet outside our hotel at 9am but I unfortunately overslept because the hotel bed was too comfy, especially compared to the bed we were forced to sleep in for the past 2 weeks. I even managed to sleep through NZL banging on my door to wake me up. I was eventually woken up at around 11am by the cleaners knocking on my door. NZLC had to come fetch me at around 12pm where I had to shamefully walk to the Jane Street headquarters by myself. I had unfortunately missed both breakfast and the Jane Street estimation where I was told New Zealand came last place.

I showed up just in time to join the second round of a Jane Street trading game, where I had no idea what was going on and was immediately scammed out of 10 blue chips by NZL3, where I then had to sit for a few minutes while everyone else traded. I did not have a chance to redeem myself as the second round also happened to be the last, and we were soon forced onto a boat where we were served even more fancy food while travelling through London.

Fun Fact: The members of the New Zealand IMO team are not only extremely talented mathematicians but also extremely skilled photographers

Notice the food and drinks on our table, they were very good.

I had been given a puzzle which I had somehow managed to disassemble by picking it up from the corner and removing one piece which resulted in the entire puzzle falling apart. We then struggled to reassemble the puzzle and eventually gave up and decided to join the Canadians at the foosball table and was shocked to discover that this one Canadian team was actually a foosball prodigy, passing the ball back and forth to himself and scoring points left and right. Soon our boat ride ended and we were forced back to Jane Street.

After a couple of talks we were finally let loose in the Jane Street headquarters, where we were fed loads of food. I decided to embark on a journey of exhausting my positive marginal utility and minimising my opportunity cost by eating as much of the Jane Street food as I could before throwing up.

I also took loads of pictures of the employee’s laptop screens and have figured out Jane Street’s latest trading strategies which I am unfortunately not allowed to share.

Afterwards I decided then that I was going to learn how to make a flexahedron and in the process I encountered a girl from the Canadian team who was severely brain rotted “did you just say skibidi glue” – real quote from her. I thought this was quite funny although at the moment I was actually flabbergasted. I also remember the song All Star came on and she said it was “based” which I agree with. She also had to drop a couple of hints because the origami required to make the flexahedron was a little too tricky for me. We all ended up getting superglue over our hands which made our fingers feel like they were made out of plastic.

Afterwards some members of the canadian team joined us for 斗地主 and it was funny to see the Canadians learn the rules of this game for the first time and being confused by the 三带一, 三带二 and the 飞机 (my personal favourite).

When we got back to the hotel, we ended the night by playing Codenames with the German team.

That was our last day in England, and looking back now, the best and most fun part of the IMO was definitely spending time playing games with my team and people from different countries. My biggest regret is being too scared to meet new people and make friends. I wish I could go back and meet as many new people as I possibly could.

Other Photos

Puzzles

• Given two potatoes, prove that there is a loop of wire which fits around both.
• Does there exist a triangle which can be tiled with 13 triangles, where each tile is congruent to each other and similar to the original triangle

Q&A’s

How has attending this event demonstrated greater knowledge of available career paths in science and technology?

Through the numerous guest speakers and visits to Jane Street headquarters I learnt that a lot of the problem solving and puzzle solving I love about maths olympiad can be found in places like trading, research and data science. So although I am still unsure of what career paths I wish to pursue in the future, I am aware of a few options that I can look into in the future.

How has attending this event enthused or inspired you to pursue science and technology careers?

I am a lot more optimistic about one day having to give up competing in contest maths because I’ll still be able to chase the things I enjoy in a science technology career if I simply find the right problems to work on and the right place to work at.

Has attending this event changed about how you feel about science or technology?

Not much.

How has attending this event and participating with like-minded students been of benefit to you?

It hasn’t really provided me with any real benefit, but it was super exciting meeting mathematicians from all over the world.

Now that you have had time to reflect about your experience, what have you learnt about yourself?

• I didn’t realise just how much I liked maths.
• I tend to be very careless, sloppy and at times manic
• I still have a lot of space to improve my mathematical abilities

What did you enjoy about your experience?

• The people
• Everything besides the Lime Tree food and the heat.