That’s part of my question- are there feeder schools/programs for the competitions? And if so, is this a nature thing or nurture thing that can be taught? How frequent is it that a kid from a normal public school participates at a high level?
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Elite math kids usually begin training in elementary school. There are math circles in most large US cities, often rin by universities, that nurture this talent, and annual testing with more frequent regional competitions. Those performing at the top often choose to attend a smaller set of high schools, including Exeter. Yes, public schools are represented as well.
It’s a nurture thing. Some of the schools in east coast and CA have this math programs inside or outside the school and the kids do this math training every day after school, practicing day in, day out.
For math, the greatest concentration of such students in the US come from two high schools: Exeter and Princeton International School of Mathematics and Science (a HS few heard about).
Both nature and nurture play a role. Exeter has the former US national team coach and the latter is affiliated with a Chinese HS that produces the most IMO medalists in recent years (and staffed with some experienced teachers from that Chinese HS). Because of their reputations, these two schools also attract talented students who’re more inclined to participate in such competitions.
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Since you asked about demographics, in our area, the participants are overwhelmingly male, and Asian American. One of my kids was literally the only Caucasian girl in a room of 500 at an event.
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Agree with @roycroftmom. Exeter actively recruits for math talent, and its students can represent 4-5% of all winners. Perhaps more surprising is that Lexington High School, an open-admission public high school, achieves similar results. There is a reason for this as well, in that about 25 years ago an MIT graduate created an after school enrichment program called The Math Club for kids as young as 8 or so, and really nurtured their math talent. Then they found an exceptional teacher in middle school who has coached multiple national MathCounts champions.
This nurturing matters. The kids who excel at this have been training for this in the same way other kids train for sports. As with sports, most kids enjoy the activity, some enjoy the competitive nature of it, and for some it becomes toxic.
But some may ask, given the extreme concentration, are these really the strongest math kids in the nation? That’s a bit like asking if the winner of some Olympic sport such as fencing is really the world’s best given that most people never try out fencing. And the answer in both cases is that yes, these are the people who have shown the best skill in this activity. There could be others out there with potential to be better, but we don’t know who they are.
The first AMC exam is the AMC-8, taken by 8th graders and below. In a typical year, perhaps only 150k students take it nationwide. Now the reality is that this is still a hard exam, really suitable only for the top 10% of students or so. But with only 150k students taking it, that leaves many potentially strong math students that were never identified. And by the time it gets to high school, it is too late to catch up.
So back to my earlier point. Admission to elite colleges is is not nearly as random as it seems, as students who were fortunate to attend a place like Lexington High School (skews rich) have a completely different set of opportunities to shine than someone who attends Boonies High School. The competition to shine remains intense as qualifying for USAMO is still very hard. But the rules of the elite admission game are skewed in ways that most applicants don’t understand.
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Nurture is definitely part of the program. My son was the Rensselaer Medalist at his HS (best math and science student) and he never competed. Never heard of them. He’d been tracked ahead since 3rd grade. A lot of competitors come out of a few schools, and for whatever reason, many of them are Chinese. @hebegebe would know more. That’s my conclusion from a very brief look a few months ago when I first learned about USAMO.
EDIT: beat to the punch. 
China has dominated IMO (and IPhO) competitions for a while. The Chinese national CMO competions are arguably significantly more difficult than USAMO because of greater competition there to be the “final 6” (some even argue that they’re more difficult than IMO competitions). Some of these Chinese students have immigrated to the US, or are attending HS in US, which make them eligible for USAMO.
Thank you for answering my questions everyone. Very interesting. I assume there are college-level competitions that the students pour into, and serious bragging rights are involved, then. Very much like with recruited sports.
The AOs must two-track math students- the stealth-recruited mathletes and everyone else. The schools probably have a number of mathlete slots to fill. The odds of getting into those schools as a math student go way down if you don’t compete, but it probably doesn’t impact anyone else in the applicant pool much.
Cate- I don’t know if I agree that the odds of getting in go down if you aren’t a math competitor. My kid was a math major at MIT- he was not a math competition kid in HS. MIT doesn’t care what you put down as a prospective major, so it’s not like they are comparing the physics kids to the physics kids and the chemistry kids to the chemistry kids. It’s all the same applicant pool.
My kid had other interests besides math. I don’t think he would have been a “world class” math competitor in HS but fortunately, there are other ways to explore your interests besides competitions!
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Yep - I am pretty much convinced that for an Asian American male to be admitted to HYPSM as a STEM major, they need to be high ranking in these math contests and/or ISEF type competitions, or have a serious hook. These buckets are calibrated to a very fine detail. My high stats ORM son is neither of these and we will not be wasting any $$ applying to these universities .
The question is not really about how good they do in math or physics competition. The question is really are they really talented in math and physics and go on to be exceptional mathematicians or physicists or they just use this to get in “elite” schools.
You can do all the cramming training to be good in competition, but in career…
There’s a physics competition too?! (Smacks head)
[quote]
CateCAParent,
Thank you for answering my questions everyone. Very interesting. I assume there are college-level competitions that the students pour into, and serious bragging rights are involved, then. Very much like with recruited sports. [/quote]
Yes, its called “The William Lowell Putnam Mathematical Competition”.
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LOL. And Biology: https://www.usabo-trc.org/
F=ma and then USAPhO
Yes, there are competitions in physics, biology, chemistry, computer science. The depth of talent is extraordinary. Many of the math competitors we knew had studied college math before they even began high school. Lots of talented kids out there.
Also explains why most people don’t really understand the degree of competition to get one of the academic spots at an Ivy. It has very little to do with winning your high school math award
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Or being valedictorian of your high school.
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And ACT/SAT math is nothing. 
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Some are talented, but most are not, just more practicing.
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