Problem 11.1. Find all values of $a$ such that the equation

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

has a unique solution.

Alexander Ivanov, Emil Kolev
11.1. Setting $y=2^{x}$ we have to find all values of $a$ such that the equation

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

has exactly one positive root. Obviously $a=0$ is not a solution. For $a \neq 0$ the equation has a positive root $y_{1}=a^{2}$ It is unique if either $y_{2}=3 a-2 \leq 0$ or $y_{2}=y_{1}$. In the first case we obtain $a \leq \frac{2}{3}$ and in the second one we have $a^{2}=3 a-2$, whence $a_{1}=1$ and $a_{2}=2$.

Finally, $a \in(-\infty ; 0) \cup\left(0 ; \frac{2}{3}\right) \cup\{1\} \cup\{2\}$.