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[quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous][quote=Anonymous]The person I know who attended Cambridge for maths was a USAJMO qualifier in HS, and had a several top 10 and recognition awards in less-popular but still prestigious national-level pure and applied, individual and and team, exam and project, math contest 
 award, and a research internship in HS.

That's about top 100 to 200 in USA in-grade-level performing HS math student. [/quote]

It may be true that there are a good number of high performing math olympiad kids studying math at prestigious university math programs. However, competition math and the math you study in college are quite different. Competition math is not everyone's cup of tea and you don't need to have invested in becoming a math competition champ to be good at the kind of math one does in college or graduate school or as a career mathematician. [/quote]

"Competition math" (aka  advanced, more abstract, deeper math) is the closest thing to college/grad/mathematician work than anything else done in high school. [/quote]
No, much of competition math is fun tricks and skills particular to the competition format. Taking actual math courses is the closest to college math work.[/quote]

OK, so I see the problem is that you don't understand the difference between "competition math” and ”math competitions”.  The " fun tricks and skills" are how the winners get crazy fast at the contests to get perfect scores. Yes, that is not helpful for higher level math. But that's the A1 sauce on the steak.

The meat of it is learning a ton of math, applying it in novel situation without being told which formula to use, learning how to prove results so you can answer questions like " how many solutions does this constraint have?",  constantly sitting for tests that each cover 4 years of math in random, order,  not just last week's lesson, and being happy solving problems that each take 5 to 100 minutes to solve.

People who wave off "competition math" are people who failed to even start learning it, looking for a excuse. Or people are legitimately in their own little world doing deep math study, not just Multivariable calculus.  Is there what you are talking about? 
Someone like Jacob Lurie, who won Westinghouse with a pure math paper in high school. Who, by the way also got a perfect score in the IMO? 
[/quote] "competition math" is what's tested in "math competitions". They're pretty much interchangeable. Speed is mostly irrelevant (with some middle school competitions being exceptions) - for almost all highschool students, double time would not significantly increase their competition scores. But you're dead on regarding everything else.

A talented student who spends their time on competition math is much better prepared for STEP than an equally talented student who spends their time on freshman/sophomore math (calc 3, diff eq, linear algebra)[/quote]

Speed is what separates the mid-high performers  (top 1000 per grade) from the top performers (top 100 per grade) on pre-Olympiad AMC, AIME, ARML, etc. (Most of the fast students can probably also do harder problems, but that's not on the tests.)

Of course the average students can't do any of it beyond the warmup problems.
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