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:
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Solution:
The smallest and largest numbers in the original list are and , respectively, so the original range is and the doubled range is . If is still the smallest number after the list is expanded, the largest number in the new list will be .
Let and represent the two additional integers. For the mode of to remain unchanged, neither a nor b can equal , or , nor can and be equal in value. Suppose . For the median of to remain unchanged, a must be less than and must be greater than . It follows that the sum will be maximized if and . The maximum possible sum of the two additional numbers is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions