Just curious about how people usually self study these subjects.
1) Is it the most efficient to read through the chapters, and write down theorems, definitions, and take notes of important parts and work out all the proofs and examples? And then make sure you can re-prove everything after finishing a chapter? Like for example, studying analysis, topology, etc. And the same with physics. Or is it better to just do all the problems and move on? What would be the most efficient method of studying considering time and being able to understand it 100%? Like, I know it's good to use multiple books too, and the optimal method would be to work out everything, but it seems too time-consuming to be able to learn enough and go into the research field and publishing faster.
2) And how long does it usually take one to finish a textbook? What are the advantages to taking a class than self-studying assuming one would put in as much work as needed if one had to take a final exam?
3) And where exactly does talent play in the process? Is it just about understanding things quicker and applying it more efficiently? I find that I can do math and physics with relative ease; are there people with varying degrees of talent and how pertinent is it to have the "most" talent in terms of being one of the top in math and physics research career?
4) By the way, just one more question, how important is like physics or math competitions to physics or math careers in research? I only got interested in these subjects very recently, so although they were my best subjects in school and I understood them easily, I never really practiced for olympiads or anything. And the only thing I did regarding competitions was math team and contests like ARML.