Problem:
A sample of integers is given, each between and inclusive, with repetitions allowed. The sample has a unique mode (most frequent value). Let be the difference between the mode and the arithmetic mean of the sample. If is as large as possible, what is (For real is the greatest integer less than or equal to .)
Solution:
Let be the mode and be the mean. We may assume, without loss of generality, that . For to be as large as possible, , since if , then increasing by increases by no more than . As a result, certainly does not decrease.
Given that , we must make as small as possible. Now must occur in the sample at least twice, for otherwise it could not be the unique mode. If occurs exactly twice, then every other number in the sample must occur once. In this case, will be smallest if the other values are . This leads to a mean of
If occurs exactly times, then every other value can occur at most twice, and will be smallest if the other sample values are . We then have a mean of
If the mode occurs times, the smallest possible mean is
If the mode occurs times, the smallest mean is
and if the mode occurs times, the smallest mean is
Finally, if occurs exactly times, with , then
The above shows that the smallest occurs when occurs exactly times, and this is . Thus the largest value of is , and .
The problems on this page are the property of the MAA's American Mathematics Competitions