10.2. Find all values of parameters $a, b, c$, for which the system of equations $\left\{\begin{array}{l}a x+b y=c \\ b x+c y=a \\ c x+a y=b,\end{array}\right\}$

has at least one negative solution (when $x, y<0$).
Answer. When $a+b+c=0$.

Solution. Assume the contrary, that $a+b+c \neq 0$ and the system has a negative solution. Add all the equations: $(a+b+c)(x+y)=a+b+c$ and divide by $a+b+c \neq 0$, we get
$x+y=1$, which is impossible for $x, y<0$. Therefore, the condition $a+b+c=0$ is necessary. If this condition is satisfied, then, as it is easy to verify, $x=y=-1$ is the desired negative solution of the system.

Criterion. Necessity of the condition $a+b+c=0: 4$ points.

Sufficiency of it: 3 points.