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Figure Skating Scoring


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Each of the 9 referees on the figure skating championship estimates the program of 20 sportsmen by assigning him a place (from 1 to 20). The winner is determined by adding those numbers. (The less is the sum - the higher is the final place). It was found, that for the each sportsman, the difference of the places, received from the different referees was not greater than 3. 

What can be the maximal sum for the winner?
 

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I assume we're looking for the lowest (best) score the winner could have.
If we're looking for the highest (worst) score the winner could have, we arrange the scoring so that everyone gets the same score of 94.5 = (1890/20). If half points are not allowed then we arrange for equal numbers of scores of 94 and 95. The winner's score would then be 94.

Worst case is the score differences of adjacent competitors is exactly 3.

The the winner's score is w
Second place score is w +3
Third place score is
w +6,
...
Last place score is
w +57

Sum of all scores is 20 w + 570

Sum of all points given is (9 judges)x(1+2+3+...+19+20) = 9x210 = 1890

20 w + 570 = 1890

w = 66

 

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