[ [ Evaluation + example $\quad]$

In a corridor 100 meters long, 20 carpet strips with a total length of 1000 meters are laid. What is the maximum number of uncovered segments (the width of the strip is equal to the width of the corridor)?

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First, let's provide an example: take eleven long paths, each 90.5 meters long, and the remaining nine short paths, each 0.5 meters long. Place the eleven long paths on top of each other, leaving a 0.5-meter gap from the edge of the corridor, and in the remaining 9 meters of the corridor, leave the first half empty and cover the second half with a short path. Thus, there are 9 uncovered segments before each short path and 1 before the long ones.

Now, let's show that it is impossible to achieve more uncovered segments. First, note that the difference between the number of uncovered segments and the number of covered segments, in absolute value, does not exceed one. Therefore, it is sufficient to prove that the number of covered segments cannot exceed ten.

For each covered segment, find the number of carpet paths used in that segment. Let the maximum of these numbers be \( N \). Then the number of covered segments does not exceed \( 1 + (20 - N) = 21 - N \). Thus, it remains to prove that \( N \geq 11 \). To do this, we need to show that there is a point covered by at least eleven paths. Suppose that each point is covered by no more than ten paths. Since the total length of the paths is exactly ten times the length of the corridor, the entire corridor is covered, and thus there are no uncovered segments, which is not the case we are interested in. Therefore, there is a point covered by at least eleven paths. Consequently, in the corresponding covered segment, at least eleven paths are used, so \( N \geq 11 \).

## Answer

11 segments.

Can 100,000 six-digit phone numbers be chosen such that when the \( k \)-th digit ( \( k=1,2, \ldots 6 \) ) is simultaneously erased from all these numbers, all five-digit numbers from 00000 to 99999 are obtained?

## Solution

We will prove that it is sufficient to choose all phone numbers for which the sum of the digits is divisible by ten. First, note that when the \( k \)-th digit is simultaneously erased from all these numbers, all five-digit numbers are obtained, and each exactly once. Indeed, the number \(\overline{a_{1} a_{2} a_{3} a_{4} a_{5}}\) is obtained by erasing the digit \( d \) if and only if the number \( d + s \), where \( s = a_{1} + \ldots + a_{5} \), is divisible by ten, and for any integer \( s \), there is exactly one digit \( d \) such that \( s + d \) is divisible by ten. Therefore, the total number of selected numbers is equal to the number of five-digit numbers, i.e., 100,000. Thus, we have found 100,000 numbers that meet the requirements of the problem.

## Answer

Yes, it is possible.