1. On each page of the book, its number is written. The page numbering starts from one. Vasya tore out all the even-numbered sheets from the book (there are two pages on each sheet of the book). The numbers of the remaining pages in the book together contain exactly 845 digits. How many pages could there have been in the book initially? List all possible options and prove that there are no others. (20 points)
Solution: Among every four consecutive pages, two pages with numbers that have remainders of 1 and 2 when divided by 4 remain. The remaining pages have single-digit numbers 1, 2, 5, 6, 9, double-digit numbers $10, 13, 14, 17, \ldots, 97, 98$, and some three-digit numbers $101, 102, 105, 106, \ldots$. Among the first 100 pages, exactly half remain, five of which are single-digit, meaning 45 double-digit pages remain. For the numbering of the remaining pages up to 100, $5 + 45 \cdot 2 = 95$ digits were used, so 750 digits were used for the three-digit numbers, which means a total of $750 : 3 = 250$ three-digit numbers remain. In each set of four numbers, exactly two three-digit numbers remain, so the last two remaining numbers will be in the 125th set of four - these are the numbers $101 + 124 \times 4 = 597$ and $102 + 124 \times 4 = 598$. The next two pages could have been torn out, in which case there were 600 pages in total, or they could have not been torn out, in which case there were 598 pages.

## Answer: 598 and 600