I took part in math competitions as part of a math club in High School during the 70s. It was fun, made me think and was a bit of a social outlet. I don't think any one test was that important - we took a bunch of standard and experimental national tests and monthly meetups had their own problems.

All that said, my impression is current math clubs and competitions have been taken over by kids just wanting credit for college rather than the kids of my era, who were there 'cause they actually enjoyed math.

If the only purpose you can see for math competitions is find a few very talented kids, yeah, well then they would qualify as counter-productive.

> my impression is current math clubs and competitions have been taken over by kids just wanting credit for college

I work directly with students in this field and my experience is that most are involved for the interest, challenge, or social aspect.

If you are involved with “pay to play” programs like AoPS, you probably see more resume-grooming.

Nobody I know would continue to donate their time if they perceived it was primarily fluff.

Good Hear!

Outside the few hundred top mathematicians, how many advance the field much?

Depends what you mean by advance. If nobody but those few hundred mathematicians was even aware of mathematics then what would be the point?

Every now and again, a tardy student writes down an impossible to solve problem and solves it because he thought it was just homework.

> Every now and again

As far as I can tell, this has only happened once, in 1939 [1].

[1]: https://www.snopes.com/fact-check/the-unsolvable-math-proble...