1989 AIME Problems/Problem 11
Problem
A sample of 121 integers is given, each between 1 and 1000 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. What is the largest possible value of
? (For real
,
is the greatest integer less than or equal to
.)
Solution
It is obvious that there will be values equal to one and
values each of
. It is fairly easy to find the maximum. Try
, which yields
,
, which yields
,
, which yields
, and
, which yields
. The maximum difference occurred at
, so the answer is
.