2014 AMC 12B Problems/Problem 11
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?
Solution
We start off with the fact that the median is , so we must have , listed in ascending order. Note that the integers do not have to be distinct.
Since the mode is , we have to have at least occurrences of in the list. If there are occurrences of in the list, we will have . In this case, since is the unique mode, the rest of the integers have to be distinct. So we minimize in order to maximize . If we let the list be , then .
Next, consider the case where there are occurrences of in the list. Now, we can have two occurrences of another integer in the list. We try . Following the same process as above, we get . As this is the highest choice in the list, we know this is our answer. Therefore, the answer is
See also
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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All AMC 12 Problems and Solutions |
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