The Tournament Schedule
In a round-robin chess tournament, each of the 6 players plays every other player exactly once. A win earns 1 point, a draw earns 0.5 points, and a loss earns 0 points. After the tournament, the scores of the players are all distinct integers or half-integers. The organizer claims that the sum of all scores must be 15. A player, Anna, finished with 4 points. Another player, Boris, claims that if Anna scored 4, then the maximum possible score of the player who finished last is at most 1.5. You are asked to verify these claims and analyze the possible score distributions.
1. Verify the organizer's claim that the sum of all scores is 15. Justify your answer.
Read the worked answer
In a round-robin tournament with 6 players, each pair plays exactly one game. The number of games is (\binom{6}{2} = 15). In each game, regardless of the result, the total points awarded to both players sum to 1 (either 1+0 for a win/loss, or 0.5+0.5 for a draw). Therefore, the sum of all scores is equal to the number of games, which is 15. So the organizer's claim is correct.
2. Evaluate Boris's claim: If Anna scored 4 points, is it true that the last-place player's score is at most 1.5? Provide a proof or counterexample.
Read the worked answer
We need to check if there exists a tournament with 6 players where one player (Anna) scores 4, and the last-place player scores more than 1.5. Since scores can be half-integers, the next possible score above 1.5 is 2. We attempt to construct a tournament where Anna has 4 points, and the last player has 2 points. The total sum is 15. Suppose the scores are: Anna 4, and the other five players have distinct scores summing to 11, with the smallest being 2. The possible distinct half-integer scores for the other five could be 2, 2.5, 3, 3.5, and 4? But Anna already has 4, and scores must be distinct? The problem says "all distinct integers or half-integers" – so all six scores are distinct. Anna has 4, so others cannot have 4. The other five scores must be distinct half-integers, sum to 11, and the smallest is at least 2. Let's list possible distinct half-integers less than 4 (since Anna has 4, and others could be above 4? But if someone is above 4, that's possible. However, we need the last place to be at least 2. Let's try to assign scores to the other five: they must sum to 11, be distinct, none equal to 4, and the minimum is at least 2. The smallest possible set of five distinct half-integers with minimum 2 is {2, 2.5, 3, 3.5, 4} but 4 is taken. Next is {2, 2.5, 3, 3.5, 4.5} sum = 15.5 >11. Or {2, 2.5, 3, 3.5, 4} not allowed. {2, 2.5, 3, 4, 4.5} sum=16. So to sum to 11 with five distinct half-integers, minimum at least 2, and none equal to 4, we need smaller numbers. But if minimum is 2, the smallest possible sum of five distinct half-integers (including 2) is 2+2.5+3+3.5+4=15, which is >11. So impossible. Therefore, the last-place score cannot be 2 or more. The maximum possible last-place score is 1.5. So Boris's claim is true. But we must also ensure that such a tournament exists with Anna 4 and last 1.5. For example, scores: 4, 3.5, 3, 2, 1.5, 1? Sum = 15.0? 4+3.5+3+2+1.5+1=15.0 yes, all distinct. So feasible. Thus the claim holds.