We are in a very similar situation with our 7yo son, in some respects. He's specifically much much better in mathematics than his other subjects, and he clearly just naturally thinks like a mathematician. He's in 1st grade just finishing 5th grade math, and we anticipate him finishing AP Calculus BC (covering 1-1.5 years of college calculus) in 6th grade at age 11-12.
Since we are in a (free public) virtual school, he can go at any pace. Basically we are homeschooling using the "canned" courses from k12.com, which are designed for average students but which are designed to be compressible, to cover material quickly for the strong student, so he can go through all the material without gaps.
We have 2 Maths PhDs, but 1 income (by choice) and 3 kids (2 not yet in school), so we are very much focused on bringing up, and especially educating, our children ourselves directly, but have to minimize costs. The schools are mostly really awful here, in various ways, so we have few options.
Yes, of course one has to keep going in maths. We don't have a detailed plan, just some rough ideas. We'll supplement with AoPS courses, for all topics not on the standard path to calculus, and the competition preparation courses, whenever he is ready. (The few hundred dollars per course is an okay expense. Question: are there any other "schools" of AoPS's ilk that are worth looking into?) There is a school in the state (but far away) that has more advanced, or different, courses such as multivariable calculus, differential equations, discrete mathematics, etc., and these courses are available for free (within the state), so that will cover a couple of years maybe. After that, there is the possibility of courses, or maybe reading courses, and the local university. It is a fairly average state university, but a lot of the faculty have PhDs from Ivy League/Oxbridge type places, so there is plenty that a smart schoolkid can learn from them. And we can plain old fashioned homeschool using books and our own mathematical knowledge. Maybe he could do some research. There are logistics to work out with all this. It's just a vague plan. One concern is I was wondering if taking university courses (while officially being a seconday school student) could disqualify you from competing in certain maths competitions. We are not going to worry too much about credit, as long as university entrance (and high school graduation) conditions are satisfied, and as long as universities at least somehow take into account all the "extracurricular" maths.
As for maths competitions, whatever their drawbacks, I think one has to compete. They are a way to see how one measures up against others in your region or country (or the world). There's competition to get into universities and to get jobs, so competition can't be avoided. And a string of very good competition results may be regarded as more signicant than rapid progress and high marks in easy schoolwork. These competitions (or sequences of competitions) really can identify people as being not just top 1% or top 0.1%, but even top 0.01% or rarer, and that kind of identification can help.
When I was a kid, I never heard of people preparing for maths competitions. I just thought they were fun, and did well. But I see these days there are competition preparation courses and math clubs/circles. While "teaching to the test" would be a sad thing, I gather these preparation activities are just a good way to learn some mathematics that's not in school, and to interact with similar kids, so that's definitely something we'll look into.
So after finishing calculus in 6th grade, our son won't be twiddling his thumbs for the following 6 years waiting for the next maths course to show up at uni. There's plenty of maths he can do in the meantime, even if it takes some scrambling and improvising. If he has to repeat some material, hopefully it's at an elite (Ivy League/Oxbridge type) university, where it's presented at a much higher level. (I was looking at the Princeton University website once where it said words to the effect, my paraphrasing, "yeah, sure, you mighta taken calculus before, but you haven't taken our calculus", and they do have a point.) We don't yet know if he'll be good enough for those places, but maybe competitions over the next few years will give us a rough idea where he stands.
As to the issue of whether to start university early, here's why not for us. Our son is fairly good at all his subjects, but he is absolutely not the kind of kid (in contrast to some on this forum) who could accelerate multiple years in all subjects. (Actually he's 1 year accelerated across the board, so he could conceivably start uni at 17 instead of 18.) Instead he's specifically very good at maths and less good at the non-mathy subjects, so he'll probably continue those at the regular pace. And there's the usual considerations such as maturity, social eptness etcetera. But another consideration for maths is that when it comes to competing for entrance at an elite institution, it's very hard for a 15 or 16 year old to compete with a, say, top 0.01% 18 year old, which is what I'm guessing it takes to get into these places, though I could be totally wrong about that.
By the way, does anyone know what it takes to get into maths at an elite institution? Is it based purely on merit? Or do you, as some have suggested on this forum, have to fluff your CV with extracurricular activities like volunteering at the homeless cat shelter and playing polo?
One final thought. There are a lot of jobs where mathematical ability is important, but very few jobs as a research mathematician. So you have to have your child prepared for this uncertainty.
I'm sure I've forgotten to say several things, but I need to sleep now.
