Monday 1st August 2016

Barnard Patel: IMO 2016 Hong Kong Report

Introduction

The International Mathematical Olympiad (IMO) is an annual maths competition in which over 100 different countries participate, each sending a team typically comprised of six secondary school students. The 57th IMO took place in Hong Kong from 6-16 July 2016, with New Zealand sending a team of six contestants:

Helping our team to prepare for the competition was team leader Michael Albert along with deputy leader Stephen Mackereth. In addition, Phil Truesdale was the team manager, overseeing the organisation of travel arrangements for the IMO, which as a result went smoothly. I was lucky enough to be selected for the New Zealand International Mathematical Olympiad (NZIMO) team, which is chosen after an almost year-long process of training, selection problems and contests. It was a unique and exciting experience to be able to travel to Hong Kong, especially as I had never visited this part of the world before, and participate in a maths competition on an international scale, meeting with – and competing against – some of the top maths students of various countries from all over the world.

IMO Training in Auckland

Before going to Hong Kong, I stayed in Auckland for 5 days (Monday 4 July to Friday 8 July), where I was kindly billeted by a host family, to whom I am thankful. During this time, a training programme was held at Auckland University, who generously allowed the NZIMO team to use one of their rooms for the week, with the team preparing together for the IMO. From Monday to Thursday, the daily routine was the same: we would sit a mock IMO for 3 hours in the morning, before having a lecture in the afternoon, either from our leader or deputy leader, as well as getting our mock IMO papers back and going over the answers. The mock IMO papers each consisted of three problems, of varying difficulty: Generally, there was one ‘easy’, one ‘medium’ and one ‘hard’ (by Olympiad standards – even the ‘easy’ problems were fairly difficult). These were taken from the 2015 IMO shortlist (from which the problems in last year’s IMO were chosen), and hence provided a good representation of the questions we would be likely to get at our IMO.

Hong Kong

Arrival in Hong Kong

The first thing that struck me about Hong Kong was the heat. This was to be expected, as we were going to Hong Kong at the height of summer, and temperatures soared to almost 40°C, something I wasn’t used to, given that I live in Wellington. When we landed at the airport, we were greeted by organisers of the IMO, as well as our team’s guide, Joanna, who would help us around Hong Kong for the duration of our stay. We then travelled to where we would be staying: The Hong Kong University of Science and Technology (HKUST), which hosted the IMO (including the contest itself as well as other activities) and accommodated all 109 teams. The campus was very nice, in terms of being ultra-modern, as well as huge, sprawling across a giant hill. As a result, travel around the campus – for instance, from our rooms to the cafeterias – involved a significant amount of time on elevators. For example, to get from our rooms to the rest of the campus (where the cafeterias, lecture theatres, bus stop and everything else relevant were located), we needed to take a minimum of three different elevators, travelling past around 30 different floors. This took a bit of getting used to, especially since the elevators in different buildings had their own numbering schemes, which often counted in different directions: we would go up 21 floors from ground only to end up, confusingly enough, on ‘LG7’.

We arrived on the morning of Saturday 9 July, and had nothing planned for the day, allowing us to acclimatise to the heat and the time zone. When we arrived at our rooms, which were quite nice and thankfully fitted out with both air conditioning and a fan, we were each greeted by a backpack filled with various items of IMO memorabilia. We were also given food vouchers, which were redeemable at most of the cafeterias on site at HKUST. The vouchers came in $10 and $20 denominations, with no change given when used. As a result, it became customary to pool the vouchers together when purchasing food and strive to obtain a total cost which was ‘zero mod 10’, that is, a multiple of 10, by any means: this included buying more food and drink than was necessary. However, it turned out that such efficiency was not required, as some of us did not even use up all of our food vouchers – I still had hundreds of dollars in food vouchers remaining on the day we departed Hong Kong. Nevertheless, the idea of purchasing food together was quite a good one because it allowed each of us to sample a wider range of the Hong Kong cuisine.

Opening Ceremony

On Sunday 10 July, we took a minibus from HKUST to the city centre of Hong Kong, enjoying an impromptu excursion into Hong Kong outside the university. We walked around, visiting a few different sites such as shopping malls, parks (see photo below) and even a velodrome.

The team and deputy leader in a Hong Kong park. The background illustrates the contrast between the greens of trees and bushes, and the greys of towering skyscrapers. Such contrast seemed to be commonplace in Hong Kong, which was quite interesting.

That afternoon, we attended the IMO opening ceremony. This was at an external venue outside HKUST, and featured a range of different musical performances, including the “57th IMO Overture”, “Mathematical Installation Percussion” and “57th IMO Theme Song”. One of the main events of the opening ceremony was the ‘team parade’, in which, as the name suggests, each team walked up onto the stage with their national flag. It was intriguing to see all of the different countries that we would be competing against in the IMO, as well as the order in which these countries joined the IMO (this was how the countries were organised).

Contest Days

The IMO is divided into two days of competition. On each day, the contestants are given 4½ hours to solve three problems. The first of these two days was on Monday. The anticipation was palpable that morning, and when it came to sitting in the exam room waiting to begin, I was very excited (if a bit nervous also). The question paper was in an envelope, while a folder held individual plastic sleeves in which we were to place our solutions to each of the problems. We also had a range of coloured cards on our desk, corresponding to things like ‘Water’, ‘More paper’ and ‘WC’. The set up was very professional. Quite suddenly, as I was examining the various items on my desk, the competition began. Question one was a geometry question – this was pleasing as I had solved a geometry question one on the last mock IMO that we had, so I felt that this would be a good opportunity to attain a complete solution to a problem. It didn’t take long for me to determine the main idea of the problem – using the radical axis theorem to prove that three lines are concurrent – which reduced the problem to proving that three quadrilaterals in the diagram were each cyclic, a fairly standard task in Olympiad geometry. However, although I thought I managed to write a complete solution, unfortunately I made a mistake in the write-up, with my reasoning in slightly the wrong order. As a result, I failed to get a perfect 7, instead ending up with a 5, indicating that my solution, while complete, had a flaw. As I did not make substantial progress towards finishing problems two and three, which were each much more difficult than the first one, my score for day one totalled 5, which was not ideal, but it could have been worse.

On Tuesday, we had day two of the IMO. The first question was number theory, while the second and third were algebra and combinatorics, respectively. These three branches, together with geometry, comprise the four broad areas of Olympiad Mathematics. The problems that we were presented with can be found here: https://www.imo-official.org/problems.aspx. As before, I concentrated my efforts primarily on the first problem. However, I did not manage to attain a completed solution. The problem essentially asked for (my wording) the minimum number of consecutive positive integer terms b (where b is at least 2) in the sequence an = n2 + n + 1, such that each term shares at least one prime factor with another term. Using the Euclidean algorithm, a fundamental technique in Olympiad Mathematics number theory, I managed to show that b ≠ 2, 3 and maybe 4 and 5 also (my reasoning for the latter two may have been slightly incomplete). I therefore conjectured that the answer was b = 6. This turned out to be the right answer! However, disappointingly, I failed to find or prove the existence of such as construction. As a result, I was awarded 2 points out of 7, for a partial solution. As I spent very little time on problems 5 and 6, instead attempting to finish of problem 4, my total for day two was just 2. As a result, my overall score was 7. Although I might have hoped for an honourable mention at least (attaining 7 points on one problem), so I wasn’t too pleased with my performance, I nevertheless enjoyed the academic challenge of the IMO questions, as well as the overall experience of competing in such an illustrious, albeit difficult, contest. The New Zealand team ended up 53rd, which is just in the top half (109 countries in total). The full results can be seen here: https://www.imo-official.org/year_country_r.aspx?year=2016.

Excursions

With the contest days completed, the team leaders, deputy leaders and co-ordinators had a busy few days before them, marking the papers and agreeing on fair scores. However, for the contestants, the work was behind us, and now we had a couple of days to visit some of the attractions that Hong Kong has to offer. On Wednesday, we went to Disneyland (see photo below); This was a very enjoyable day. Although the buses were (an hour) late picking us up in the morning from HKUST, the experience once we arrived was great, and we still had ample time to try most of the rides we wanted to go on. There were a whole range of rides, to suit all of our team members, some of whom seemed to prefer the “It’s a small word” type of ride (gentle, slow-moving and filled with Disney happiness and singing) – which there is nothing wrong with – while others, me included, had particular fun on the more thrilling rides, such as the Star-wars themed “Hyperspace mountain” rollercoaster.

The NZIMO team, along with our guide, at the entrance of Disneyland. (It was drizzling – yet hot as ever – in the morning, as shown above, but the weather improved considerably, and it was an extremely sunny afternoon.)

On the second day of excursion, Thursday, we visited a few different locations around Hong Kong, all of which were very different to each other. First, we went to ‘The Peak’ (see photo below), one of the most popular attractions of Hong Kong, which provides a great view of the cityscape. Next, we visited a local school, St. Stephen’s College, where we embarked on a ‘heritage trail’, which consisted of learning about the history of the school and in general its location in Hong Kong. In the afternoon, we went to the Stanley markets, where the clusters of stalls sold a variety of different wares. Although this day was perhaps not quite as enjoyable as Disneyland, it was valuable in that we gained a better understanding of what different parts of Hong Kong, away from HKUST, were like.

One of the buildings at The Peak, shown above, resembles the mathematical symbol for pi, π; the official 57th IMO Hong Kong logo utilises this likeness.

Closing ceremony

On Friday afternoon, after a short lecture in the morning about the efficiency (i.e. minimising perimeter for a given area) of various pentagonal tiles, we went to the Hong Kong Convention and Exhibition Centre for the closing ceremony. Similar to the opening ceremony, there were several musical performances, such as an impressive marching band. However, there was of course also the presentation of medals to the recipients. This year, the cut-offs were 16 for bronze, 22 for silver and 29 for gold. We arrived at the ceremony an hour or so early, but this was good in that it provided plenty of time to give away New Zealand souvenirs to as many countries as we could (we did, after all, each have what could very well have been over 100 pens, keyrings etc. to offload). At the opening ceremony and then on the coach back to HKUST, I managed to exchange souvenirs with contestants from a wide range of countries, which was very nice, giving away almost all of my New Zealand souvenirs (which I was quite happy about, given that we were leaving the next day).

After the closing ceremony, we enjoyed the IMO dinner, which was conveniently held at the same venue. The food itself was extremely fancy, probably more so than any meal I had ever eaten in my life before the IMO. For instance, the entrée (there were several courses) was entitled “Smoked Scottish Salmon and Oscietra Caviar, Mozuku Condiments” (pictured alongside). As a result, the IMO gave me the experience of trying new foods that I would otherwise never have had the opportunity to eat.

To be honest, I didn’t particularly like the salmon, but the following courses just got better and better. In particular, the dessert, “Exotic Fruit with Lime Sherbet”, was wonderful. I have a picture below, and as you can see, I had already started eating before I remembered I was supposed to be documenting my experience! The lime sherbet was especially tasty.

Conclusion

The IMO was a fantastic experience. It was a privilege to be selected to represent New Zealand, and I am deeply grateful to all who made the 2016 NZIMO team’s participation at the 57th IMO in Hong Kong possible. This of course includes the people involved in the selection and training process for the NZIMO team, including team leader Michael Albert, deputy leader Stephen Mackereth, and team manager Phil Truesdale. The team selection process, including the January camp and subsequent training, was extremely beneficial to the development of my Olympiad Mathematics abilities: Merely one year ago, I had not ever done an Olympiad Mathematics problem! In particular, I am appreciative of Stephen reading and giving me feedback on my assignments every week leading up to the IMO. The way in which this training has improved my problem solving abilities will likely be helpful for whatever I chose to study in the future. In addition, the IMO itself was, overall, organised very well, especially given the large quantities of participants involved. It is not feasible to name all of the many volunteers, including the guides, HKUST staff and other helpers, but it is important that some mention is made of these people who ensured the smooth running of the IMO. Furthermore, particular thanks should be given to our sponsors, who made our participation at the IMO a reality, most notably to the Royal Society of New Zealand, who were the principle sponsor for our journey. This funding allowed me and rest of the team the rare opportunity to represent our country and test ourselves against the very best in the world, as well as travel to an exciting destination in Hong Kong. This will no doubt be an experience that I will look back on as being very beneficial, in many ways not limited to solely the development of my mathematics ability.