6000th Brilliant Problem Solved - Problem

Geometry Level 5

Find the largest constant \(m\) and the smallest constant \(M\), such that given any 13 distinct real numbers, there always exists 2 numbers \(x\) and \(y\), with \(x > y\), such that

\[m < \dfrac {x-y}{1+xy} < M.\]

If the value of \(m+M\) can be written as \(a+b\sqrt{c}\), where \(a\), \(b\) and \(c\) are integers, and \(c\) is square-free, find \(a+b+c\).

To commemorate my 6000th solved problem, which was this, I have decided to post this problem.

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