2014 AMC 10B Problems/Problem 18
Contents
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 1
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
Solution 2
Note that let so . To maximize the value of where ranges from to , we let any elements be so . Now we have to let one of above values = hence now let , hence is the answer.
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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