Can we all agree to never use that word? Actual prodigies are measure zero in the general population. Terry Tao is a prodigy. The rest of us are just normal schmucks.
Prodigy suggests that talent comes naturally to and is dismissive of the massive amount of work that goes into preparing for something like the IMO.
As someone who attended the IMO, I can tell you it varies widely. For some people, they screw around a lot, attend math camp but spend a lot of time playing games, dick around doing normal teenager things, get in trouble for stupid teenager things, don't study too hard, and nevertheless do well enough to attend the IMO. And then some people are responsible, study a lot, work hard, keep their heads down, and also attend the IMO.
Talent does come naturally to them. Instead of arguing about labels, it's better to think about talent quantitatively. If the USA's IMO contestants were generally slackers, they'd still be comfortably in the top 100 in their birth year (in the entire country) in the AMC/AIME combined score, just by showing up when the teacher told them to. (Well, back in the early 2000's. Maybe competition is stiffer nowadays.) And the gulf in talent between them and the worst kids in that top 100 list is large.
I didn't end up doing IMO due to weird school situations, but I'd won some prestigious math competitions (two other winners I befriended one year went on to become IMO gold medalists) and was able to solve some IMO problems in middle school on my own. At the time I felt I was just different from others in terms of mathematical ability.
My general feeling now is that talent as perceived by others is mostly a function of hard work. Looking back, the level of persistence I was able to apply to mathematical problems at such an early age without any real supervision was incredible. I was almost entirely self-taught from age 7-10 and without being on a strict curriculum, this included some college level math. I think this was a function of a lot of things that don't have much to do with natural ability.
At least a few IMO gold medalists have younger brothers who are good enough to be involved in math activities in high school but are not remotely close to the same level. They are in fact different in terms of mathematical ability. You would be more aware of the value of natural ability if you had put work in, but then couldn't come close to winning prestigious math competitions.
My own brother was exactly that way (good enough to be one of the best in school and hold his own at an elite university, but not close to good enough for any kind of actual competition) but my understanding is that he was only ever trying, never doing.
There's a huge difference between learning something because some external pressure dictates that you learn it and truly immersing yourself in something and just letting go. Math isn't that exciting for most people, so the vast majority of people never cross over to the latter. So the extremely small percentage of people that are able to cross the chasm and deal with math natively can stand out.
I agree with what you're saying in general. I don't know whether you really quantitatively disagree with me. I think without any deep immersion like you describe, the IMO team members would still have done well enough on the AMC/AIME to qualify for the USAMO. Do you disagree with that? (Again, my difficulty scale is based off the early 2000's.)
I mean, my basis for that estimate is that I know a bunch of people without such immersion who got that far when I was a kid, who were on the regular school education track, didn't self-educate outside that, and didn't have much external pressure either.
Edit: But I don't mean to imply these people absolutely never thought about math or numbers outside of the classroom, because it's interesting, and they most certainly did, just no traversal through a curriculum, or deep immersion.
Things exist on a spectrum and I don't think we're really on the same page on what it means to be immersed - it's about how your subconscious mind decides to prioritize things, not about any kind of curriculum. I spent a lot of time in my life cleaning and organizing stuff, but I'm not good at it and not getting better at it either. I'm pretty sure this isn't because some lack of natural ability prevents me from being good - for whatever reason, this is just something that I don't find interesting. This is how most people approach math - as a chore to be done and forgotten. Even regular good-at-math people (which most people at elite schools would qualify) that aren't remotely close to competing at the likes of IMO and Putnam are qualitatively different in this regard compared to average students because they are broadly speaking more immersed in their studies. For most people, it's not merely that they don't think about math outside of the classroom - their mind is also too busy thinking about anything but math while doing math.
The flaw with this model is that it doesn't really explain the world, because if you compared most kids who do well on math contests to the average high school honors student, you'd have to argue that their performance in almost every subject besides Band and PhysEd is the result of mental prioritization. Also, observing that an underperformer just didn't mentally prioritize it the same is basically mind-reading; it isn't a refutable observation. We can see the weakness of the argument by using it to explain the effect of age on performance, without finding direct evidence to refute it. But the actual cause is that kids get smarter as they get older. It also fails to explain why kids rank so differently at contests like 24, versus later contests that test a mostly different sort of ability. That kids who do well on math contests like thinking about math can't be taken as a factor in more than a banal sense.
And imagine trying to make the same argument for basketball. Kids with low field goal percentages just don't want it enough?
