I'm not sure that this is the right exchange for this question. It asks about the possibilities of research in mathematics and computer science.
I am very interested in mathematics and theoretical computer science. Over the past few years I've gained lots of experience from math competitions, programming competitions, software development internships, and talking to professors, and I've learned that I do not enjoy writing code; I enjoy writing "beautiful" and elegant code in practice, just as I enjoy beautiful and elegant proofs in mathematics, but I do not enjoy writing hundreds of trivial imperative lines of code to make some company money. I've learned that I like to think critically about problems, and I like finding the quickest, most efficient, most elegant solutions. I enjoy the mathematical (or theoretical) side of computer science.
So, recently, after coding only with imperative languages my whole life, I discovered Haskell, a functional programming language. It is extremely close to mathematics: Everything is a function. Ideas are defined rather than executed; instead of giving a computer step-by-step instructions, the computer is given a definition. This discovery reaffirmed my passion for mathematics and the mathematical and theoretical side of computer science.
I've also been frequenting a website called Project Euler, which is filled with 450+ mathematics-based programming problems. This is essentially the epitome of my passion. I love solving these problems with Haskell (and sometimes, when I'm clever enough, with merely a pen and paper).
I know that many of the questions (if not all of them) have already been solved and researched in both the field of mathematics and the field of computer science, and hundreds of their variants have been as well. However, many of these problems were initially proposed and solved hundreds of years ago by mathematicians (such as Euler and Gauss), and most of the relatively newer problems were all proposed and solved by the 1980s.
So my question is, are there still problems like this that I can think about for a living? If so, which field and sub-field is closest to this type of research? If not, would I be involved in these problems by teaching and then introducing them to students of mine, or hosting competitions?