Is high school contest math useful after high school?

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.

That said:

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?

Since you are posting it is here I will interpret you question to ask whether the high school math stuff turned to be relevant for me when I got to uni level math.

My answer then is that for me it was not relevant at all. Of course it depends a lot on the high school in question and the university in question. I got kicked out of several schools so I went from good ones to bad ones but I found the approach to maths in all of them algorithmically oriented. Just learn a technique and apply it to death, mostly without any understanding of they whys and hows. (I'm sure there are better high schools somewhere).

Once I got to university the mathematics was rigorous from day 1. I never followed calculus courses. It was $\epsilon-\delta$ analysis from the get go, which was great. I never found that I lacked in computational techniques. I just picked them up as they were derived rigorously.

Now that I've been teaching at universities for several years I find that I assume very little prior knowledge from the students. I can't rely on them knowing anything deep (a usual experiment I run is to ask at the beginning of a calculus course if 0.999... is equal to 1.000.... or not. Across different countries most students say they are different. Roughly 20% of the students know the correct answer, but of those perhaps 1-2 students are able to explain why). Basically, I do assume that they can perform algebraic manipulations and that they know about divisibility of integers, as well as very basic familiarity with the elementary functions. I assume basic arithmetic with the rationals and I presume that the students are under the false impression that they can compute with real numbers.

To conclude, it certainly depends on the high school and university in question. For me, it is absolutely useless.

High school math competitions require you to learn how to solve problems, especially when there is no "method" you can look up telling you how to solve these problems. Problem solving is a very desirable skill for many jobs you might someday wish to have.

Math competitions are great. You learn a lot. It will make you a smart aggressive problem solver. But be warned: that is not all there is. As you continue your study of mathematics, you will see this is true in spades.

I will simply say that my high school competitions (c. 1987-88) fostered in me a love of competitive problem solving that has never left me to this day. More importantly, being in math clubs brought me in contact with other math enthusiasts, from whom I learned much. This was especially important to me, having grown up in a working-class city where loving math was not exactly commonplace.