9.5. After watching the movie, viewers rated it one by one with an integer score from 0 to 10. At any given time, the movie's rating was calculated as the sum of all the given scores divided by their number. At some point in time $T$, the rating became an integer, and then with each new voting viewer, it decreased by one. What is the maximum number of viewers who could have voted after moment $T$?

(O. Dmitriev, R. Zhenodarov)
Answer: 5.

Solution. Consider a moment when the rating has decreased by 1. Suppose that before this, $n$ people had voted, and the rating was an integer $x$. Thus, the sum of the scores was $n x$. Let the next viewer give $y$ points. Then the sum of the scores becomes $n x + y = (n + 1)(x - 1)$, from which $y = x - n - 1$. The maximum possible value of $x$ is 10, and the minimum possible value of $n$ is 1; therefore, the maximum value of $y$ (on the first such step) is 8.

With each subsequent step, the value of $x$ decreases by 1, and the value of $n$ increases by 1. Consequently, on the second step, the value of $y$ does not exceed 6, on the third step - 4, and so on. Since any rating is not less than 0, the number of steps does not exceed 5.

It remains to show that five steps are possible. Suppose that at moment $T$ the rating is 10 (with 1 voter), then the second viewer gives 8 points, the third - 6, the fourth - 4, the fifth - 2, and the sixth - 0. Then the rating sequentially takes the values $9, 8, 7, 6,$ and 5.

Comment. Only the answer - 0 points.

Only an example is provided where 5 viewers voted - 2 points.

It is proven with full justification that more than 5 viewers could not have voted (but the example is missing or incorrect) - 5 points.