8.3. On the island of knights and liars, each resident was asked about each of the others: is he a knight or a liar. In total, 42 answers of "knight" and 48 answers of "liar" were received. What is the maximum number of knights that could have been on the island? Justify your answer. (It is known that knights always tell the truth, while liars always lie.)
Solution: Each of the $n$ residents gave $n-1$ answers; in total, there were $n(n-1)$ answers, which, according to the problem, equals $42+48=90$. Hence, $n=10$, meaning there are 10 residents on the island. Let the number of knights be $x$, then the number of liars is $10-x$. Answers of "liar" arise in two cases: when a knight speaks about a liar (there are $x(10-x)$ such cases) and vice versa (the same number of cases). We have the equation $2 x(10-x)=48$, from which $x=6$ or $x=4$. Both options are possible.

Note: Instead of the equation $2 x(10-x)=48$, the equation $x(x-1)+(10-x)(9-x)=42$ (by the number of "knight" answers) or even an equation of the type $\frac{2 x(10-x)}{x(x-1)+(10-x)(9-x)}=\frac{48}{42}$ might arise. All of them are equally valid.

Answer: 6 knights.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| A correct approach to the solution resulted in an incorrect answer due to calculation errors | 6 points |
| A correct equation (system of equations) that fully describes the problem condition; this equation (system) is not solved | 4 points |
| The number of residents on the island is correctly found (in the absence of further progress) | 3 points |
| It is shown that the number of knights could have been 4 or 6 (at least one of these numbers) and it is not proven that there are no other options | 1 point |
| Answer without justification | 0 points |