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3 problems match “round-robin tournaments”. Clear

Matches in a round-robin group ●○○○○ 10 ptsA group has four teams, and each team plays every other team in the group exactly once. How many matches are played in the group?
Participants in a chess tournament ●●●●○ 100 ptsEach participant in a chess tournament played one game against each of the other participants. A win earns one point, a draw earns half a point, and a loss earns no points. At the end of the tournament, it was determined that each participant had earned exactly half of their points in games against the three competitors who finished in the last three places. How many participants were there in the tournament?
Maximum rounds guaranteeing a four-team condition ●●●●● 200 ptsThere are 18 teams in a single round-robin tournament. In each round the 18 teams are divided into 9 pairs, and the two teams in each pair play one match. The teams are regrouped from round to round, and over 17 rounds each team plays exactly one match against each of the other 17 teams. Suppose that after $n$ rounds, played according to any feasible schedule, there always exist 4 teams that have played exactly 1 match among themselves in total. Find the maximum possible value of $n$ .