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 
. Now, 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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| All AMC 10 Problems and Solutions | ||
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