CIIM 2013 Problem 6

Let $(X,d)$ be a metric space with $d:X\times X \to \mathbb{R}_{\geq 0}$ . Suppose that $X$ is connected and compact. Prove that there exists an $\alpha \in \mathbb{R}_{\geq 0}$ with the following property: for any integer $n > 0$ and any $x_1,\dots,x_n \in X$ , there exists $x\in X$ such that the average of the distances from $x_1,\dots,x_n$ to $x$ is $\alpha$ i.e. $$\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.$$