Competition problem — Combinatorics

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Combinatorics Difficulty 3.4 AMC 10/12 Find the answer

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:

Pick one

Solution

Let the scores be $A$ , $B$ , $C$ , and $D$ . Then we have $A + C = B + D$ , $A + B > C + D$ , and $D > B + C$ . Call these equations $(1)$ , $(2)$ , and $(3)$ respectively. Then adding $(1)$ and $(2)$ gives $2A + B + C > B + C + 2D \implies 2A > 2D \implies A > D$ . Now subtracting $(1)$ from $(2)$ gives $A + B - A - C > C + D - B - D \implies B - C > C - B \implies 2B > 2C \implies B > C$ . Finally from $(3)$ , since $D > B + C$ and $C$ is non-negative, we must have $D > B$ , so putting our results together we get $A > D > B > C$ , which is answer $\boxed{\text{E}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.