Monday 5th October 2020

Ishan Nath: IMO 2020 Report

Introduction

The 61st International Mathematical Olympiad was held this year in St. Petersburg Auckland Christchurch, from the 8th to the 18th of July 19th to the 28th of September 2020. The IMO is a prestigious mathematical tournament held annually, taking six of the best young mathematicians from every country around the globe and having them duel it out in the ultimate mathematics competitions consisting of six ultra-extra-mega difficult problems, and nine hours total to solve them. This year, our delegation consisted of Hamish, Ethan, and me, who would be taking the test in Christchurch, and Rick, Nathaniel, and Phillip, who were stuck in Auckland. Ross and Josie, our leader and deputy, would be accompanying the Christchurch squad, while May, our observer A, took care of the other half of the team. Included in this report are a variety of problems I found interesting, for the reader to enjoy.

Pre-IMO Training

Due to this year's IMO being postponed till September, the squad and team were given much more training than usual. Some other members of the squad were also eligible to participate in the Cyberspace Mathematical Competition this year. Even with the ongoing crisis, online competitions such as the CMC, USEMO and FEOO allowed us to continue with our maths training. Some cool problems from training or competitions from this year are:

“Find all primes p such that (7p-1 -1)/p is a perfect square”

“Two players play ‘nim with a pass’. There are 2021 piles in front of them, 2020 of which have a distinct positive number of stones between 1 and 2020, and 1 of which has a special passing stone. The players take turns removing any number of stones from the same pile, and the person who removes the last non-passing stone wins. Who wins?”

“Let O be the center of equilateral triangle ABC, and let P be a point in the plane. Prove (without bashing!) the area of the triangle with side lengths PA, PB and PC depends only on length PO.”

As well as RMM 2020 P3, USEMO 2019 P5 (even though it took place in 2020), APMO 2020 P3 and P5, Canada National Olympiad 2020 P4, EGMO 2020 P3, CMC 2020 P2, and USA TST 2020 P4.

September 18th

Due to the ongoing COVID-19 pandemic, the format of the IMO has changed. This year, Team New Zealand will be staying in the country to do the exams, with two test centres in Auckland and in Christchurch. This was due to the uncertainty of Auckland’s alert level at the time of the IMO. Therefore, this year I only have one flight to get to the IMO: Rotorua to Christchurch, a big plus. Unfortunately, this means I am unable to play Big Al’s Casino on the international flights, a major disadvantage. I arrive at the airport, and am greeted by Hamish and Josie, and eventually Ross as he shows up in his rented Corolla. I learn that Ethan will be joining us only for the exams, but will be staying for the day of the excursion.

As was established last year as an NZMOC tradition, we head off for dinner at Nando’s. Unluckily for us, we do not have a certain Tony Wang present and are therefore unable to receive everything at a discount. We then received our official NZMO-sponsored t- shirts, which have “Russia-Auckland” on the back, which was pretty unfortunate seeing as we were in neither of those places. Later in the afternoon, Ross tells us what not to do in an exam, such as not drawing a diagram for a geometry problem. I then proceed to solve a geometry problem on the NZMO2 without drawing a diagram. As we are acclimating to IMO conditions, we head to bed at around 1 AM.

September 19th

In the morning, we have a team meeting to discuss the events and take a few photos. We then head outside for some fresh air. During this time, Josie steps on our treasured frisbee, and it receives a minor fracture. Josie then decides to step on it again, doubling the number of frisbees we now own.

“Twice the number of frisbees, twice the fun” - Ross, 2020

We return, and I start talking about the overwhelming likelihood of there being some russian combinatorics problem on the exam. Ross then gives me the following problem:

“A cube has all its faces painted either black or white, and is placed on a chessboard, such that the painted face touching the board is the same colour as the cell it touches on the chessboard. We are allowed to roll the cube such that the face that touches the board is the same colour as the cell it touches. What is the maximum number of cells the cube can move onto?”

Afterwards, I struggled on an inequality:

“Let a, b, c be positive reals such that (ab)2 + (bc)2 + (ca)2 = 3. Prove that (a2 -a+1)(b2 -b+1)(c2 -c+1) ≥ 1”

I failed to solve it, although I was 99% certain that there would be no inequality on this year's competition, considering their dying relevance in the mainstream olympiad curriculum. It was then time to travel to the university, for our last mock exam, this time meant to directly simulate the IMO. It turns out problem 2 is a fairly easy complex bash, while problem 3 is vacuously false due to the nonexistence of regular decahedron. The problem reads:

“A dice is a regular decahedron, with 1 real number on each face. D10 is a dice with each face having a unique integer between 1 and 10. Find all pairs of dice such that for any real r, the probability of rolling the two dice and having their total sum to r is the same as the probability of rolling two D10 and having their total sum to r.”

September 20th

It was the day before the IMO, and as tradition has it, we were supposed to abstain from any mathematics. We briefly go over the previous day’s mock, before diving into some of Ross’ Dumb Puzzles, as they were aptly named. Here are five, ordered from least dumb to most dumb.

  1. Let’s agree that when you are at the North pole you can’t look North. Similarly, when you are on the South pole you can’t look South. Suppose you can look North and you can look South, but you cannot look East nor West. Where are you?
  2. In many London Underground stations there are two up escalators but only one going down. Why?
  3. What is the next word in the sequence: In, The, Paul, Paul, Paul, Paul, Paul, Paul, Paul, Paul, Paul, Paul, Paul, Paul, Paul, …?
  4. What can go up a chimney down but not down a chimney up?
  5. How can you go from 98 to 720 using just one letter?

Indeed, puzzle 5 turned out to be the hardest problem that I received during my time in Christchurch, with my inability to solve the problem being the butt of many jokes to come.

September 21st-22nd

It was finally the contest days. We began with our usual routine of going out and then solving some dumb puzzles, but this time we had 4.5 hours of math to do at the end of each day. We arrived at the contest room at around 6:30PM, leaving about an hour for us to banter and for Josie to talk about the many ways we could get disqualified during the exam.

Upon flipping day 1’s paper, total fear and disparity struck me. The first thing I saw was a geometry question 1 with some weird angle conditions. Right under it was an inequality, the last thing I expected to see - and the last thing that I prepared for! Problem 3 was a nice combi problem, but I only wanted to start attacking it after I finished the first two problems. Fortunately, after five minutes I guessed the relevant intersection point for question 1 (the circumcenter of PAB) and the problem’s difficulty subsided. Then, it was onto the second problem. After a while, I remembered what weighted AM-GM was, and the problem turned into an asymmetric homogeneous inequality, which then fell to many techniques, including my brutish “Hurr-durr multiply by (a+b+c+d)3 ” method. I was left with 2.5 hours for problem 3, and fortunately I stumbled onto the graph-theoretic interpretation fairly quickly, leading to me finishing all problems with around 20 minutes to go. If Geoff’s rule did still hold intact, problem 4 and 5 were to be some permutation of (combinatorics, number theory), which were the subjects I liked the most. Thus, solving these two put me in a good position to solve 5 problems, and maybe even 6 if I was lucky enough.

Day two arrived, and when I turned over the paper, what I saw matched my expectations, apart from the appearance of a combi-geo problem 6. I had guessed it had to be either algebra or geometry, given the problem distribution, but it didn’t really faze me, since I enjoyed combinatorics the most. After solving problem 4, another graph-theoretic problem, my attention turned to problem 5. Immediately I went down the wrong track, trying to do things involving PNT or weighted averages of sequences. As the time ticked down, I became more and more agitated. At 30 minutes left, I had to make the choice between fruitlessly attacking problem 5 for the rest of the exam, or trying to solve problem 6 in the last half hour. Although it was combinatorics-flavoured, the problem had an aura of difficulty surrounding it, and I chose to do the former. Alas, nothing transpired until the last 5 minutes, when I finally thought of the solution! With five minutes on the clock, in the words of William Han, “I write with a speed that I have never written with before, my handwriting degenerating into an almost indecipherable scrawl”. I left the exams with 4.5 problems solved, and it was up to Ross and Josie whether I could get the other 0.5.

September 23rd

After the second exam, I continued to stress over the validity of my solution to problem 5, however there was nothing to be done by me at this stage, and all the time for us to go do excursions and other fun activities. At this point Ethan was now residing with us. We decided to play a card game which was secretly an escape room, and then decide what other events to do. After this, the 2019 shortlist and 2020 problems are released, allowing discussion. Obviously the first thing everyone does online is complain about problem 2. The shortlist last year featured some good problems, including: A4, A6, A7, C4, C6, C7, G3, N3, N5 and N6. I have yet to do C9 or N7, but I am promised they are exceptional. Problems 3 and 5 on this IMO were my favourites, even with my rocky relationship with problem 5. First on our excursion agenda is mini golf, which, like many things in my life, I give up on halfway through. Somehow I end up ∞ over par on one of the holes, and am being declared the supreme loser.

“How are you so bad at mini golf” - Everyone

“Let’s round that down to a 5”

Our next expedition is via a 4-seat bike-car hybrid, which we push to a top speed of 25km/h. Our day ends with me hopelessly trying to figure out what letter can be added to 98 to make it 720, and being belittled by Ross for not knowing the solution. Tomorrow I was returning home, meaning today was my last day as a contestant in an olympiad. I took a moment to soak it in, before attempting to solve puzzle 5 again, in vain.

Post-Christchurch

Thankfully on my return back to Rotorua, school was finished so I could relax for the next two weeks. However, the one thing I was anxious about were the medal cutoffs, which were due at 2am on the 28th of September. Still used to the late nights, I dutifully stayed up past midnight, and waited for the official marks to appear on the website. At 1:30AM, through the painfully slow internet of imo-official.org, we all got to see our marks. New Zealand came 47th equal this year, a jump up 11 places from last year, and moreover, I scored a gold medal, with 35/42 marks total! New Zealand also achieved two bronze medals and three honourable mentions, gaining 102 marks total, our fourth best ever.

Reflection

This IMO was very different to the previous ones that I’ve been to. One big factor that has changed has been the social factor of the event. Previously, contestants could congregate together in one big hall full of board games, or they could compete in football matches with teams from all over the world and get to meet everyone else in person. This year, the atmosphere was very different, with most contestant communication happening through online messaging services such as discord and telegram. Even though I got to communicate with overseas members through online games, the different time zones and online factors meant that the contestant interaction this year was toned down compared to previous years. I do have to commend the organisers, however, who tried to keep the IMO spirit alive with their guest lectures, which were of great quality this year since great mathematicians and presenters from around the world could interact with the participants, without being hindered by the location of the contest. They also organised chess matches, which unfortunately team New Zealand could not participate in since I misread the time of the match and thus everyone woke up too late :(. I am grateful that I was able to have an IMO experience though, and had a week with two of the team members, Ross and Josie in Christchurch, considering other countries couldn’t even participate. Even if it wasn’t the ordinary IMO experience you would expect, spending a week with other great mathematicians and talented minds was more than enough, and I would highly recommend attending the IMO, no matter the circumstances.

Now it’s time to thank everyone. First, I’d like to thank Ross, Josie, May, Phil, our guide Lidiya, the NZMOC, and everyone who has supported me in my 4-year journey towards attending this IMO. I’d like to thank the organisers of IMO 2020 for pulling through in these hard times and ensuring an IMO could still happen. Finally, I would like to especially thank the Royal Society of New Zealand for giving me the opportunity to participate in this event.

Finally, one last problem for the extra-curious.

“Let ℤ2 denote the set of points in the Euclidean plane with integer coordinates. Find all functions f : ℤ2 → [0,1] such that for any point P, the value assigned to P is the average of all the values assigned to points in ℤ2 whose Euclidean distance from P is exactly 2020.”

The team