Does one have to be good at competition math to become a great mathematician in academia? So I'm seventeen and currently a junior in high school. I love mathematics very much, it's developed into a strong passion over the past few years, after witnessing the president of the math club at my school give Euclid's proof on the infinitude of primes I've obsessed over finding beautiful arguments and proofs. The dilemma I've been having recently is regarding my relatively lackluster performance at mathematics competitions. I've worked my way through quite a few math books. (currently reading spivak's calculus on manifolds, have read his calculus, Shilov's linear algebra, apostol's calculus volumes and apostol's mathematical analysis.) I usually don't have insurmountable trouble doing the exercises here, but show me a problem on the AMC12 and I'll take an ungodly amount of time trying to make sense of it. My interests don't particularly lie in the discrete sort of mathematics that those sorts of competitions test though, but shouldn't the bit of mathematical maturity I seem to have be able to help me reason through these tests? Do these sorts of problem solving skills help me become a better mathematician? One last thing to note is I'm not a very competitive person so I feel that probably strongly affects the way I approach these competitions.
 A: I hear you @Eric. I personally am not a fan of mathematical competitions either. Actually I am not fond of any sort of strangled time restriction on problem solving. Yes, you need to learn to meet deadlines if you intend to work in industry, but the highest echelon in mathematics to me is research and that takes time, hard work and it comes with perpetual, repetitive failures. 
I always thought I lacked the computational skills my fellow students had when I was in high school. Actually, I still doubt my capabilities in that regard. But I've been told and have learnt, that does not decide whether you will be of any importance when it comes to mathematics. 
Yes, mathematical competitions are very useful. They provide great incentive to budding mathematicians and lure bright young minds into the field. But performing substandard in these time-restricted tests does not mean you are less capable than your fellows. 
The crucial point is that you love mathematics. The subject in its purest form. You seem like you have a serious curiosity to learn math and have a taste for the beauty that lies in its precision, truth and  subtlety. That is your strength. Pursue what you like, what whets your curiosity and keep doing what you enjoy, the results and the recognition will follow. 
I am a critic of short mathematics tests. I am in college and our papers are only 2 hours long and sometimes include MCQ questions which I despise. I came across a Graph Theory paper online once which required students to solve 4 rather difficult and long problems but were given 6 hours to do so. I would kill to be sitting for papers of that form because like you I am not a fan of rushing a solution to a problem. I mean, you miss all the fun. The joy lies in wrecking your brain to come up with a "solution". Not to come up with an answer following a known, taught algorithm.
So where were we? Right. You'll be just fine. Don't be discouraged. You have a long way to go. And keep reading. You've got your hands on some good literature. Might also want to read some number theory books. Search for reference request questions on Elementary Number Theory, there are plenty on this site. 
Good Luck!!   
A: To more directly answer your question, I used to think that math competitions were useless beyond something to write on a college application. It turns out that a ton of finance companies give interview tests with these sorts of problems, so yes it is important if you ever want to be a quant.
