Question 5
There are $n$ cities in a country, where $n \geq 100$ is a positive integer. Some pairs of cities are connected by (two-way) flights. For two cities $A$ and $B$, a path is a sequence of distinct cities $C_0, C_1, C_2, \ldots, C_k, C_{k+1}$, such that there are flights between $C_i$ and $C_{i+1}$ for every $0 \leq i \leq k$, with $C_0 = A$ and $C_{k+1} = B$.
A long path between $A$ and $B$ is defined as a path such that no other path has more vertices. Similarly, a short path is defined as a path with the fewest vertices. In particular, if $A$ and $B$ have a direct flight, that is the shortest path.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). For a given $n$, find all possible numbers of flights in the country.
A long path between $A$ and $B$ is defined as a path such that no other path has more vertices. Similarly, a short path is defined as a path with the fewest vertices. In particular, if $A$ and $B$ have a direct flight, that is the shortest path.
Assume that for any pair of cities $A$ and $B$ in the country, there exist a long path and a short path between them that have no cities in common (except $A$ and $B$). For a given $n$, find all possible numbers of flights in the country.