6. In the Lemon Kingdom, there are 2020 villages. Some pairs of villages are directly connected by paved roads. The road network is arranged in such a way that there is exactly one way to travel from any village to any other without passing through the same road twice. Agent Orange wants to fly over as many villages as possible in a helicopter. For the sake of secrecy, he will not visit the same village twice, and he will not visit villages in a row that are directly connected by a road. How many villages can he guarantee to fly over? He can start from any village.
Solution. It will not be possible to fly over all villages if some village is connected by roads to all others (

However, it is possible to fly over all villages except one. Let's form a flight plan: choose the first village randomly, and each subsequent one randomly from those that can be flown to from the previous one according to the rules. At some point, it will not be possible to add the next village. This means that among the remaining villages (if any), only those connected by a road to the last added one. Now, if there is more than one, remove the last village from the plan and add the remaining ones. They are not connected by roads to each other, otherwise there would be two paths between connected villages: directly and through the removed village. Moreover, more than one village cannot be connected by a road to the village preceding the removed one (otherwise there would be two paths between the removed and the preceding one). Therefore, if there are at least two, they can be added to the flight plan according to the rules. Now it remains to fly over the villages according to the plan.

Criteria. The task is worth 7 points. Proving that all cannot be flown over - 2 points.

Showing how to fly over all except one - 4 points.

Lack of explanations for traversal, but the solution is correct - up to -3 points.