Problem:
Suppose that is a finite set of positive integers. If the greatest integer in is removed from , then the average value (arithmetic mean) of the integers remaining is 32 . If the least integer is is also removed, then the average value of the integers remaining is 35 . If the greatest integer is then returned to the set, the average value of the integers rises to 40 . The greatest integer in the original set is 72 greater than the least integer in . What is the average value of all the integers in the set ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Answer (D): Let be the number of elements of , let be the sum of all the elements, let be the greatest element, and let be the least element. The information given in the problem leads to the following equations:
Subtracting the first equation from the third gives , and because , this implies , so . Thus
Adding the first and third equations and subtracting the second gives . Because , the requested average is . One such set is .
The problems on this page are the property of the MAA's American Mathematics Competitions