Agakhanovo $H . X$.

Different numbers $a, b$ and $c$ are such that the equations $x^{2}+a x+1=0$ and $x^{2}+b x+c=0$ have a common real root. In addition, the equations $x^{2}+x+a=0$ and $x^{2}+c x+b=0$ have a common real root. Find the sum $a+b+c$.
The common root $x_{1}$ of the first two equations satisfies the equation $(a-b) x_{1}+(1-c)=0$, that is, $x_{1}={ }^{a-b} /{ }_{c-1}$. Similarly, the common root $x_{2}$ of the last two equations is ${ }^{c-1} / a-b$. Since $x_{1} x_{2}=1$, then $x_{2}-$ is a root of the first equation, that is - the common root of the equations $x^{2}+a x+1=0$ and $x^{2}+x+a=0$. Hence, $(a-1)\left(x_{2}-1\right)=$ 0. But for $a=1$ these equations do not have real roots. Therefore, $x_{2}=1$, then $a=-2, b+c=-1$.