Problem:
Two integers are inserted into the list to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since the median remains unchanged, one number less than and one number greater than is added. The inequalities are strict since the mode doesn't change, which means that only can appear twice. The current range is . The new range is therefore . To maximize the smaller number, we can set it equal to . The larger number is forced to be . Their sum is .
Thus, is the correct answer.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions