If $X$ is a connected, compact metric space with distance function $d : X^2 \rightarrow \mathbb{R}^+$, is it true that there exists a positive real number $a$, dependent on $X$ and $d$, such that for any $n$ and for any $x_1, x_2, \cdots, x_n \in X$, there exists $y$ such that $$\frac{1}{n}\left(d(x_1,y) + d(x_2,y) + \cdots + d(x_n,y)\right) = a?$$

Motivation: trivial when $X$ is the boundary of a circle, a tricky contest problem when $X$ is the boundary of a square (these examples all use Euclidean distance)