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 $D^{}_{}$ 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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1989 AIME Problems/Problem 11

Problem

Solution

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