There are quire a few very organized competitions including:
Algebra in Action
http://www.colstate.edu/mathcontest/problem.php?CategoryID=1&LinkID=Current
Middle School Madness
http://www.colstate.edu/mathcontest/problem.php?CategoryID=2&LinkID=Current
As far as I know, homeschoolers are accepted, although I know that is *some* competitions, homeschoolers may need to find a team to join. The only other "homeschool caution" is that sometimes there are strict grade definitions, and sometimes they care about things like the number of community college courses one has taken, etc. so do look at each contests rules. I suspect that one may want to keep the came grade level for all competitions
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