2014 AMC 10B Problems/Problem 18
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
solution by crastybow
See Also
2014 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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All AMC 10 Problems and Solutions |
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