I am currently a high school senior and, naturally, I have participated in many math competitions in my years. In fact, in preparation for such competitions, I have probably spent upwards of $250$ hours practicing my problem solving skills by working on past AIME and USAMO exams, with varying success.
However, as I transition from high school to college, I would like to continue practicing my problem solving skills but in topics that are more college-level / "real" math. As I'm sure is the case, math professors don't sit around trying to find the side lengths of a triangle given some random information about it.
So, what is your advice? I have tried looking into Putnam exams and I find that they are pretty fun to do, but I'm not sure if they're relevant to college mathematics or not.
Well a year ago I was in a similar situation as you. Namely in your post I see that you want to do "real" mathematics to prepare for college.
In that case here is how I proceeded and what I have learned on that journey.
First and foremost in mathematics computation almost always comes last, of course with respectful exceptions in some applied fields. Having that said you will have to be familiar with proving.
When it comes to proving things there is no better way than to read proofs from experienced mathematicians.
For me the turning point from high-school math to rigorous mathematics was the book named Planimetry by A.P Kiselev. You can easily locate it here.
It is a book about plane geometry, it is written very clearly and does not leave any concept undefined. I have found that through careful reading I have learned what does it mean to prove something without giving it much thought. It will not be required that you study this book or do exercises (which will of course aid your intuition) since you need to read proofs so that you can understand what does it mean to prove something.
On other hand you can grab a book dedicated to proving itself. It is called _How to Prove It). I have not read it myself but many recommend it as well as How to Solve It by Polya.
Once you have acquainted yourself with proofs you will need a method, a trick as Terrence Tao says. Many mathematical problems in exercises sections can be solved by using some well established pattern and some thought.
Now for about learning. When one learns mathematics you must not only know the definitions and theorems, but as well as what are consequences of those theorems and how do definitions relate to one another.
For suggested readings on basics of some fields I would suggest either Spivak or Apostol calculus and Linear Algebra Done Right by Axler
And one last tip for the trip is this: Mathematics is not nearly as much about rigor and theorems as it is about patterns and generalizations. Many times an insight into a pattern of a problem and solution will help you much more than any theorem did, please remember that and do not repeat my mistakes.