69th Belarusian Mathematical Olympiad

There are $n\ge 5$ cities in some country. Some of the cities are connected with each other by roads, and the next three conditions are satisfied: 1) there is at most one road between any two cities; 2) not all cities are connected with each other; 3) there are exactly $k\ge 1$ roads between any four cities. Find all $n$ and $k$ at which this situation is possible.