Head A.C.

Dima calculated the factorials of all natural numbers from 80 to 99, found the numbers that are the reciprocals of these factorials, and printed the resulting decimal fractions on 20 infinite tapes (for example, on the last tape, the number $\frac{1}{99!}=0, \underbrace{00 \ldots 00}_{155 \text { zeros }} 10715$.. was printed). Sasha wants to cut a piece from one tape, on which $N$ digits are written in a row without a comma. What is the largest $N$ for which he can do this so that Dima cannot determine from this piece which tape Sasha has damaged?
$N=155$. Let on ribbons, on which numbers $1 / k!$ and $1 / l!$ ( $k < l$ ) are written, there are pieces of 155 characters that are indistinguishable. Since $1 / 100! < \frac{1}{10^{156}}$, therefore $N<156$.

Thus, a piece of 156 characters is always sufficient to determine from which strip it was cut. On the other hand, on the strips with the numbers 1/98! and 1/99! there are identical pieces of 155 characters: $1 / 99!=0, \underbrace{00}_{155 \text { zeros! }}$...00

![](https://cdn.mathpix.com/cropped/2024_05_06_caf98da37e6d9925b347g-55.jpg?height=125&width=2032&top_left_y=352&top_left_x=0)
on both strips there is a piece $\underbrace{00 \ldots 00}_{153 \text { zeros }} 10$.