Data10’s experience was similar to the experience that a colleague’s son had, where a teacher recognized that he was “operating on a different level” and arranged for him to be greatly accelerated. He wound up taking a graduate math class as his first math class in college. When this sort of thing happens, the student inspires the teacher to take action. But the teacher has to be oriented in this way to begin with. The local school was on record in the newsletter to the parents as opposing acceleration in math, because then the student would need to take college courses in high school, and would “miss out on the social experiences” in both high school and college. I could not make up something like this! In my high school, the teachers actually liked smart students. This is not universal.
In this geographical area, a student who takes university courses in math will be taught almost exclusively by non-English speakers (going very deep–up to senior-level college courses), which adds a complication to letters of recommendation. Also, since rather advanced mathematics is taught in their countries of origin, a high school student in their classes (up through sophomore or junior level in college) will be studying what they studied in high school, and therefore not looking very unusual to them (just not held back, like most American students).
I knew a graduate student from Taiwan who needed to take an electricity and magnetism course at the university. To help her out, I showed her the texts for various E&M courses. When I showed her the senior-level E&M text, she started to laugh, and said, “Oh, I think this is easy. I learned this in high school.” She took the graduate-level course and did fine (A/B), with only one minor blip that was directly due to a language problem.
In terms of chances overall, I have posted on other threads that I have seen an element of randomness in admissions. This is not to say that they are totally random, just that an admissions outsider (me) can have pretty good knowledge of the whole picture for a few top students, and yet find the outcomes impossible to predict (which probably makes chancing pointless for me to do).
When a student I know pretty well is admitted to some “top” schools but not others, or not admitted to any as sometimes happens depending on where one draws the cut line for “top,” I always wonder whether the relative (competitive) weaknesses that the admissions staffers saw in the applications were 1) real, 2) due to omissions from the application that the student might have filled in, but failed to fill in, 3) due to inaccurate impressions that the recommenders or GCs created inadvertently, or 4) based on misinterpretation of statements in the application that might have been interpreted correctly by another reader. (“Not enough beds,” “can’t admit 15,000 more students,” etc., I get that.)