I like math and academic contests. I got a lot out of these contests in middle and high school. I like helping kids learn how to do math contests and build problem-solving skills. It's some fun times. If anyone has any questions, let me know. I'm happy to help.
Math League is a nice nationwide contest that goes from grades 4 through High School. At the elementary and middle school levels, it's a one-time 30 minute contest with 30 or 40 questions. It's a good introductory contest for those grades because you do it once, score it and you're done. At the High School level, it's a bit more involved and consists of 5 monthly contests each with 6 problems.
I run the Math League contest for my local elementary school and have introduced several other elementary schools to the contest. Back in High School in Texas, I did Math League and almost won the state my senior year.
The Math League organization keeps track of top scorers from 6th grade through High School and gives out awards for top individuals and schools in the region and the state.
MathCounts is a nationwide competition with "stages" (similar to competitions like Academic Decathlon): Regional, State, National. At any stage, there are three contests: Sprint (individual, 30 questions in 40 minutes); Target (individual, 8 questions given 2 at a time with 6 minutes for each pair); and Team (team, 20 questions, 10 minutes).
At the State level and beyond, there is also a Countdown competition which is an individual 1-on-1 ladder style competition for speed and accuracy. The national MathCounts Competition has been on TV (on ESPN2) the last few years.
The school where I help coach has been one of the top schools in the region for many years and usually goes to State. We'll see how they do this year, I think it's a little more up in the air. MathCounts didn't exist when I was in middle school which is too bad because I would have loved it.
MOEMS (Math Olympiads for Elementary and Middle School) is a world-wide competition oriented towards building problem solving ability. There are five monthly contests, each contest consists of five problems requiring an exact answer.
MOEMS is probably my favorite competition, as it integrates really well with the classroom instruction. One day a month you spend working on the problems and then talking about them with the students and discussing problem solving approaches. MOEMS also gives out tons of awards to the students: certificates, patches, pins, medals, trophies. And students like awards!
There are many other contests like ARML, the AMC contests run by MAA (AMC-8/10/12, AIME, USAMO), USA Math Talent Search, JETS (Junior Engineering Technical Society) TEAMS, and some smaller ones. Most of these are for the more advanced students, but the kids who like this kind of stuff will love these. I usually point them out to the kids in case they're interested.
I did ARML for three years as part of the Texas team and actually placed third in the Individual contest my senior year. I also did JETS teams and won the National Math contest my senior year. I did AMC (called AHSME in my day) and AIME and did okay, though I had a friend who won the USAMO. Just to show that I have some nerd cred in this area :-)
Thoughts and Summary
I have a back-burner plan to organize some sort of Sacramento area-wide math contest one of these years. The best bet would be to get the local universities involved. It's going to take some gumption and persistence, and I'm just about tapped out of those right now.
My dream would be for California to one day have statewide academic competitions similar to the ones that Texas's UIL (University Interscholastic League) runs along with the sports competitions. These contests run the gamut from math and science to accounting to drama and literature. I did some of these in high school and they were great!
The CIF (California Interscholatic Federation) is only about sports. Lame-o-rama. If only I could win the lottery, I'd fund something like this. Anyone have a few million they'd like to donate?
(ETA: More math contest info and a summary)