There are several different mathematical problem solving contests for Irish students.
PRISM runs two contests that are open to all Irish post-primary students - one for junior cycle students and one for senior cycle students. These are multiple choice format problems designed to test your basic mathematical problem solving ability. You can find sample problems here.
This contest takes place each year around the first week of May. In any given year about 200 students take part in this contest, many of whom have attended one of the extra currciular mathematical enrichment programmes that are offered in NUIM, UCD, UCC, UL and NUIG. The problems of the Irish Mathematical Olympiad are typically a bit more difficult than the PRISM contest problems. They involve essay type solutions where careful and clever reasoning are the key. You can find some past Irish Olympiad papers here.
This is the highest level, as far as preuniversity maths contests go. The IMO takes place each year in mid July, when the best student problem solvers from around the world gather in the host country (which changes from year to year) to tackle 6 very challenging mathematical problems. Each country sends up to six students along with a two or three accompanying persons. The Irish IMO team is selected on the basis of the IrMO results.