1. It looks like all the competitions (AMC, Math League, Math Olympiad, etc.) require a team. What happen to homeschooled kids or public schools with no math team? First of all, many competitions don’t require a team. As a matter of fact, one of the competitions you mentioned, AMC, does not require a team. For the competitions that require a team, my ds will usually team up with her friends. 2. Do the whole team in each school advance to district to regional/state to national altogether, or will they pick the best individuals from each team to form new team to compete at higher rounds? In other words, do kids need to go to the schools with better teams? This depends on the competition. Generally, to be safe, it is best for your teams to belong to “good” team. 3. Are there any US math contests for individuals? Plenty--AMC 8/10/12/AIME/USA(J)MO/BAMO(Bay Area only contest)/USAMTS to name a few. Here is a more comprehensive list: https://artofproblemsolving.com/wiki/index.php/List_of_mathematics_competitions 4. Which competitions are better for 2e kids? I heard Math Count is easier but more intense, higher time pressure and hence is more disadvantageous to 2e. I’m not sure which ones are better for 2e kids, but MATHCOUNTS is very time-pressing. There are others contests like BAMO or USAMTS that are less about time and more about understanding.