120. Cross out 100 digits from the number

$$
12345678910111213 \ldots 979899100
$$

so that the resulting number is
a) as large as possible;
b) as small as possible.
120. In this number, there are 192 digits; in the number obtained from it by erasing 100 digits - 92 digits.

a) The first digits of the number we are interested in should be as large as possible. We can make the number start with 5 nines, borrowing these nines from the numbers $9, 19, 29, 37$, and 49; in this case, we will have to erase $8+19+19+19+19=84$ digits. After this, the next digit cannot be made a nine, as for this we would have to erase another 19 digits (to "reach" the nearest nine - the digit of the number 59); in this case, 103 digits will be erased, i.e., more than needed.

The nearest eight to the remaining digits belongs to the number 58; to make it follow immediately after the nines, we would have to erase another 17 digits, which is also impossible (since $84+17=101>100$). Therefore, the best we can do is to erase another 15 digits preceding the seven in the number 57. Thus, so far we have erased a total of $84+15=99$ digits; we are allowed to erase one more digit. Clearly, it is advisable to erase the five in the number 58; thus, the required number is:

b) We can achieve that the remaining number starts with 5 zeros (these zeros appear in the numbers 10, 20, 30, 40, and 50); in this case, 86 digits will be erased. We cannot reach the next zero (in the number 60); however, by erasing just one five, we can make it so that after 5 zeros in the obtained number, a one follows. The next digit after this cannot be made a one either; however, by erasing another five, we can make the next digit a two. This reasoning can be continued; thus, we will see that by erasing a total of $86+1+1+1+1=90$ digits, we will arrive at a number starting with the combination of digits: 000001234, followed by the digits $55565758596061 \ldots$ Clearly, the digit following the first nine digits written above in the resulting number can be a five, but it cannot be less than 5; to make it a five, we do not need to erase any more digits. After this, we finally get the opportunity to reach a zero by erasing 10 preceding digits (which is still allowed); therefore, the number we are interested in is:

$00000123450616263 \ldots .100$

(this number starts with five zeros, so it actually contains 87 matching digits).