1989 AIME Problems/Problem 11

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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 $D^{}_{}$ be the difference between the mode and the arithmetic mean of the sample. What is the largest possible value of $\lfloor D^{}_{}\rfloor$? (For real $x^{}_{}$, $\lfloor x^{}_{}\rfloor$ is the greatest integer less than or equal to $x^{}_{}$.)

Solution

It is obvious that there will be $n+1$ values equal to one and $n$ values each of $1000, 999, 998 \ldots$. It is fairly easy to find the maximum. Try $n=1$, which yields $924$, $n=2$, which yields $942$, $n=3$, which yields $947$, and $n=4$, which yields $944$. The maximum difference occurred at $n=3$, so the answer is $947$.

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See also

1989 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions