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An easy way of solving this is to note that if each participant beat each person who finished below them, this would fit the given criteria, and in this case the 3rd beat the 7th.
To show this more rigorously, let x_i be the score of the person who finished in ith place. Then x_2 ≤ 6, since clearly x_2 cannot be 7, and if x_2 = 6.5 this would imply x_1 = 7, but then person 1 would have beaten everyone, hence x_2 could not be 6.5. Now players 5, 6, 7 and 8 played exactly 6 games among themselves, hence x_5 + x_6 + x_7 + x_8 ≥ 6. Since we are given that x_2 = x_5 + x_6 + x_7 + x_8, we conclude that x_2 = 6. To finish, note that since the sum x_5 + x_6 + x_7 + x_8 is exactly 6, neither player 5, 6, 7 or 8 could have beaten or tied any of players 1, 2, 3 or 4, since then this sum would be greater than 6. It now follows that the 3rd beat the 7th. |
