Here are a few (okay, MORE than “a few”) contests and resources to look into:
For elementary kiddos:
1. MOEMS (Math Olympiads for Elementary and Middle School) are great (http://www.moems.org/ ). The contests run Nov-March. Their books are fantastic.
2. Continental Math League are also great (http://www.cmleague.com/product-category/cml/ ). A few schools I know do them. They also run Nov-March.
3. Noetic Learning Math Contest (http://www.noetic-learning.com/mathcontest/ ). Again, a few schools in Cherry Creek do these. The contests are one in November, one in April.
4. Colorado Math League (http://www.mathleague.com/). We do this one. It’s all about Number Sense. Fantastic contest! And books are also great. Contest is in April.
5. Math Kangaroo is another great one, in March. Kids can take it at very “centers” around Colorado. http://mathkangaroo.org/MathKangaroo/Results.xhtml
6. The Rich Morrow Math Challenge (http://www.themathchallenge.org/ ). This was started by my mentor. Only for Cherry Creek School District Students. Contest is the first Saturday in December.
7. Beast Academy. Great curriculum for advanced math (and contest math). (https://www.beastacademy.com/ )
For Middle School
8. Many of the above contests and resources also have middle school editions.
9. Art of Problem Solving (or AoPS).( https://www.artofproblemsolving.com/ ) This is THE first place to go. It has a wealth of resources, and sometimes it takes time to find them all. The MathCounts Trainer is great. So is “For the Win.” All the old AMC contests (and many others) are available, with solutions (both official and ones uploaded by users/students).
10. The AMC8 is in November. It’s the beginning of the AMC ladder of contests, which ends with the IMO (International Math Olympiad) where each country sends its best six students. All the old AMC8 contests are available (with solutions) here: (https://www.artofproblemsolving.com/wiki/index.php?title=AMC_8_Problems_and_Solutions )
11. MathCounts is really the top middle school math contest program. Each school brings 10 students to their “chapter” competition (in February), where individuals and teams compete. It’s SO great! Old school handbooks are all over the internet–the most recent is here: https://www.mathcounts.org/resources/school-handbook and others are very “Google-able”
12. Colorado Math League. Same as #4 above, but the middle school contests are in February. Again, their books are fantastic. http://www.mathleague.com/
13. The Purple Comet is a great team contest in April. (http://purplecomet.org/ )
14. The Cherry Creek Math League starts in September, and all Cherry Creek Middle Schools participate. Each middle school has contests and practices every Monday, and each school should have more details.
15. The Colorado Math Circle in Boulder (http://www.coloradomath.org/ ) has monthly meetings which are fantastic.
16. The Denver Math Club (started by my former students) also has monthly meetings. Get on the mailing lists for these last two clubs to get their updates http://denvermathclub.weebly.com/
17. My website (https://coloradomathcamp.com/ ) has more links to books/videos/websites that I like.
18. The AoPS book curriculum!! If I have one recommendation for middle school advanced math students, it is to BUY some AoPS books and work through them. They are all great. Full of great problems. Art of Problem Solving, Volume 1 is a good place to start: https://www.artofproblemsolving.com/store/item/aops-vol1
In addition to these contests, there are a variety of other camps and “math days” and “talent searches” you may want to check out in the future.
19. The “Center for Bright Kids” (https://www.centerforbrightkids.org/ ) has tons of resources, so I encourage you to check them out.
20. The Westerly Creek Math Tournament in Denver (in May). I don’t know much about it, but I’ve heard great things about it.
21. I know a bunch of families who have loved the Epsilon Camp (http://epsiloncamp.org/ ), which is coming up at the of July. Also, IdeaMath and MathZoom and A-Star Math and MathPath have great camps.
22. The Julia Robinson Math Festival at CU Denver. Not sure if it’s an annual thing, but last year’s (in September) was a big success.
23. The CSU Math Day (http://www.primes.colostate.edu/mathday/index.shtml ) seems to be mainly for High School students, but again last year’s (in November) was a success.
24. The UNC Math Contest (https://uncmathcontest.wordpress.com/ )is again for high schoolers, though middle schoolers can participate, and some have done very well. The first round is in November, and the final round is in January.
25. Mu Alpha Theta is a website with WAY too much info about contests….but perhaps you can get more ideas there http://www.mualphatheta.org/
There’s SO much more out there…… but this should give you a good start!