Added: On the pragmatic side of things, participation in structured extra-curricular activities is a "good thing" to be able to note in your future applications to college. In that sense, if you are not presently engaged in extra-curricular activities (for lack of interest, say, in sports, etc), but you like math, participating in math competitions would we a good match. If you are interested in pursuing a degree in math (or math-related field), such participation/dedication also shows that your interest in math, in particular, extends beyond the classroom.
Learning math isn't just about learning "things": content, definitions, rules, theorems, etc. That is, it isn't just about "what you learn or know" but also
- math is about "how you learn, how you think, how you solve problems, and how you use the "what" of what you're learning"
- and math is about the "why" and how to demonstrate that you know "how to..." (e.g. solve problems, construct proofs), and "why" (justifying, fully understanding why things work)...
So preparing for math competitions, to the extent that doing so makes you a more agile thinker, develops creativity, intuition, problem solving strategies, developing facility with proofs and logic, etc., will all help you in the long run, in math, and in life. Some of the "tricks" you may pick up and many of the things you simply memorize may not stick, and likely will not help much, in the long run. But the "activity" of doing math and engaging with challenging problems will transform you, your mind and your brain.
[Personally, I like to think of "mathematics" as a verb, as well as a noun!]
So, the short answer: Yes, preparing for math competitions will be useful - in that you will benefit, and it is relevant in that you will learn, and learn how to learn, you will learn how to better perform under time constraints, you will have the opportunity to engage with others who love doing math and meet new people, particularly if your "eye is [not only] on the prize.
Besides, if it's fun, and you like what you're doing, isn't that reason enough?