Limerick-based student wins silver at International Mathematical Olympiad

A Co Limerick-based student has become only the second person to win a silver medal for Ireland at the world's most prestigious second-level mathematics competition.

Lucas Bachmann (19), a 6th year student at Glenstal Abbey in Co Limerick, was one of 621 contestants from 112 countries who competed in the International Mathematical Olympiad (IMO) in Bath, England.

The annual competition attracts the brightest mathematical brains on the planet. It is regarded as a source of national prestige in Asia in particular, with China, South Korea, North Korea and Thailand prominent among the countries participating.

Mr Bachmann was part of the six-person Irish team selected after sitting two mathematical papers at home. He was born and brought up in Switzerland and then lived in China before his family relocated to Ireland for work purposes four years ago.

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He previously won a bronze medal at the competition and hoped to win gold this time around, but was stumped by question about positive integers.

“It was one of those simple questions that had a trick to it in order to solve it. Now I know the trick,” he said. “Everything I have done over the last four years has truly cemented me in participating in this. Computer science had been my favourite subject. That was not a subject I was offered at school in Ireland. Mathematics took over for me then.”

Training camps

In the three weeks leading up to the Olympiad, he did mathematics training camps with the other Irish participants.

“I would say that it is not something that any ordinary student could come in and ace. You need a lot of preparation to even understand the questions.”

Having now studied in three countries, Mr Bachmann says maths in Irish schools is too easy and involves “remembering all the information before exams and then spitting it out”. He said this was not in keeping with the analytical thinking needed to take part in the Olympiad.

The only previous time a student from Ireland gained a silver medal at the Olympiad was Fiachra Knox in 2005. Ireland has participated since 1988.

Laura Cosgrave from Midleton College in Co Cork and Tianyiwa Xie from Alexandra College in Dublin received honourable mentions for their efforts at the competition in Bath.

Three members of the Irish team were born in China, meaning four of the six participants began their mathematical education abroad.

Ireland finished 71st out of 112 countries, an improvement on past performances.

Dr Bernd Kreussler, of Mary Immaculate College in Limerick, who runs the Irish Olympiad, said the predominence of foreign-born students participating for Ireland in recent years reflects on how maths is taught in Irish schools.

“Students who have had their basic education in a school in other countries are much better in dealing with these kind of problems,” he explained.

“In standard school book situation when you are given a problem, you know exactly what to do. You have to be very creative to solve the problems in the Olympiad. This is obviously not something we are well trained at in Ireland.”

Try it at home:

The International Mathematical Olympiad examination took place over two days in Bath, England. There are six questions in total.

Three questions were asked on the first day and three on the second and each problem is worth 7 points. Students were given 4½ hours for both papers.

Problem 1: Let Z be the set of integers. Determine all functions f : Z ! Z such that, for all integers a and b, f(2a) + 2f(b) = f(f(a + b)):

Problem 2: In triangle ABC, point A1 lies on side BC and point B1 lies on side AC. Let P and Q be points on segments AA1 and BB1, respectively, such that PQ is parallel to AB. Let P1 be a point on line PB1, such that B1 lies strictly between P and P1, and \PP1C = \BAC. Similarly, let Q1 be a point on line QA1, such that A1 lies strictly between Q and Q1, and \CQ1Q = \CBA. Prove that points P, Q, P1, and Q1 are concyclic.

Problem 3: A social network has 2019 users, some pairs of whom are friends. Whenever user A is friends with user B, user B is also friends with user A. Events of the following kind may happen repeatedly, one at a time: Three users A, B, and C such that A is friends with both B and C, but B and C are not friends, change their friendship statuses such that B and C are now friends, but A is no longer friends with B, and no longer friends with C. All other friendship statuses are unchanged. Initially, 1,010 users have 1,009 friends each, and 1,009 users have 1,010 friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.

Problem 4: Find all pairs (k; n) of positive integers such that k! = (2n .. 1)(2n .. 2)(2n .. 4) (2n .. 2n..1):

Problem 5: The Bank of Bath issues coins with an H on one side and a T on the other. Harry has n of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly k >0 coins showing H, then he turns over the kth coin from the left; otherwise, all coins show T and he stops. For example, if n = 3 the process starting with the configuration THT would be THT-HHT-HTT-TTT, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration C, let L(C) be the number of operations before Harry stops. For example, L(THT) = 3 and L(TTT) = 0. Determine the average value of L(C) over all 2n possible initial configurations C.

Problem 6: Let I be the incentre of acute triangle ABC with AB 6= AC. The incircle ! of ABC is tangent to sides BC, CA, and AB at D, E, and F, respectively. The line through D perpendicular to EF meets ! again at R. Line AR meets ! again at P. The circumcircles of triangles PCE and PBF meet again at Q. Prove that lines DI and PQ meet on the line through A perpendicular to AI.