So I'm currently a senior in high school in the U.S. and I'm hugely interested in mathematics. I was accepted to the University of Chicago (my top choice) and I'll be matriculating there in the fall. My background in math is a bit of an odd one. I used to hate it as a kid, but I always read novels and wanted to be a writer. In my freshman year, however, I came upon a senior who was the president of the mathematics club. He showed me Euclid's proof of the infinitude of primes and I was so stricken by the beauty of the proof that I basically took to math like nothing else existed, and I've spent most of the time in the intervening years exploring mathematics on my own, taking courses at the local uni in real analysis (currently in a course developing the Lebesgue integral and it's awesome), linear algebra, geometry, and discrete math. My question though involves math competitions. While I seem to definitely have some facility for proof based mathematics (I've done pretty well in the proof based courses at my local uni and I seem to manage to do most of the exercises in the math books I read) I seem to have a terrible time with problem solving especially at math competitions. I've never managed to move onto AIME from the AMC for example. The time constraints make me feel uncomfortable and while I'm interested in the topics covered on these tests (I'm hugely interested in geometry, less so in elementary number theory) I seem to not be able to solve these problems in the time constraints. Should I go back to basics and try to cultivate more skill in problem solving? is it really all that important a skill for a mathematician to cultivate?