Lately I have been getting into solving problems in some of the math journals I enjoy reading. More and more I find that solvers employ a theorem or identity that makes solving the problem much easier. Sometimes, that identity or theorem is one I am not familiar with.
For instance a few weeks ago I saw the Stolz-Cesaro Theorem used on a problem in the Fibonacci Quarterly. It was used in a very slick way, and was a theorem that up to that point I was unfamiliar with.
My question is: what are some of the best theorems, identities, inequalities, etc.. that the consummate problem solver should have at their disposal?