I think we can safely stipulate that there is no universal route to contest success or Luke's other example of 800 math SATs.
But, I can answer your question that, yes, I'm sure that at least some of the students are receiving supplemental tutoring. Not necessarily contest-focused, but still.
Anecdotally: the two friends I had from undergrad who were IMO medalists (about 10 years ago) had both gone through early math tutoring programs (and both had a parent who was a math professor). All of my undergrad friends who had 800 math SAT had either received tutoring or had their parents buy them study materials (most of them did not look back fondly on the experience).
Remember, for any of these tests, there's a point where even a small amount of training to the test overwhelms a good deal of talent. Familiarity with problem types, patterns, etc can vastly improve performance.
I have no way to evaluate the scope of your restrictions on doing "super-well" or the particular that the tutoring start at an "early age" (although at least one of the anecdotal IMO cases did a Kumon-type program that started at pre-school).
Are there some people who don't follow that route? Certainly. However, I do think that it's important to be aware of other factors that may be present.
Series: How to Purchase AI Risk Reduction
A key part of SI's strategy for AI risk reduction is to build toward hosting a Friendly AI development team at the Singularity Institute.
I don't take it to be obvious that an SI-hosted FAI team is the correct path toward the endgame of humanity "winning." That is a matter for much strategic research and debate.
Either way, I think that building toward an FAI team is good for AI risk reduction, even if we decide (later) that an SI-hosted FAI team is not the best thing to do. Why is this so?
Building toward an SI-hosted FAI team means:
Both (1) and (2) are useful for AI risk reduction even if an SI-hosted FAI team turns out not to be the best strategy.
This is because: Achieving part (1) would make SI more effective at whatever it is doing to reduce AI risk, and achieving part (2) would bring great human resources to the cause of AI risk reduction, which will be useful to a wide range of purposes (FAI team or otherwise).
So, how do we accomplish both these things?
Growing SI into a better organization
Like many (most?) non-profits with less than $1m/yr in funding, SI has had difficulty attracting the top-level executive talent often required to build a highly efficient and effective organization. Luckily, we have made rapid progress on this front in the past 9 months. For example we now have (1) a comprehensive donor database, (2) a strategic plan, (3) a team of remote contractors used to more efficiently complete large and varied projects requiring many different skillsets, (4) an increasingly "best practices" implementation of central management, (5) an office we actually use to work together on projects, and many other improvements.
What else can SI do to become a tighter, larger, and more effective organization?
They key point, of course, is that all these things cost money. They may be "boring," but they are incredibly important.
Attracting and creating superhero mathematicians
The kind of people we'd need for an FAI team are:
There are other criteria, too, but those are some of the biggest.
We can attract some of the people meeting these criteria by using the methods described in Reaching young math/compsci talent. The trouble is that the number of people on Earth who qualify may be very close to 0 (especially given the "committed to AI risk reduction" criterion).
Thus, we'll need to create some superhero mathematicians.
Math ability seems to be even more "fixed" than the other criteria, so a (very rough) strategy for creating superhero mathematicians might look like this:
All these steps, too, cost money.