KöMaL 2013 A600

Show that for every closed polygonal disk $\mathcal{K}$ there exists a real number $\alpha$ such that for arbitrary positive integer $n$ and any points $A_1,\ldots,A_n\in\mathcal{K}$ there is a point $X\in\mathcal{K}$ satisfying $\frac{|XA_1|+\ldots+|XA_n|}{n} = \alpha$.