Problem:
In a certain cross-country meet between two teams of five runners each, a runner who finishes in the position contributes to his team's score. The team with the lower score wins. If there are no ties among the runners, how many different winning scores are possible?
Answer Choices:
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Solution:
With exactly runners contributing to their teams' scores, the sum of the scores of the two teams is . The lowest winning score is . Every winning score must be less than half of . Thus, there are possible winning scores, provided that all scores from through are possible.
In fact, all integers between and are possible scores. Note that if a certain finishing order results in score for Team , then there is a runner from Team in that finishing order after whom the next finisher is from Team . If the positions for these two runners were interchanged, then the resulting finishing order would give Team a score of .