To fulfil my tigercopter parenting need to live vicariously through my children, I need a steady supply of mathematics competitions. ETA just kidding

I found this article which I agree with
http://www.artofproblemsolving.com/Resources/articles.php?page=pc_competitionsSince we are homeschooling, these need to be competitions one can enter as an
individual rather than being part of a school, team or club, so that seems to eliminate certain competitions.
So far I've found some. (USA-centric list here. Feel free to list ones from other countries.)
Math Kangaroo
http://www.mathkangaroo.org/mk/default.htmlThis is for grades 1-12 in the USA (and many other countries), though it seems more popular at elementary school level. The harder questions look appropriately challenging for the grade level, so it looks like a "real" competition.
Mathnasium Trimathlon
http://mathnasiumtrimathlon.com/This is just for grades 2-5 in the USA.
The questions look very easy for the best students, so I don't know if this is one the really mathy kids go for. (Mathnasium is a tutoring center, catering more to the struggling student, though the competition is open to all.)
American Mathematics Competitions
http://www.maa.org/math-competitionsI understand this is the main competition series, right? (Our oldest, DS7, won't start these for another year or two, and us parents are "aliens" so we don't know much about the AMC competitions yet.) Apparently you have to take it in a B&M school, but we found this link to search for locations.
http://amc-reg.maa.org/amc_external/SchoolSearchByZipCode.aspxAnd what about MATHCOUNTS
http://mathcounts.org/Again, as "foreigners" us parents have no experience with this but would it be correct to say that this is the other main mathematics competition (for grades 6-8).
I have a question.
It looks like MATHCOUNTS is very much aimed at schools/teams/clubs. How does participation work for unaffiliated individuals? What other mathematics competition are there for individuals that we should know about? Actually I'm (half) joking about the tigercopter parent need living vicariously (okay there's some truth to it), but we actually
do want to be
selective about what competitions to enter. We don't want something too boring/routine or with numerous regular "competition meetings". Ideally they should be competitions held once a year where kids in the top 0.1% get to see
where they are in the top 0.1%.
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?