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?
Here are some for younger kids:
Problem of the Week or Month—Submit Your Answer Competitions
Ole Miss Math Challenge Problem of the Week
http://mathcontest.olemiss.edu/index.phpChallenges include Problem of the Week, Algebra in Action, Middle School Madness, and Elementary Brain Teaser. The contest is free, but students must register to submit their answers because they need to create an account to compete. Each Monday new problems are posted; solutions are due Friday by midnight eastern time. Once the submission deadline has passed, you can still submit answers to any past problem and receive a response to your submission, but your name will not be added to the list if your answer is correct. The Challenge Mode provides a list of the most challenging problems for each contest.
The ABACUS International Math Challenge Problem of the Month
http://www.gcschool.org/program/abacus/index.aspxEight problems for each of three age groups are posted monthly from September through April. Students should submit their solutions and their reasoning for their solution via e-mail.
