When Is This Possible?
Let be a positive integer between and (inclusive). Suppose players are playing a tennis tournament. A game is played between two teams of players, where each team has two players. The teams aren't fixed; which means if and are on the same team at some game, at the next them they might be on opposite teams. It turns out that every player has each of the other players playing against him (i.e. on the opposite team) exactly once. Find the largest possible value of for which such a tournament is possible.
Details and assumptions
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Remark that
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This problem is inspired by a problem from the Russian olympiad.
The answer is 2009.
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1 solution
Observe that if a number of n players would play such that they all play with each other then for any configuration of players that will respect the conditions of the problem there must a number of (n-1)! different pairs of players that play against each other. Secondly it can be said that for any game that is played as there are 2 teams and in each team are 2 players there will be 4 distinct pair of players that play against each other and therefore as the number of pairs is (n-1)! and at each game there will be taken 4 pairs from that total they anyways being completely distinct the number of pairs of teams such that these pairs do not appear in more than 1 game is divisible by 4. Therefore it can be said that for any tournament which has the propriety that by a number of games between teams of players all the players play against each other exactly once as stated in the problem implies that it has a number of pairs of players which divide 4. The largest such number until 2014 is of 2009 players and is therefore the answer to the problem.
I think more interesting in such problems however is deepening their structural proprieties and therefore understanding their construction. In this case it wouldn't be enough to study just the general proprieties of such a tournament but understanding how it's different components affect and influence each other in the given structure under such constraints anyways.