Choose 13 real numbers $ x_1,x_2,\ldots,x_{13}\in\mathbb{R} $ with $ x_i\neq x_j $ if $ i\neq j $. For these 13 numbers there exist at least two numbers amongst them such that
$ 0 < \frac{x_i-x_j}{1+x_ix_j} \leq 2-\sqrt{3} $
It's a tricky one to prove, unless you are one endowed with a certain amount of intuition (exempli gratia, Uli, et alii). Simple trigonometry and the ever overlooked Pigeonhole Principle are key tools to solving it.
