Question 114
In a country, there are $n \gt 1$ cities; some pairs of them are connected with two-way direct flights. There is a unique (perhaps, non-direct) route between every two cities. The mayor of each city $X$ found the number $f(X)$ of the enumerations of all cities by $1, 2, \ldots, n$ such that along each route starting at $X$, the city numbers increase. All the mayors except one noticed that their resulting numbers are all divisible by $2016$. Prove that the remaining mayor's number is also divisible by $2016$.
(F. Petrov)
(F. Petrov)