Problem:
A list of positive integers has the following properties:
Find the sum of the squares of all the items in the list.
Solution:
Notice that the median not being in the list implies that the list has an even length, because otherwise the middle item in the sorted list would be the median.
Because is the unique mode, there must be at least two s in the list. Because the list contains only positive integers and sums to , there cannot be more than three instances of . If there are three s, then the remaining numbers must sum to . There are three possibilities for the ordered list in this case:
The first possibility violates the condition that the mode of the list is unique, and the remaining two possibilities violate the condition that the median not appear in the list. This implies that there cannot be three s in the list. Hence there must be exactly two s in the list, and because is the unique mode, all the remaining numbers must be distinct.
Note that if there were eight or more numbers in the list, then the minimum possible value of their sum would be , which is greater than . Hence there are either four or six numbers in the list. If there are six numbers, then four of them must be distinct numbers summing to , and the only two collections of four such numbers are and . In both cases the median of the resulting list is not an integer, so neither case is possible. Therefore there must be four numbers in the list. This leaves the following four possibilities:
The first two lists violate the condition that the median not appear in the list, and the third list violates the condition that the median must be an integer. The only remaining possibility is . This list does indeed satisfy all three properties. The requested sum of squares is .
The problems on this page are the property of the MAA's American Mathematics Competitions