Problem:
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers through . Each throw hits the target in a region with a different value. The scores are: Alice points, Ben points, Cindy points, Dave points, and Ellen points. Who hits the region worth points?
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Answer Choices:
A. Alice
B. Ben
C. Cindy
D. Dave
E. Ellen
Solution:
Ben must hit and . This means Cindy must hit and , because she scores using two different numbers, neither of which is or . By similar reasoning, Alice hits and , Dave hits and , and Ellen hits and . Alice hits the .
Ellen's score can be obtained by either or . In the first case, it is impossible for Alice to score . So Ellen's is obtained by scoring and , and Alice's total of is the result of her hitting and . The others scored and .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions