5. In a hat, there are 11 slips of paper with numbers $0,1,2,3,4,5,6,7,8,9,10$ (one with each number). Each of the eleven students takes one slip. Then they announce in alphabetical order the sentence "so far, at least $k$ students have told a lie," where $k$ is the number on the drawn slip. How many of these 11 statements can be true at most?
## Solution.

If the statement of a student who drew $k \geq 6$ is true, then at least 6 students before that student gave false statements, so the maximum number of students who gave true statements is $11-6=5$.

If all statements of students who drew $k \geq 6$ are false, the maximum number of true statements is 6.

Let's show that it is possible to arrange the numbers in such a way that there are exactly 6 true statements.

Let the students, in alphabetical order, have drawn the following numbers:

$$
0,10,1,9,2,8,3,7,4,6,5
$$

Then the first, third, fifth, seventh, ninth, and eleventh students gave true statements, while the others were lying.

Therefore, the maximum number of true statements can be 6.