1. Find all roots of the equation

$$
a^{2} \cdot \frac{x-b}{a-b} \cdot \frac{x-c}{a-c}+b^{2} \cdot \frac{x-a}{b-a} \cdot \frac{x-c}{b-c}+c^{2} \cdot \frac{x-a}{c-a} \cdot \frac{x-b}{c-b}=x^{2}
$$

where $a \neq b \neq c$ are arbitrary given values.
# Solution

Note that the degree of the equation does not exceed two.

It is easy to verify that $a, b, c$ are roots of the equation:

$$
\begin{aligned}
& a^{2} \cdot \frac{a-b}{a-b} \cdot \frac{a-c}{a-c}+b^{2} \cdot \frac{a-a}{b-a} \cdot \frac{a-c}{b-c}+c^{2} \cdot \frac{a-a}{c-a} \cdot \frac{a-b}{c-b}=a^{2} \\
& a^{2} \cdot \frac{b-b}{a-b} \cdot \frac{b-c}{a-c}+b^{2} \cdot \frac{b-a}{b-a} \cdot \frac{b-c}{b-c}+c^{2} \cdot \frac{b-a}{c-a} \cdot \frac{b-b}{c-b}=b^{2} \\
& a^{2} \cdot \frac{c-b}{a-b} \cdot \frac{c-c}{a-c}+b^{2} \cdot \frac{c-a}{b-a} \cdot \frac{c-c}{b-c}+c^{2} \cdot \frac{c-a}{c-a} \cdot \frac{c-b}{c-b}=c^{2}
\end{aligned}
$$

An equation of degree not higher than two has three distinct roots if and only if it is degenerate into an identity, meaning the answer to the problem is any number $(x \in \mathbb{R})$.

Answer: $x$ - any real number.