Problem:
What is the sum of all real numbers for which the median of the numbers , and is equal to the mean of those five numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The mean of the given numbers is
The median depends on the value of .
If , then the median is . If the mean and median are equal, then , which is equivalent to .
If , then the median is . If the mean and median are equal, then , which is equivalent to . But this is outside of the given range.
If , then the median is . If the mean and median are equal then , which is equivalent to . Again this is outside of the given range.
Therefore the only value of for which the mean equals the median is , so the requested sum is also .
The problems on this page are the property of the MAA's American Mathematics Competitions