- #1

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Choose 13 real numbers [tex]x_1,x_2,\ldots,x_{13}\in\mathbbb{R}[/tex] with [tex]x_i\neq x_j[/tex] if [tex]i\neq j[/tex]. For these 13 numbers there exist at least two numbers amongst them such that

[tex]0 \; < \; \frac{x_i-x_j}{1+x_ix_j} \; \leq \; 2-\sqrt{3}[/tex]

Isn't that cool?!

(I think I have a proof, but feel free to give it a go and post something ).