The 20th Irish Mathematical Olympiad was held in five universities around the country yesterday, and approximately 500 secondary school pupils, each handpicked for their skill in maths, took part. This is no run-of-the-mill competition. Each student, aged between 15 and 18, has been specially trained by university lecturers for the past year and for many, this is their second and third time trying out for the national team.

"We have been holding this competition for the last 20 years and it has been very successful, " said one of the organisers, Dr Richard Watson of NUI, Maynooth. "Every year, we select the top 500 maths performers in the Junior Cert and approach their school principals asking if the student would like to take part.

"We then organise maths courses in the five universities, which are usually for three hours on a Saturday morning for 13 weeks between November and May. A lot of the topics covered by the Olympiad are not on the Leaving Cert, so it's important that they are properly prepared."

Students who did well in the Problem-Solving for Irish Second-Level Mathematicians (PRISM) event last year were also invited to try for the Olympiad. For some students, the commitment is too much and they drop out. For others, however, it becomes an important part of their lives. Watson recalled one student who last year travelled from Donegal to Maynooth every weekend for the entire course. He eventually made it into the national team.

"These are no ordinary pupils, but are in fact extraordinarily gifted, " said Watson. "I'm not being modest when I say that I would not be able to work out the problems in [yesterday's] exam papers. It's one thing setting the questions; it's quite another tackling them. And those that do well in the Olympiad will often go on to study mathematics in universities like Oxford and Cambridge. This is very important to them."

The Irish Mathematical Olympiad was first held in Ireland in 1987 by Professors Tom Laffey of University College Dublin (UCD) and Finbarr Holland of University College Cork (UCC). The five centres are based in these colleges along with NUI, Maynooth; NUI, Galway and University of Limerick (UL).

The Olympiad yesterday was divided into two exams, each lasting three hours and comprising five questions. Competition to get into the top six is intense: this year the International Maths Olympiad is taking place in Vietnam from 19-31 July and is funded for by the Department of Education and the host country.

"It is very rare for a student taking part in the Irish Olympiad for the first time to make it into the national team, " said Watson. "Mostly it is students trying out for a second or third time that make it. For some of them, this means taking part during their Leaving Cert year, which can put them under a lot of pressure. But there is a big incentive."

So far, Ireland has yet to do particularly well in the international event, with countries such as China, Russia and Iran usually taking the honours. "They harvest gifted children from a very young age, which we wouldn't really do, " said Watson. However, four members of the team last year were awarded honourable mentions.

This year's winners will be announced by this Friday, as work has to start straight away for the International Olympiad. "It is a real pleasure for lecturers like myself to teach such bright children, " said Watson. "We're used to repeating ourselves over and over to maths undergraduates. With these kids, you never have to repeat yourself. They get it straight away."

Test yourself (SAMPLE QUESTIONS FROM 2006 PAPERS)

1. P and Q are points on the equal sides AB and AC respectively of an isosceles triangle ABC such that AP = CQ. Moreover, neither P nor Q is a vertex of ABC. Prove that the circumcircle of the triangle APQ passes through the circumcentre of the triangle ABC.

2. Prove that a square side of side 2.1 units can be completely covered by seven squares of side 1 unit.

3.ABC is a triangle with points D, E on BC, with D nearer B; F, G on AC, with F nearer C; H, K on AB, with H nearer A. Suppose that AH = AG = 1, BK = BD = 2, CE = CF = 4, B = 600 and that D, E, F, G, H and K all lie on a circle. Find the radius of the incircle of the triangle ABC.

4. Two positive integers n and k are given, with n =2. In the plane there are n circcles such that any two of them intersect at two points and all these intersection points are distinct. Each intersection point is coloured with one of n given colours in such a way that all n colours are used.

Moreover, on eachc circle there are precisely k different colours at present. Find all possible values for n and k for which such a colouring is possible.