10.1. Each of the 10 people is either a knight, who always tells the truth, or a liar, who always lies. Each of them thought of some number (not necessarily an 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?

(O. Podlipsky)
Answer: 9 knights.

Solution. Estimation. Note that none of the knights could have said the phrase "My number is greater than 10," otherwise the number they thought of would indeed be greater than 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 is at least one liar, and there are no more than 9 knights.

Example. Let's show that there could have been 9 knights. Suppose the first person thought of the number 1.5, the second -2.5, ..., the ninth -9.5, and the tenth person thought of the number 5. Then for $k=1,2, \ldots, 9$, the $k$-th person could have said the true phrases "My number is greater than $k$" and "My number is less than $k+1$" (i.e., he is a knight), while the tenth person is a liar, saying the phrases "My number is greater than 10" and "My number is less than 1."

Remark. The example given above ceases to be valid if the tenth person thinks of a number outside the interval $(0 ; 10)$, as then exactly one of his answers is true.

Comment. It is proven that there are no more than 9 knights (or, equivalently, there is at least one liar) - 3 points.

The example showing that there could have been 9 knights, with the indication of which person said which phrases - 4 points.

If the example provided is not fully described (for example, it is not specified what number the liar 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.