# Problem 1. Maximum 16 points

Solve the equation for $a \neq 0 ; a \neq \pm 1$

$\frac{\left(x-a^{2}\right)\left(x-a^{3}\right)}{a(a+1)(a-1)^{2}}-\frac{(x-a)\left(x-a^{3}\right)}{a(a-1)^{2}}+\frac{(x-a)\left(x-a^{2}\right)}{(a+1)(a-1)^{2}}=x^{3}$
# Solution

The left side of the equation is a polynomial $P(x)$ of degree no higher than two.

We compute $P(a)=a ; P\left(a^{2}\right)=a^{2} ; P\left(a^{3}\right)=a^{3}$.

Notice that for $a \neq 0 ; a \neq \pm 1$, the numbers $a ; a^{2} ; a^{3}$ are distinct.

From this, it follows that $P(x)=x$.

It remains to solve the equation $x=x^{3}$, whose solutions are $x_{1}=0 ; x_{2,3}= \pm 1$.

Answer: $x_{1}=0 ; x_{2,3}= \pm 1$.

## Grading Criteria

Correct answer and correct justification - 16 points.

Correct answer, but insufficiently justified (for example, it is claimed that the equation takes the form $x=x^{3}$ without justification) - 8 points.

The equation $x=x^{3}$ is obtained, there is justification, but the root $x_{1}=0$ is lost - 6 points. The equation $x=x^{3}$ is obtained, but there is no justification; if the root $x_{1}=0$ is lost - 3 points

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