The 1998 International Mathematical Olympiad

As our Canadian team arrived back at Toronto airport after the 39th International Mathematical Olympiad (IMO) in Taiwan, I felt a mixture of emotions. I was proud of our six team members for their exemplary conduct and fine performance. I was relieved that we had not lost any members of our delegation into the crowds of Taiwan where few people speak English. However, I was also saddened to reflect that I would have to say goodbye to the students with whom I had worked closely over the last year.

The road to Taiwan was the culmination of a number of challenging steps for each of these students, and involved a large number of dedicated individuals who gave up much of their time in support of a program that celebrates the best in high school mathematics.

Choosing The Team.

The early hours of morning of January 9th, 1998 found me waiting at the Kitchener bus stop for the last students to arrive from Montreal for the Winter Training Camp at the University of Waterloo. January 1998 was the time of the Ice Storm whose devastating effects disrupted many lives across Ontario and Quebec. Most of the students had arrived on time for the Winter Camp. However, two of the students, who had planned to take the train from Montreal, had had to take the bus instead. The students arrived tired but seemingly happy, and I drove them to their accommodation and listened to stories about life without power in the middle of winter.

The Winter Training Camp is not the first step to the International Mathematical Olympiad, but it is an important early one. Students are selected on the basis of their performance in a variety of competitions. The 1998 Winter Camp featured a group of 15 students from across the country and a team of trainers and support people: Bill Sands from the University of Calgary, who was there to observe the format of the Winter Camp and to plan for the Summer Camp in Calgary, Dorette Pronk, a Postdoctoral Fellow at Dalhousie, who had agreed to act as Deputy Leader Observer for the Canadian team at the 1998 Olympiad, and my wife Kristin Lord, who was Deputy Leader Observer in Argentina the previous year. There were a couple of "guest" lectures given by David Jackson on combinatorics and by Mary Thompson on extremal problems as well. At the suggestion of some alumni of the Olympiad system, we brought together some older students from previous years, now undergraduates at Waterloo, with our students at the Winter Camp for a closing evening of five-pin bowling. The suggestion of an evening of bowling turned out to be fortuitous, as the freewheeling format, which allowed Richard Hoshino and others to propel bowling balls down the alley in much the same way that Roger Clemens pitches fastballs, was an excellent counterweight to the difficult academic challenges earlier in the camp.

After the Winter Training Camp, the results of competitions such as the Canadian Mathematical Olympiad and the Asia Pacific Mathematical Olympiad were pooled to determine the group of six students who would represent Canada at the International Mathematical Olympiad in Taiwan. The selection committee chose Adrian Birka (Niagara Falls, Ontario), Adrian Chan (Toronto, Ontario), Jimmy Chui (Toronto, Ontario), Mihaela Enachescu (Cote St. Luc, Quebec), Jessie Yin Lei (Windsor, Ontario) and Adrian Tang (Scarborough, Ontario) as its team. It was of some concern that the team concentration was so heavily based in Ontario. However, the selection committee was unanimous in feeling that these were the students whose performance had been of such high caliber as to strongly recommend them for the IMO team. It was also interesting to note that three of our students were named Adrian. We joked about the advantages of having this name in the selection process. Of course, as a statistician I knew that coincidences not only can appear in random selection, but can be expected to appear.

Gathering Storm Clouds.

Meanwhile, back at the ranch (Ottawa), the indefatigable Graham Wright was working hard on a challenging problem. Jessie Lei, one of our team members, was due to travel to the IMO on a Chinese passport. In itself, this did not represent a major obstacle to going to Taiwan. However, the delay in processing Jessie's visa application was a serious obstacle to our plans, as it was looking as if her visa would not be ready in time. It did not help to hear that the team from the People's Republic of China would not be going to Taiwan because of the use of the term "Republic of China" in Taiwan's correspondence and promotional material for the IMO. Fortunately a timely intervention by Ed Wang of Wilfrid Laurier University helped save the day. We were grateful to Ed and his mother for their deft navigation of the tricky business of obtaining a Taiwanese visa.

The Path To the I.M.O.

For the six students chosen for the 39th IMO in Taiwan, the intensive training began at the University of Calgary on June 30. Georg Gunther, a veteran of the Olympiad training program, flew from Cornerbrook to be a part of the first week of training. It was Georg who introduced us to the idea of the evening debriefing session, where at the end of each day's activities we gathered together to talk informally about how the training was going. I was delighted that Arthur Baragar of the University of Nevada at Las Vegas, who was a member of Canada's very first team at the IMO in Washington in 1981, agreed to serve as Leader Observer and to assist in the training in Calgary. Arthur provided the students with some useful background in number theory and also shared his passion for geometric number theory. Richard Guy of the University of Calgary also served as a guest lecturer and an inspiring role model for the students. Thanks must go to Richard and his wife for generously making their house available one afternoon. We sat around his living room and watched a video of animated mathematics on television. There is something about seeing an animated construction of the Simpson line that is highly convincing.

After a week of activities, Georg Gunther had to head home. However, the cavalry rode over the hill in the form of J.P. Grossman, a former gold medal winning IMO constestant and now a doctoral student at M.I.T. J.P. is a top-notch trainer with a wealth of original problems and ideas to share with the students. He was a daily reminder to them that the colour gold was not too high to reach. On July 5, we gathered our belongings in a van rented by Bill Sands and moved from the University of Calgary campus to the beautiful Kananaskis research station in the foothills of the Rocky Mountains. It only took me a few minutes to realise that the Kananaskis site would be excellent for training in mathematics. In a lovely setting with no outside distractions other than the scenery, we were free to work on challenging math problems and go for hikes.

Mihaela Enachescu and Adrian Tang at Johnston Canyon in the Rockies
Photo by Kristin Lord

The IMO.

As Leader Observer and Leader respectively, Arthur Baragar and I had to leave for Taiwan earlier than the students. So on July 9th, we were on our way, wishing our students the best of luck. J.P. Grossman and Dorette Pronk stayed with the students to finish the training. My first job as Leader was to sit on the jury and to work on the selection of the six problems for the competition. Fortunately, the jury members did not start from scratch; a problem selection committee narrowed down the choice to around thirty problems before we arrived. Once the activities of the jury began, Arthur and I sequestered, and could not contact any of our contestants until after the final problem session of the competition was completed. We did get a chance to see how our team was doing at the opening ceremonies for the IMO. As no contact was allowed between jury members and contestants, the jury observed the opening ceremonies from the back of the auditorium. The contestants and dignitaries from around the world were seated in the front. The Canadian uniforms were distinctive. So we had no trouble identifying our team at the opening ceremonies. A total of 79 countries participated. That was down from the previous year in Argentina where 82 countries had assembled for the IMO.

There were a few surprises at this year's IMO. One surprise was an earthquake which shook parts of the island for a few moments while we were there. Sadly, four people in another part of Taiwan died from debris that fell as result of the earthquake. However for the IMO delegates centred in Taipei, the earthquake was only a minor incident at the time it happened. Another surprise that I would not wish to be repeated at a future Olympiad was that many of the students had to sleep on bunk beds with no mattresses. While this is a commonplace practice in Taiwan, many students from North America, Australia and Europe were not accustomed to it.

The 1998 IMO was heavy on number theory and geometry but was reasonably balanced on difficulty. Problem 6 was universally admired by the jury. It asked the contestants to consider all functions f from the set of positive integers to itself such that

f( t2 f(s) ) = s (f(t))2

for all positive integers s and t. The problem was to determine the least possible value of f(1998).

Each year, medals are handed out at the IMO according to the following rules:

1. No more than 50% of the students should receive medals.

2. Among those receiving medals, the gold, silver and bronze medals are to be awarded as close to the ratio 1:2:3 as possible.

The Canadian team received the following medals:

Gold: Adrian Chan.
Silver: Mihaela Enachescu.
Bronze: Jimmy Chui, Adrian Tang.

Jessie Lei was awarded an Honourable Mention, which goes to any student who does not receive a medal but who gets a perfect score on at least one of the problems.

I was delighted with the performance of our team. I also especially pleased by the fact that Mihaela Enachescu was tied with one other woman from the U.S.A. as the highest scoring woman in the competition. Mihaela narrowly missed a gold herself.

The IMO is a competition between individuals, not countries. Therefore students are ranked and honoured in the closing ceremonies of each IMO. Nevertheless, international rivalries being what they are, countries sum the scores of their contestants and compare their summed scores scores with those of other countries. For those who keep track of such things, 1998 was a good year as Canada was ranked 20th out of the 79 countries.

There are many other stories of the 1998 IMO, most of them involving feet for some unexplained reason: the former male team member who showed up one day at the Winter Training Camp wearing high-heeled shoes, the sad story of a cheap frisbee that was artfully and cleverly decorated only to be crushed ignominiously by an awkward teenage limb, and the team member who bravely hiked through Johnston Canyon with bare feet. But perhaps these stories are not mine to tell.

Christopher Small

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