9.1. It is known that $a^{2}+b=b^{2}+c=c^{2}+a$. What values can the expression

$$
a\left(a^{2}-b^{2}\right)+b\left(b^{2}-c^{2}\right)+c\left(c^{2}-a^{2}\right) ?
$$

take?
Answer: 0.

Solution. From the condition, it follows that $a^{2}-b^{2}=c-b$, $b^{2}-c^{2}=a-c$, and $c^{2}-a^{2}=b-a$. Therefore, $a\left(a^{2}-b^{2}\right)+b\left(b^{2}-c^{2}\right)+c\left(c^{2}-a^{2}\right)=a(c-b)+b(a-c)+c(b-a)=0$.

## Grading Criteria:

+ a complete and justified solution is provided

the correct answer is obtained by considering specific examples

- only the answer is provided
- the problem is not solved or is solved incorrectly