I am not quite sure if I understand you.
First off, I put 3 question marks to make it apparent that I wasn't actually 100% sure. Instead of thinking that I am assuming things, you shouldn't assume about things yourself.
2. The link I sent about "misguiding answers" is not a suggestion that Guest is doing it all wrong. He did miss a lot of steps and as you know, there have been anynomous peolpe who have been deliberately sending wrong answers.
3. AIME 2019? Interesting which one, I or II? Both of them absolutely don't require such "continued fractions"? If I am mistaken, again, let me know.
4. There is a difference between a theorem and a theory. A theory is something big, like the "BIg Bang Theory". I am pretty sure not anyone would name such a small thing in math, a theory.
5. So you said something about derivatives? Could you explain this "theorem" to me because it looks like it requires calculus and requires more understanding for the person who asked the question.