9.5. At the summit, 2018 politicians gathered. Every two of them were supposed to hold private negotiations. At some point, it was found that among any four, there is one who has already talked to the other three. What is the maximum number of negotiations left to be held?
Answer: three.

Solution. Let A and B not have talked. Consider any other pair, B and C. If they also did not talk, then in the quartet A, B, C, D, there would be no one who talked to three others, which contradicts the condition. Therefore, any pair of non-talking participants must include A or B.

Suppose A did not talk to anyone besides B, whom we will call C. Consider quartets of the form A, B, C, D, where D is each of the other politicians. Since A did not talk to B and did not talk to C, in each such quartet, D must have talked to all the others. Thus, A and B must have talked to everyone except C. Therefore, the only unheld talks could be between B and C, which does not contradict the condition. In total, at most three meetings need to be held: A - B, A - C, and B - C.

Evaluation criteria.

«+» A complete and well-reasoned solution is provided

«士» A generally correct reasoning is provided, containing minor gaps or inaccuracies

«Ғ» The correct answer and an example of a trio who did not talk to each other are provided

«Ғ» The solution is missing, but there is progress (for example, it is proven that each person could not have talked to no more than two participants)

«-» Only the answer is provided

«-» The problem is not solved or is solved incorrectly