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 is .
There are three possibilities for the median: it is either , or .
Let's start with .
has solution , and the sequence is , which does have median , so this is a valid solution.
Now let the median be .
gives , so the sequence is , which has median , so this is not valid.
Finally we let the median be .
, and the sequence is , which has median . This case is therefore again not valid.
Hence the only possible value of is .
The problems on this page are the property of the MAA's American Mathematics Competitions