In a country with $n$ towns, the distance between the towns numbered $i$ and $j$ is denoted by $x_{ij}$. Suppose that the total length of every cyclic route which passes through every town exactly once is the same. Prove that there exist numbers $a_i,b_i$ ($i=1,\ldots,n$) such that $x_{ij}=a_i+b_j$ for all distinct $i,j$.