Problem:
A list of 11 positive integers has a mean of 10 , a median of 9 , and a unique mode of 8 . 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 8 occurs 5 times, the smallest possible sum of the first 10 numbers is
If 8 occurs 4 times, the smallest possible sum of the first 10 numbers is
If 8 occurs 3 times, the smallest possible sum of the first 10 numbers is
If 8 occurs 2 times, the smallest possible sum of the first 10 numbers is
Thus the largest possible value of the 11th number is .
The problems on this page are the property of the MAA's American Mathematics Competitions