69. $N$ oligarchs built a country with $N$ cities, each oligarch owns exactly one city. In addition, each oligarch built several roads between the cities: any pair of cities is connected by at most one road from each oligarch (between two cities there can be several roads belonging to different oligarchs). In total, $d$ roads were built. Some oligarchs would like to form a corporation, combining their cities and roads, so that from any city of the corporation it would be possible to reach any other city of the corporation by its roads, possibly passing through cities of other oligarchs. However, it turned out that no group of fewer than $N$ oligarchs can form a corporation! What is the largest $d$ for which this is possible?
69. Answer: the maximum number of roads is $\frac{N(N-1)(N-2)}{6}$.

Let's number the oligarchs and their cities from 1 to $N$ respectively.

Estimation. We will say that a road is liked by an oligarch if it belongs to this oligarch or one of the cities at the ends of the road belongs to this oligarch. Note that a road cannot lead out of a city belonging to the owner of that city, as in this case, the owners of the cities connected by this road can form a corporation. Therefore, any road is liked by exactly three oligarchs. We will associate each road with a triplet of oligarchs who like it.

Consider an arbitrary triplet of oligarchs $A, B$, and $C$. We will prove that this triplet is associated with no more than one road. Suppose this is not the case. The chosen triplet of oligarchs can be associated with only three roads (if such exist): the road $B C$ belonging to oligarch $A$, the road $A C$ belonging to $B$, and the road $A B$ belonging to $C$. It is easy to see that whichever two of these roads the triplet $A, B, C$ is associated with, this triplet of oligarchs can form a corporation. Therefore, each triplet of oligarchs is associated with no more than one road, i.e., there are no more than $C_{N}^{3}$ roads.

Example. Let the road between cities $i$ and $j$ belong to the oligarch numbered $k$ if and only if $k > i, k > j$.

Let's check that this example satisfies the condition of the problem. Suppose this is not the case and some group of oligarchs can form a corporation. Let $M$ be the highest number of an oligarch in this corporation. Then no road leading out of city $M$ belongs to the members of the corporation, which contradicts the fact that from this city one can reach any city of the corporation.

Note that in our example, any triplet of oligarchs $i < j < k$ is associated with the road $i j$ belonging to the oligarch $k$, i.e., there are a total of $C_{N}^{3}$ roads.