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
This problem needs a solution. If you have a solution for it, please help us out by adding it. It is obvious that there will be n+1 values equal to one and n values each of 1000, 999, 998... . It is fairly easy to find the maximum. Try n=1 (yields 924) n=2 (yields 942) n=3 (yields 947) and n=4 (yields 944). The maximum difference occered at n=3, so the answer is 947.
