In the UK there are three organisations that run maths competitions:
- the main one, UKMT, which organises both individual and team events, and runs the UK's IMO participation. There are three levels of individual competition - Junior, Intermediate, Senior - and at each level the first round is computer-marked multiple choice, and a follow-on is human-marked writing of proofs. At Senior level the proof-writing follow-on is BMO1; for those who do well at that, there's a line of further proof-writing follow-ons leading up to IMO team selection. Besides that track, the Senior level also has another follow-on, the Senior Kangaroo - computer marked but answers are 3 digit numbers - for the next 3000 or so after the 1000 who go to BMO1 (Bit that should be in the UBT: DS didn't quite make it to BMO1 but did get to the SK and had fun; we await results). All the events they run are timed. Levels have upper but not lower age (well, school year) limits, so young ones can enter all three levels if they like.
http://www.ukmt.org.uk/- the Mathematical Association, which organises a Primary Maths Challenge in two rounds (timed multiple choice)
http://www.m-a.org.uk/- the Scottish Mathematical Council, which organises take-home write-proofs style competitions in four levels (Primary, Junior, Middle, Senior) - each child can only enter one.
http://www.scottishmathematicalcouncil.org.uk/In this context, our answer to "how do you choose which ones to enter" has been "just enter them all" - but that obviously only works because there aren't ridiculously many and entering them is logistically easy for us. It does get a bit "from the sublime to the ridiculous" sitting the UKMT Senior a few days before the Primary Maths Challenge, but he enjoys it and I reckon it's good exam technique practice to do a few of these timed things at various levels of difficulty each year.
