1A. Given are two quadratic equations $x^{2}+a x+1=0$ and $x^{2}+x+a=0$. Determine all values of the parameter $a$ for which the two equations have at least one common solution.
Solution. Subtracting the second equation from the first, we get

$$
(a-1) x+1-a=0 \Leftrightarrow(a-1)(x-1)=0
$$

If $a=1$, then we have the same equation, so there is at least one common solution.

If $a \neq 1$, then $x=1$, and by substitution into the equations, we get $a=-2$. Therefore, the equations have at least one common solution only when $a=1$ and $a=-2$.

2AB. In the set of real numbers, solve the system of equations

$$
\left\{\begin{array}{l}
2 a^{2}-2 a b+b^{2}=a \\
4 a^{2}-5 a b+2 b^{2}=b
\end{array}\right.
$$