Problem:
A list of positive integers has a mean of , a median of , and a unique mode of . What is the largest possible value of an integer in the list?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
The numbers in the list have a sum of . The value of the 11th number is maximized when the sum of the first ten numbers is minimized subject to the following conditions.
If the numbers are arranged in nondecreasing order, the sixth number is .
The number occurs either , or times, and all other numbers occur fewer times.
If occurs times, the smallest possible sum of the first numbers is
If occurs times, the smallest possible sum of the first numbers is
If occurs times, the smallest possible sum of the first numbers is
If occurs times, the smallest possible sum of the first numbers is
Thus the largest possible value of the th number is .
The problems on this page are the property of the MAA's American Mathematics Competitions