Question 27
In a sports gymnastics competition with 50 participants, each participant is identified with a number from 1 to 50. The competition has 13 judges, and each of them ranks the participants from best to worst, at their discretion. Then he assigns 1 point to the best, 2 to the second, ..., 50 to the last. It turned out that for each pair of participants $(i, j)$, with $i \lt j$, there were exactly 6 judges who thought that $i$ is better than $j$. This means that in the scores of those 6 judges, the number assigned to $i$ is less than the number assigned to $j$, and in the scores of the remaining 7 judges the number assigned to $i$ is greater than the number assigned to $j$.
The final score for each competitor is the sum of the 13 numbers assigned by the judges. Decide if with this information the final score of each of the 50 competitors can be determined with certainty.
If the answer is affirmative, determine the final score for each of the 50 competitors; If negative, explain why.
The final score for each competitor is the sum of the 13 numbers assigned by the judges. Decide if with this information the final score of each of the 50 competitors can be determined with certainty.
If the answer is affirmative, determine the final score for each of the 50 competitors; If negative, explain why.