Problem 1. Sasha wrote down the numbers from one to one hundred, and Misha erased some of them. Among the remaining numbers, 20 numbers have a one in their notation, 19 numbers have a two in their notation, and 30 numbers have neither a one nor a two. How many numbers did Misha erase?

$[4$ points] (A. V. Shapovalov)
Answer: 33.

Solution: Among the numbers from 1 to 100, the digit 1 appears exactly twenty times: the number 1 itself, ten numbers from 10 to 19, the numbers $21, 31, \ldots, 91$ (eight of them), and the number 100. Therefore, none of these numbers were erased. Similarly, the digit 2 appears exactly nineteen times: the number 2 itself, ten numbers in the third decade, and $12, 32, 42, \ldots, 92$ (eight such numbers). Thus, none of these were erased by Misha either. In total, there are $19+20-2=37$ such numbers (we subtract 2 because the numbers 12 and 21 were counted twice). In total, $37+30=67$ numbers remained, and Misha erased $100-67=33$ numbers.