Zeev, after reviewing the performance of practically all of the countries in the IMO, I don't think Finland was ever as big on competitive math as we are. Your point is well taken and I certainly would not want to lose that experience. I haven't looked deeply into their compulsory phase of education but I have yet to see any reference to advanced math starting there. This is troubling since that is generally when it does start for the brightest.
I have studied AMC and Math League results before, usually by starting with the list of high scorers and tracing them back to their respective schools. What I usually find is that the school has a teacher interested enough in math to establish a math club or the school already has a long tradition of a math club. I guess where I am going with this is a discussion of how to best support competitive mathematics. I can throw a couple things out there but we should probably go and look at what the winners do.
Going on my own experience, to support the top students in these subjects (not just math) schools should devote one period (an elective) to the task, starting as early as 4th grade, but 7th grade would be a really good start. Some schools do have this but very few and it isn't consistent year to year. Most gifted students, after they have experienced gifted classes in elementary school avoid the experience in secondary school and opt instead for honors classes. Gifted pedagogy seems as broken as all the rest of pedagogy. Targeted gifted classes (math, music, computer science, etc.) would be much more effective in my opinion and draw those students back to their passions.