Question 11
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 $x.$)