1. Vasya wrote natural numbers on the pages of an 18-sheet notebook. On each page, he wrote no fewer than 10 different numbers, and on any three consecutive pages, no more than 20 different numbers. What is the maximum number of different numbers Vasya could have written on the pages of the notebook?

(A. Khryabrov)
# Answer: 190.

Solution. Consider two consecutive pages, neither of which is the first page. Let their numbers be $n$ and $n+1$. Then on page $n-1$, there are at least 10 different numbers, and on pages $n-1, n$, and $n+1$, there are no more than 20 different numbers. Therefore, pages $n$ and $n+1$ add no more than 10 new numbers to the numbers on the previous pages.

On the first three pages, Vasya wrote no more than 20 different numbers. On the fourth and fifth pages, no more than 10 new numbers, on the sixth and seventh pages, no more than 10 new numbers, and so on. On the last three pages, no more than 20 different numbers. In total, we get that Vasya has no more than $20 + 15 \cdot 10 + 20 = 190$ different numbers in his notebook. Let's provide an example showing that Vasya could write 190 different numbers. On the first page, write the numbers from 1 to 10, on the second and third pages - the numbers from 11 to 20, on the fourth and fifth pages - the numbers from 21 to 30, and so on. On pages numbered $2k$ and $2k+1$, write the numbers from $10k+1$ to $10k+10$. It is easy to see that on any three consecutive pages, there are no more than 20 different numbers, and in total, the numbers from 1 to 190 are written.