Eva wrote down consecutive natural numbers: 1234567891011. . What digit did she write on the 2009th place?

(M. Volfová)
There are 9 single-digit numbers (1 to 9) and it takes 9 digits to write them. There are 90 two-digit numbers (10 to 99) and it takes 180 digits to write them. There are 900 three-digit numbers (100 to 999) and it takes a total of 2700 digits to write them. To write all single-digit and two-digit numbers, 189 digits are needed; the 2009th digit was used in some three-digit number.

The number of digits used to create three-digit numbers is $2009-189=1820$. Since $1820: 3=606$ (remainder 2), the 2009th digit in the sequence is the second digit of the 607th three-digit number. The first three-digit number is 100, the second three-digit number is $101=100+1$, the third three-digit number is $102=100+2 \ldots$ Similarly, the 607th three-digit number is $100+606=706$ and its second digit is 0. Therefore, Eva wrote 0 at the 2009th position.

Scoring. 1 point for determining the number of digits needed to write single-digit and two-digit numbers (i.e., 189); 3 points for determining that the sought digit is in the 607th three-digit number; 2 points for finding the digit 0 in the number 706.