Problem 9.1. Find all real values of $a$ such that the system

$$
\left\lvert\, \begin{aligned}
& \frac{a x+y}{y+1}+\frac{a y+x}{x+1}=a \\
& a x^{2}+a y^{2}=(a-2) x y-x
\end{aligned}\right.
$$

has a unique solution.

Peter Boyvalenkov
9.1. If $x \neq-1$ and $y \neq-1$, we easily get that $y=a$. Plugging it in the second equation gives

$$
a x^{2}-\left(a^{2}-2 a-1\right) x+a^{3}=0
$$

If $a=0$ the system has a unique solution $(0 ; 0)$. If $a \neq 0$, we consider the following two cases.

Case 1. -1 is a root of $(*)$. Then (*) gives that $a=1$ or -1 . In both cases the system has no a solution.

Case 2. The equation (*) has a double root. Then $a=-1, a=1$ or $a=-\frac{1}{3}$. In the first two cases we get that $y=-1$ or $x=-1$, i.e., the system has no a solutions. If $a=-\frac{1}{3}$ the system has a unique solution $(x ; y)=\left(\frac{1}{3} ;-\frac{1}{3}\right)$.

Thus the desired values of $a$ are 0 and $-\frac{1}{3}$.