Inspired by AoPS's Marathon Threads, I decided to create one on CF! The only rule of this forum is that you have post an interesting theorem, or trick related to CP or math, with a brief description and provide a source for extra research.

To start things off, I think that Proizvolov's identity is very interesting (but doesn't appear much in math problems :( ) and states if you have the numbers from $$$1-2n$$$ and split the numbers into 2 sets $$$A, B$$$ such that $$$A$$$ is in increasing order and $$$B$$$ is in decreasing order, the sum $$$|A_1-B_1|+|A_2-B_2|+\dots+|A_n-B_n|$$$ is always equal to $$$n^2$$$. Source: https://en.wikipedia.org/wiki/Proizvolov%27s_identity