Problem:
When the mean, median, and mode of the list
are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real value of ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
If were less than or equal to 2 , then 2 would be both the median and the mode of the list. Thus . Consider the two cases , and .
Case 1: If , then 2 is the mode, is the median, and is the mean, which must equal , or , depending on the size of the mean relative to 2 and . These give , , and , of which is the only value between 2 and 4 .
Case 2: If , then 4 is the median, 2 is the mode, and is the mean, which must be 0,3 , or 6 . Thus , or 17 , of which 17 is the only one of these values greater than or equal to 4 .
Thus the -value sum to .
The problems on this page are the property of the MAA's American Mathematics Competitions