Problem:
A data set consists of (not distinct) positive integers: , and . The average (arithmetic mean) of the numbers equals a value in the data set. What is the sum of all possible values of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
First note that the mean of the elements of the data set excluding is 4. The given condition will therefore be met if , and if , then the mean will not be . Because is a positive integer, the mean of the full data set must be greater than or equal to . The only other possibilities for the mean are therefore 5 and 7 . The mean will be 5 if and only if , which implies that . The mean will be 7 if and only if , which implies that . The sum of the possible values of is .
The problems on this page are the property of the MAA's American Mathematics Competitions