You can play chess online anytime you want to, even against a human opponent of your level, even in hourly competitions. It would be nice to have the same thing in math, at least for kids. Everyone could compete on their own level, learn new things by solving problems, maintain some ELO like score. I see two issues that make this harder than chess: new problems need to be generated for each contest and the correctness of the solutions needs to be verified.
I think that having new problems is manageable by working with a huge enough database, such things already exist, even if for a different level/difficulty. But enthusiastic people could upload their own problems. Also, by the time someone meets all the problems from a given level, probably they have mastered them well enough so that they are ready to move on to the next level.
About verifying solutions, a simple solution is to use multiple choice tests, or fields where contestants can enter numerical values.
This might be adequate for young children, but probably they are not the target audience for online contests.
Another solution could be contestants checking each other's arguments and receiving some points for this as well, but this can lead to extended debates and opinion-based decisions.
Finally, for old enough students (from high school) one could require that the solutions are entered in some theorem prover, like Lean, which automatically checks for correctness. This would make the competition similar to coding contests. Another advantage of this is that the correctness of proposed competition problems can also be verified automatically (though not their level).
Are there any other obstacles for online math competitions?