TITLE: Real numbers inequality
QUESTION [5 upvotes]: Consider 9 random and distinct real numbers. Prove that there exist at least 2 of them, $m$ and $n$, for which $0 < \frac{m-n}{1+mn} < \sqrt{2}-1$.

Any ideas? I really don't know where to start from!
The friend who challenged me with this problem claims that it can be solved by using pigeonhole principle, but I have no idea how!!

REPLY [4 votes]: Hint. Given 9 random and distinct real numbers in the interval $(-\pi/2,\pi/2)$ whose size is $\pi$, by the pigeonhole principle there are at least two of them, say $x$ and $y$, such that
$$0<x-y<\pi/8.$$
Note that $\arctan(\sqrt{2}-1)=\pi/8$.

REPLY [3 votes]: Another hint:
That formula of $\frac{m-n}{1+mn}$ resembles $\tan(x-y)$.