In math contests, participants match wits against humans (the contest designers).
In mathematics, participants match wits against the mathiverse!
Different strategies are required to excel in these two fundamentally different worlds.
When you’re competing in a math contest, it is valid to think, “What mathematical knowledge are the people who designed this problem trying to test me for? They have some solution (or solutions) in mind and must have determined that the problem is solvable within the constraints of the contest.” But when you’re doing mathematics, all that goes out the window. It may well be that nobody knows the answer to what you’re asking, and until an answer is found, you may have no idea how complex the situation is. Maybe it’s beyond human comprehension!
In a math contest, if your approach isn’t getting you somewhere within the average time allotted for each problem, it probably means you should abandon that approach and try something else. But in mathematics, if an idea isn’t working, well, you have no idea if it’s a fundamentally wrong approach or if you just haven’t been able to carry it through.
Math contest problems are basically a kaleidoscope of repackaged math. In fact, all the roughly 200 or so math topics used in contests can be conveniently collected and explained in a book just a few hundred pages long. There are a number of such books available today. And if you solve many contest problems, you will become more and more familiar with the necessary math and you will improve on contests. That’s a large reason why certain schools have math teams that consistently out compete others. You could, if you were so inclined and determined, dramatically improve your contest performance with some focused study. But if you do decide to embark on such a path to learn that material (some of it quite spectacular and worth learning, to be sure), please don’t get so lost in it that you forget about the wondrous unbounded mathiverse.
It’s in the mathiverse where new dreams await you.