Problem:
Halfway through a 100-shot archery tournament, Chelsea leads by 50 points. For each shot a bullseye scores 10 points, with other possible scores being 8 , 4,2 , and 0 points. Chelsea always scores at least 4 points on each shot. If Chelsea's next shots are bulleyes she will be guaranteed victory. What is the minimum value for
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Solution:
The second place archer could score a maximum of points with the remaining shots. Therefore Chelsea needs to score more than points to guarantee victory. If Chelsea's next shots will score points, her remaining shots will score at least points. To guarantee victory,
Therefore Chelsea needs at least 42 bullseyes to guarantee victory.
If Chelsea does not make a bullseye, the maximum number of points her opponents could gain per shot would be . Chelsea must make enough bullseyes to prevent her opponents from gaining 50 points. Because , the most non-bullseyes she can afford to score is 8 , leaving bullseyes needed to guarantee her victory.
The problems on this page are the property of the MAA's American Mathematics Competitions