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Counting matches

Games in a single-elimination tournament ●○○○○ 10 ptsIn a single-elimination tournament, each game is between two players. Only the winner of each game advances to the next round. In a particular such tournament there are 256 players. How many individual games must be played to determine the champion?
Matches in a knockout cup ●○○○○ 10 ptsCup football matches are played under the knockout system: the loser of each match is eliminated, and there are no rematches. 75 teams take part. How many matches must be played to decide the cup winner?
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?
Matches in a knockout tournament ●●○○○ 20 ptsThere are 100 nations taking part in an elimination tournament, in which a team is eliminated as soon as it loses a match. How many matches must be played in total to determine the overall winner?
Minimum matches to find a winner ●●○○○ 20 ptsIn a badminton knockout-style contest with 51 players, a player is eliminated when he loses two matches. What is the minimum number of matches needed to determine the winner? Assume every match has exactly one winner and one loser, and that the schedule can be chosen freely.
Boards in a simultaneous chess exhibition ●●○○○ 20 ptsA master gives a simultaneous chess exhibition on several boards, with one game played on each board. In the first two hours, he won $10\%$ of the total number of games, and 8 opponents drew their games with him. Over the next two hours, he played the remaining opponents: he won $10\%$ of those games, lost two games, and drew the remaining 7 games. On how many boards was the exhibition played?
Tournament with two dropouts ●●●○○ 50 ptsIn a tournament, each participant was supposed to play exactly one game with each of the other participants. Two participants dropped out during the tournament, each having played only 4 games. In the end, the total number of games played was 62. How many participants were there in total?
Matches played in a football tournament ●●●●○ 100 ptsA football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. Each pair of teams plays at most one match, and teams from the same city do not play each other. After several days, every team except Team A from City A has played a different number of matches. How many matches has Team B from City A played?
Arm wrestling tournament rounds ●●●●○ 100 ptsIn an arm wrestling tournament, 510 athletes participate. A win gives 1 point and a loss gives 0 points. If the winner initially had fewer points than the opponent, the winner additionally receives one point from the loser. In each round, participants whose scores differ by at most 1 point compete. The tournament ends as soon as a sole leader is determined. What is the minimum number of rounds that need to be played?
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 $n$ .

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