Problem:
The mean, median, unique mode, and range of a collection of eight integers are all equal to . The largest integer that can be an element of this collection is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The values satisfy the requirements of the problem, so the answer is at least . If the largest number were , the collection would have the ordered form . But , and a mean of implies that the sum of all values is . In this case, the four missing values would sum to , and their average value would be . This implies that at least one would be less than , which is a contradiction. Therefore, the largest integer that can be in the set is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions