11.1. Each of 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them thought of some integer. Then the first said: “My number is greater than 1”, the second said: “My number is greater than $2 ”, \ldots$, the tenth said: “My number is greater than 10”. After that, all ten, speaking in some order, said: “My number is less than 1”, “My number is less than $2 ”, \ldots$, “My number is less than 10” (each said exactly one of these ten phrases). What is the maximum number of knights that could have been among these 10 people?
Answer: 8 knights.

Solution. We will prove that none of the knights could have said either of the phrases "My number is greater than 9" or "My number is greater than 10." Indeed, if this were possible, the integer thought of by the knight would be at least 10. But then he could not have said any of the phrases "My number is less than 1," "My number is less than 2," ..., "My number is less than 10." Therefore, there could not have been more than eight knights.

We will show that there could have been 8 knights. Suppose the first knight thought of the number 2, the second -3, ..., the eighth -9, and the liars thought of the numbers 5 and 6. Then the $k$-th knight could have said the phrases "My number is greater than $k$" and "My number is less than $k+2$," while the liars could have said the phrases: one - "My number is greater than 9" and "My number is less than 1," and the other - "My number is greater than 10" and "My number is less than 2."

Remark. The example given above ceases to be valid if the liars think of numbers outside the interval $[1 ; 10]$, as then some of their statements become true.

Comment. Proved that there are no more than $9-$ 0 points.

Proved that there are no more than 8 knights (or, equivalently, at least two liars -3 points.

Provided an example showing that there could have been 8 knights, with a correct indication of which person said which phrase - 4 points.

If the example provided does not fully describe the situation (for example, it is not specified what numbers the liars thought of, or it is not clearly indicated who said which phrase) - out of 4 points for the example, no more than 2 points are given.