Deciding what competitions your mathy kid will participate in can be pretty confusing. There are many math competitions out there, and clearly you have to choose. We started out by considering the Continental Math League: the first division they offer is '2nd-3rd Grade', and you can have your child participate as a homeschooler (it is the honor system: you basically proctor your child). I thought that Joey would breeze through this since he is doing 3rd grade math now at home and got an A+ last year in his JHU-CTY accelerated 3rd grade math course, but the reality is that the Continental Math League problems are strictly word problems, and most often not the straightforward kind. So being a four year old who is four grades ahead in math is not a guarantee of success. He did ok on the competition, but did not get a perfect score, so we are spending this year going through every problem in the practice books (one a day), so he starts recognizing problem types and developing strategies to solve them. Since this isn't completely satisfying, I started looking at other competitions, and asking around. And I made a stunning (ok, it was more like a 'duh' moment) discovery: parents who have gifted/mathy kids do NOT accelerate them for competitions. In other words, the competitions are already geared toward gifted kids (duh) and parents wait until the child has reached the typical age for that grade level to enter him/her in the competition (re-duh), even if their child is several grades ahead in math. Now that explains a lot.
I also discovered another competition that younger children can participate in: Kangaroo Math. Apparently, it is very popular, not as word problem-oriented for younger children, and the first division is 1st grade. The competitions happen once a year and are locally administrated. The only thing is that the kids have to learn to use a Scantron answer sheet, but these are available for purchase ahead of time from a variety of places, so you can show your child how to use them. They also have practice books available for purchase.
As your child gets older, the choices for math competitions expand. Apart from the Continental Math League already mentioned (which provides contests through 12th grade), students in 4th-8th grade can participate in Math Olympiads. While the website says that only homeschool associations (not individual homeschoolers) or institute teams (so I assume if there is a math club in your area, that might work) can participate, the form to register does give individual home schools as an option, so I am not sure what the real deal is (but I am trying to find out). JHU-CTY has a prep course for this contest.
Mathcounts is for 6-8th graders and is open to individual homeschoolers. The overall level of the child has to be 6-8th grade, so for instance, a child who would be younger and accelerated only in math would not be eligible. S/he would have to wait until s/he is caught up to at least the 6th grade level in all subjects. The Art of Problem Solving (AOPS) has both online classes and books available to prepare for these contests. JHU-CTY also has online prep classes.
AMC is for 8th graders and under (AMC8), 10th graders and under (AMC10) who have mastered algebra and high school geometry, and 12th graders and under who have mastered pre-calculus. There is no lower age limit to participate, and the web-site states that children as young as 8 have participated. My understanding is that you can participate as a homeschooler, but the your child has to take the test at a private school at the same time as their team or some other proctoring arrangement has to be made. Solving (AOPS) has both online classes and books available to prepare for these contests. JHU-CTY also has online prep classes.
MAA sells prep books for all kinds of math competitions.
For high-schoolers, there is also the Harvard-MIT Mathematics Contest. Unfortunately, all the competitions take place on Saturdays, which is a hindrance to Jewish students who are observant. They require you to be on a team of some sort (a team of homeschoolers who live in the same area would qualify). There is an online version of the tournament, on the same day as the face to face one.
There are many other math competitions, of course, but these seem to be the most popular and those for which prep classes and books are the most widely available.
