Competition problem — Combinatorics

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Combinatorics Difficulty 6.1 National olympiad Prove it Belarus

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.

Solution

Answer: $n = 5, k = 3$ .

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