At the high school and college level, the Olympiads for math and CS are pretty analogous. But there's really popular semi-formal coding contests which exist outside academia which don't really have a math equivalent.
I'd say math contests are more popular among high schoolers, and semi-formal coding contests more popular among college students.
Art of Problem Solving (AoPS) [https://artofproblemsolving.com/] is a really good resource, and there's a very healthy online community.
They're also similar in how olympiads are different from the "real thing" (TM).
Academia.SE discussion about this [https://academia.stackexchange.com/questions/86451/does-the-...]
As someone who did math olympiads in high school, my 2 cents is that they're a fantastic way to learn how to solve and approach problems and gain intuition. And I'd say intuition mainly comes from solving problems.
Alternatively if you're really interested in intuition, you could also look at the Math Olympiads. Pick a problem, beat your head on it, finally look at the solution, repeat. There are web sites and prep books.