ETA: I think there are competitions where one submits long written solutions after having some time (a month or so) to contemplate the problems. Anyone know about these, or have suggestions?
I think you're thinking of   
the US math talent search.  No doubt there are other contests w/similar format - Mandelbrot, perhaps?
Ah, yes, I'd better bookmark that. Also I searched and found the other one. 
http://www.mandelbrot.org/Thanks!
Does anyone know of others like this, in particular for younger (elem. school) kids? 
If it comes to choosing which one(s) to spend time on, then any preferences? 
ETA: Oh, I see 
ohmathmom listed some more of this type. Thanks!