Ireland s Participation in the 52nd International Mathematical Olympiad

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1 Ireland s Participation in the 52nd International Mathematical Olympiad Bernd Kreussler September 22, 2011 From 12 th until 24 th July 2011, the 52 nd International Mathematical Olympiad took place in Amsterdam (The Netherlands). With 564 participants (57 of whom were girls) from 101 countries, this IMO was similar in size to its three predecessors. It was the first time ever that a team from Kosovo participated in the IMO. Senegal and Uruguay sent Observers and plan to participate for the first time next year. The Irish delegation consisted of six students (see Table 1), the Team Leader, Bernd Kreussler (MIC Limerick), the Deputy Leader, Gordon Lessells (UL) and the Official Observer, Mark Flanagan (UCD). 1 Team selection and preparation At five different locations all over Ireland (UCC, UCD, NUIG, UL and NUIM), mathematical enrichment programmes are offered to mathematically talented students, usually in their senior cycle of secondary school. These classes run each year from December/January until April and are offered by volunteer academic mathematicians from these universities or nearby third-level institutions. Like last year, we contacted schools directly and asked them to nominate their best students for the enrichment programmes. This is significantly less efficient than the practice of previous years in which the Department of Education and Skills and the State Examinations Commission provided information which enabled school principals to identify those of their students who were among the best performers in the country in Junior Certificate Mathematics. As a result, participation of good students in the training programme is not as high as it should be. The selection contest for the Irish IMO team is the Irish Mathematical Olympiad (IrMO), which was held for the 24 th time on Saturday, 7 th May, The IrMO Name School Year Adam Keilthy Sutton Park School, Dublin 5 th Ewan Dalby Coláiste Bhríde, Carnew, Co. Wicklow 6 th Padraig Condon St. Gerald s College, Castlebar, Co. Mayo 6 th Kieran Cooney CBS Charleville, Co. Cork 6 th Vicki McAvinue St. Angela s Secondary School, Waterford 6 th Yunwoo Lee Presentation Secondary School, Wexford 5 th Table 1: The Irish contestants at the 52 nd IMO 1

2 contest consists of two 3-hour papers on one day with five problems on each paper. The participants of the IrMO, who normally also attend the enrichment classes, sat the exam at the same time in one of the five centres. This year, a total of 50 students took part in the IrMO. The top performer is awarded the Fergus Gaines cup; this year this was Adam Keilthy. The best six students (listed in order in Table 1) were invited to represent Ireland at the IMO in Amsterdam. As in previous years, special training camps for our contestants at the IMO took place in Limerick. Following the good experience of last year, instead of one weeklong camp we organised two shorter camps for the students. The first of these was held at MIC Limerick on 8/9 June As the LC Examinations started on these days, four of the six team members could not participate. The participants included the two team members who did not sit their LC Examinations this year and five of the best students (chosen on the basis of their performance at the IrMO) who will be eligible to participate in future IMOs. The second camp, at which the six members of the Irish IMO team participated, was held at the University of Limerick from 5 to 8 July The camps were organised as usual in a very efficient way by Gordon Lessells. The sessions with the students were directed by Mark Burke, Mark Flanagan, Eugene Gath, Donal Hurley, Kevin Hutchinson, Bernd Kreussler, Tom Laffey, Jim Leahy, Gordon Lessells, Anca Mustata and Edin Omerdic. A new feature this year was a joint training camp with the Colombian IMO team. The camp was held in the Youth Accommodation and Education Centre at Hanenbos a few miles south of Brussels. The facilities were excellent and very reasonably priced. The venue was organised by the Colombian Leader Maria Losada with help from the Belgian IMO Leader Bart Windels. During the period of the joint camp, classes were organised some together and some separate. Two Colombian trainers, Ivan Contreras and Esteban Gonzalez, and Gordon Lessells conducted the classes. There was also time for relaxation with Table Tennis being popular with members of both teams. The feedback from students about this activity was very positive. 2 The events After arriving in Amsterdam on the evening of Tuesday, 12 July, Mark Flanagan and I were driven to the Conference Hotel Koningshof in Veldhoven near Eindhoven in the south of The Netherlands. As usual, the Jury was kept separated from the contestants until after the end of the second exam. The Jury of the IMO, which is composed of the Team Leaders of the participating countries and a Chairperson who is appointed by the organisers, is the prime decision making body for all IMO matters. Its most important task is choosing the six contest problems out of a shortlist of 30 problems provided by a problem selection committee, also appointed by the host country. This year s Chairman of the Jury was Professor Hans van Duijn, Rector of the TU Eindhoven. He led the Jury meetings in a pleasant yet efficient way. There was more time than usual, until Wednesday afternoon, to study the 30 shortlisted problems without being spoiled by the official solutions. Even in this early phase, the presence of Mark as an Observer was very useful. 2

3 During a number of meetings between Thursday and Sunday, the Jury chose the six contest problems and approved all translations and the marking schemes. The working conditions at the hotel, a former monastery, were excellent. The formulation of the contest problems in the five official languages (English, French, German, Russian and Spanish) and the translation into 50 further languages was finished on Saturday evening. The most remarkable thing to report about the problem selection process is the following. During the past 15 years, there were always two geometry problems chosen for the contest. Not so this year, even though there was a large pro-geometry group in the Jury. The final vote, between a second geometry problem and the combinatorics problem which became contest problem 2, was as close as possible with a margin of one vote only. The six Irish contestants, accompanied by Gordon Lessells, arrived in Amsterdam by train from their training camp in Belgium on the evening of Saturday, 16 th July. All contestants were accommodated in the Hotel Novotel in Amsterdam. The Opening Ceremony took place on Sunday afternoon in the RAI Theatre, a short walk from Novotel. The organisers improved the traditional parade by grouping the teams by continent and interrupted it accordingly by short speeches and music from the World Orchestra. It was widely agreed that this was a pleasant ceremony. The two exams took place on the 18 th and 19 th of July, starting at 9 o clock each morning. There were three venues for the exams and the conditions were reported to have been excellent. On each day, 4 1 hours were available to solve three problems. 2 During the first 30 minutes, the students were allowed to ask questions if they had difficulties in understanding the formulation of a contest problem. The Q&A session on the first day of contest with 36 questions was relatively short, whereas the 189 questions from 72 countries on the second day constituted an unusually long one. The most frequently asked question was whether the order of weights matters in Problem 4 (of course, it does). After the first scripts arrived late Monday evening, it soon became apparent that Padraig solved Problem 1, except for a small arithmetical error which cost him an Honourable Mention. After the final Q&A session on Tuesday morning the Leaders and Observers moved to Amsterdam to stay in the same hotel as the contestants and Deputy Leaders. Most of our contestants went to the latest Harry Potter movie so that we did not see them until dinner time. We used the afternoon and evening to prepare coordination of the first three problems which was to start at 9 a.m. on Wednesday morning. Most of this work was done before the scripts of the second day were available in the evening. The scripts of the second day contained an interesting surprise: Ewan devised a very novel recursion for Problem 4 which (as we could show) could be completed to form a solution different from any of those found by the Problem Selection Committee or the Jury. Unfortunately, he didn t provide a clean proof of it and also didn t complete the solution. He only scored 2 points for this problem. During the process of completing Ewan s solution, Mark with his combinatorial knowledge was extremely helpful. He discovered that the numbers in Ewan s double recursion are known as bifactorial numbers. To get the coordinators to agree about the 2 points for Ewan s attempt, it was necessary to show how the solution could be 3

4 completed. The availability of the helping hand of an Observer was vital to achieve this. The purpose of the coordination, during which the representatives of each country meet experts appointed by the host country, is to achieve equal standards in marking the scripts of the students. This year s high level coordination was conducted by a well prepared team of about 80 experts, 20 of whom had already participated as coordinators in the IMO 2009 in Bremen. During the two days of coordination, excursions and other activities were organised for the students. They had a choice between sailing, cycling, cryptography and sports or an excursion to The Hague. Our students enjoyed these days. The final Jury meeting, at which the medal cut-offs were decided, took place on the morning of Friday, 22 nd July. During the afternoon, together with our team and the team guide, Maaike Assendorp, we participated in an excursion to downtown Amsterdam. This included a boat trip through the 17 th -century canals of Amsterdam, which are on the Unesco World Heritage List. The memorable end of this beautiful and relaxing day was marked by a very enjoyable dinner and party at the science centre Nemo. The closing ceremony took place on Tuesday, 23 rd July, at the RAI Theatre in Amsterdam. It was followed by a Farewell Party at the adjacent Strand Zuid. On Wednesday the Irish delegation returned to Dublin. 3 The problems First Day Problem 1. Given any set A = {a 1, a 2, a 3, a 4 } of four distinct positive integers, we denote the sum a 1 + a 2 + a 3 + a 4 by s A. Let n A denote the number of pairs (i, j) with 1 i < j 4 for which a i + a j divides s A. Find all sets A of four distinct positive integers which achieve the largest possible value of n A. (Mexico) Problem 2. Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. A windmill is a process that starts with a line l going through a single point P S. The line rotates clockwise about the pivot P until the first time that the line meets some other point belonging to S. This point, Q, takes over as the new pivot, and the line now rotates clockwise about Q, until it next meets a point of S. This process continues indefinitely. Show that we can choose a point P in S and a line l going through P such that the resulting windmill uses each point of S as a pivot infinitely many times. (United Kingdom) Problem 3. Let f : R R be a real-valued function defined on the set of real numbers that satisfies f(x + y) yf(x) + f(f(x)) for all real numbers x and y. Prove that f(x) = 0 for all x 0. (Belarus) 4

5 Second Day Problem 4. Let n > 0 be an integer. We are given a balance and n weights of weight 2 0, 2 1,..., 2 n 1. We are to place each of the n weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done. (Iran) Problem 5. Let f be a function from the set of integers to the set of positive integers. Suppose that, for any two integers m and n, the difference f(m) f(n) is divisible by f(m n). Prove that, for all integers m and n with f(m) f(n), the number f(n) is divisible by f(m). (Iran) Problem 6. Let ABC be an acute triangle with circumcircle Γ. Let l be a tangent line to Γ, and let l a, l b, and l c be the lines obtained by reflecting l in the lines BC, CA and AB, respectively. Show that the circumcircle of the triangle determined by the lines l a, l b, and l c is tangent to the circle Γ. (Japan) 4 The results The Jury tries to choose the problems in such a way that Problems 1 and 4 are easier than Problems 2 and 5. Problems 3 and 6 are usually designed to be the hardest problems. Problem 2, which was generally considered beautiful with a nice and short solution, turned out to be much harder than anticipated (see Table 2). P1 P2 P3 P4 P5 P average Table 2: For each problem, how many contestants achieved how many points The medal cut-offs were as follows: 28 points needed for a Gold medal (54 students), 22 for Silver (90 students) and 16 for Bronze (137 students). Since 1988, when Ireland participated for the first time in the IMO, it is only the fifth time that 28 points were sufficient to get a Gold medal. On the other hand, during the same period, it happened only four times that more than 16 points were needed to get a Bronze medal. Overall, 35.0 % of the possible points were scored by the contestants, which places this IMO at a similar, but slightly harder level than the 5

6 three past IMOs. From this perspective, the improvement of the total score of the Irish contestants in comparison with the previous two years is more valuable than one may first think. Name P1 P2 P3 P4 P5 P6 total ranking Padraig Condon Ewan Dalby Kieran Cooney Adam Keilthy Yunwoo Lee Vicki McAvinue Table 3: The results of the Irish contestants Table 3 shows the results of the Irish contestants. As mentioned before, Padraig narrowly missed a full score for Problem 1. The originality of Ewan s attempt for Problem 4 is an illustration of the true potential of Irish mathematical students. The figures in Table 4 have the following meaning. The numbers in the second column give the percentage of the maximum possible marks scored on each question while the third column gives the percentage of possible marks scored by the Irish team. The final column indicates the Irish average score as a percentage of the overall average. Problem all countries Ireland relative all Table 4: Relative results of the Irish team for each problem Although the IMO is a competition for individuals only, it is interesting to compare the total scores of the participating countries. This year s top teams were from China (189 points), the USA (184 points) and Singapore (179 points). An interesting observation is that since 1983, when the team size of six was introduced, it is only the fourth time that the score of the top team was below 190. Ireland, with 26 points in total, finished in 87 th place. Only one student, Lisa Sauermann from Germany, achieved the perfect score of 42 points. She now holds the leading position in the IMO Hall of Fame with 4 Gold and one Silver medal in front of her countryman Christian Reiher who got 4 Gold and one Bronze medal between 1999 and Lisa s extraordinary performance was specially honoured at the closing ceremony where she received a standing ovation, something not seen very often in the past. The detailed results and statistics can be found on the official IMO website 6

7 5 EGMO From 10 th to 16 th April 2012, the first European Girls Mathematical Olympiad will be held in Cambridge, UK. This event will be attended by up to 20 participating teams of four female students from European countries. This initiative is hoped to encourage more girls to engage in mathematical problem solving. This seems to be necessary as not enough girls participate in the IMO. However, when they do, they often compete at the highest level. Lisa Sauermann with her success at the IMO will certainly serve as a model for the participants of the EGMOs. Ireland has decided to send a team to this event. Due to the relatively early timing of the event, training activities have to start earlier than usual. If the mechanisms developed for EGMO become established, this may also be useful for our male trainees. The EGMO Team Leader, Rachel Quinlan (NUI Galway), is coordinating the preparation and selection of the Irish team. The organisers would welcome proposals of original and unpublished problems for this Olympiad. The deadline for proposals, which should include suggested solutions, is the 16 th December Outlook The next countries to host the IMO will be 2012 Argentina 2013 Colombia 2014 South Africa 2015 Thailand The Jury continued a discussion, which had been initiated at IMO 2010, about reforming IMO procedures. The main topics included the problem selection process and the regulations to deal with allegations of cheating. A poll indicated that a huge majority would not be in favour of taking the problem selection process completely out of the hands of the Jury, a reform which would effectively abolish the Jury in its current form. As an outcome of this emotional discussion, an Ethics Committee was founded, whose mission will be to safeguard the integrity of the IMO competition. Further discussions on these matters may occur on the IMO forum, to be found on under General Information. 7 Conclusions To be able to send a full team of six students and possibly also an Official Observer to any of the next four IMOs, efforts have to be increased in the near future to get sufficient funding. A number of other IMO teams regularly organise joint training camps which take place immediately before the start of the IMO. This year s positive experience with the joint training camp in Belgium could be a model for future training activities of the Irish team. When the IMO takes place in a distant timezone, a training camp in that region can help to reduce the effects of jet lag during the IMO exam. On the positive side, the true potential and originality of the Irish students is visible as follows. Padraig Condon essentially solved Problem 1, only narrowly missing 7

8 an Honourable Mention because of a simple arithmetical error. Ewan Dalby found his own original solution of Problem 4, or at least the main idea, based on a novel recursion, which could be completed to a full solution. The overall results of the Irish Team, however, show that our students have much less experience in problem solving than the majority of the contestants from other nations. This becomes obvious by looking at Tables 3 and 4. Building up experience in solving mathematical problems needs time. It seems to be imperative to get more students in their Junior Cycle, or even in Primary School, involved in such activity. A promising and valuable activity in this context is the PRISM competition ( Another option, which is currently under discussion, would be the participation of Ireland in the International Mathematical Kangaroo Contest. This contest consists of a multiple choice test with problems designed for the general student, not only for the highly talented. The content is covered in the school mathematics curriculum. However, the problems require creativity, logical thinking or a different perspective. One of the main aims is to introduce participants to mathematical challenges in an enjoyable way and so to help replacing the fear of mathematics by interest in the subject. The contest can also help to identify creative talent among the students. Currently, there are separate problem sets for five different age groups available (Primary School: Classes 3 4, 5 6; Secondary School: years 1 2, 3 4, 5 6). The contest was started in 1991 in France, building on an activity in the early 1980s of Peter O Halloran, a maths teacher at Sydney, Australia. At its annual General Assembly, the participating countries who form the International Association Kangourou Sans Frontières ( decide about the problems for the next contest. Each participating country organises and regulates the competition nationally and is responsible for the evaluation and collection of the results. In 2010, more than 5.8 million students from 46 countries participated in the Kangaroo Contest. Most countries report rising numbers of participants from year to year and increased enthusiasm for mathematical problem solving. Everybody who is interested in supporting the involvement of Ireland in the Mathematical Kangaroo Contest is encouraged to get in contact with the author of this report. The five training centres with their experienced and enthusiastic trainers are not sufficient to widen the scope of support of talent and interest in mathematics in Ireland. To get more students at an earlier age involved requires the active involvement of people who are willing to offer regular maths activities at a local level (for example maths teachers running afternoon maths clubs in their schools). The training centres would certainly be able to support such initiatives with their experience. The involvement of a larger number of younger students in mathematical activities would be an essential contribution to the education of Irish children. In the medium and long term such activities have the potential to help future generations to resist rote learning attitudes and keep their natural talents and curiosity alive. As a result, the much criticised performance of our students in international student assessment programmes of the OECD will almost certainly improve. Lobbying organisations of employers and some large multinational companies operating in Ireland have already expressed their desire for young people with better skills in and understanding of science and mathematics. Enhancing skills related to logical thinking, mathematical proof and problem solving, would also be a very valuable 8

9 contribution to the education of future generations of students who wish to study mathematics at university. 8 Acknowledgements Ireland could not participate in the International Mathematical Olympiad without the continued financial support of the Department of Education and Skills, which is gratefully acknowledged. Thanks to its Minister, Mr Ruairí Quinn TD, and the members of his department, especially Eamonn Murtagh and Doreen McMorris, for their continuing help and support. Also, thanks to the Royal Irish Academy, its officers, the Committee for Mathematical Sciences, and especially Gilly Clarke, for continuing support in obtaining funding. The essential part of the preparation of the contestants is the work with the students done in the enrichment programmes at the five universities. This work is carried out for free by volunteers in their spare time. Thanks go to this year s trainers at the five Irish centres: At UCC: Tom Carroll, Finbarr Holland, Donal Hurley, Ben McKay, Anca Mustata, Andrei Mustata Dima Rachinski and Martin Stynes. At UCD: Mark Flanagan, Marius Ghergu, Kevin Hutchinson, Tom Laffey and Gary McGuire. At NUIG: Javier Aramayona, John Burns, James Cruickshank, Graham Ellis, Jerome Sheahan, Emil Skoldberg and James Ward. At UL: Mark Burke, Eugene Gath, Bernd Kreussler, Jim Leahy and Gordon Lessells. At NUIM: Sonia Balagopalan, Stefan Bechtluft-Sachs, Steve Buckley, Peter Clifford, Katarina Domijan, David Malone, John Murray, Tony O Farrell, Ciarán O Rourke, Lars Pforte, Adam Ralph, Dave Redmond and Richard Watson. Thanks also to the above named universities for permitting the use of their facilities in the delivery of the enrichment programme, and especially to the University of Limerick for their continued support and hosting of the pre-olympiad training camp. Finally, thanks to the hosts for their impeccable organisation of this year s IMO in Amsterdam and the warm welcome they had for all of us. Particular thanks go to the team guide in Amsterdam, Maaike Assendorp. 9