As an “amateur” I don’t know where to begin, or end. Just so many I’ve figured out. A better proof of the Pythagorean theorem than what was in the textbook, discovered and proved the law of cosines, etc. In my Transformational Geometry class, I figured out a much simpler proof of one of the theorems in the book. Lots of stuff like that.
Too many Math proofs I’ve seen are “gotcha” proofs. “Here, do this magic step for no obvious reason and tada! it all works out!” So I like to come up with my own that offer an insight into the process of how to do the proof itself. Some of these are better than the “gotcha” ones.
As a CS professor, I’ve published lots of papers that have a lot of original work in them. There’s even a Wikipedia page that has my name in it for one of my results which impresses FtGKid2. (And note ultrafilter’s comment.)
But one of my cherished (very) minor ones is the one I came up with when I was 4. Lots of birthdays in my family around the same time. I noticed one sib just turned 8. Twice as old as me. Another turned 10. When would that be a doubling too? Hmm, when I turn 10. Oh: If the difference between siblings is “so much” then the older is twice as old as the younger when the younger is the difference in age. Neat!