Problem:
In a mathematics competition, the sum of the scores of Bill and Dick equalled the sum of the scores of Ann and Carol. If the scores of Bill and Carol had been interchanged, then the sum of the scores of Ann and Carol would have exceeded the sum of the scores of the other two. Also, Dick's score exceeded the sum of the scores of Bill and Carol. Determine the order in which the four contestants finished, from highest to lowest. Assume all scores were nonnegative.
Answer Choices:
A. Dick, Ann, Carol, Bill
B. Dick, Ann, Bill, Carol
C. Dick, Carol, Bill, Ann
D. Ann, Dick, Carol, Bill
E. Ann, Dick, Bill, Carol
Solution:
Let and represent the scores of Ann, Bill, Carol and Dick, respectively. Then
Adding and shows or . Subtracting from gives or . By , (since is nonnegative), so .
Note. All the conditions in the problem can be satisfied, for instance, with the scores and .