1987 AMC 12 Problem 17

Question

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.

Dick, Ann, Carol, Bill

Dick, Ann, Bill, Carol

Dick, Carol, Bill, Ann

Ann, Dick, Carol, Bill

Ann, Dick, Bill, Carol

Answer: E

Concepts:inequalities ordering algebraic comparison

Difficulty rating: 1830

Small hint:

Represent the four scores by

A,B,C,D

in name order

Big hint:

Add and subtract the equality

A+C=B+D

and the inequality

A+B\gt C+D

Solution:

Let

A,B,C,D

be the scores of Ann, Bill, Carol, and Dick. The conditions are

\begin{aligned} A+C&=B+D,\\ A+B&\gt C+D,\\ D&\gt B+C. \end{aligned}

Adding the first two comparisons gives

A\gt D.

Subtracting the equality from the inequality gives

B\gt C.

Finally,

D\gt B+C\ge B.

Hence

A\gt D\gt B\gt C.

Thus the correct answer is E.