## Task A-2.2.

Determine all complex numbers $z$ for which the ratio of the imaginary part of the fifth power of the number $z$ and the fifth power of the imaginary part of the number $z$ is the smallest possible.
## Solution.

Let $z=a+bi$, where $a, b \in \mathbb{R}$ and $b \neq 0$. Then,

$$
\operatorname{Im} z^{5}=\operatorname{Im}\left(a^{5}+5 a^{4} b i-10 a^{3} b^{2}-10 a^{2} b^{3} i+5 a b^{4}+b^{5} i\right)=5 a^{4} b-10 a^{2} b^{3}+b^{5}
$$

We want to minimize the following expression

$$
\frac{\operatorname{Im} z^{5}}{(\operatorname{Im} z)^{5}}=\frac{5 a^{4} b-10 a^{2} b^{3}+b^{5}}{b^{5}}=5 \frac{a^{4}}{b^{4}}-10 \frac{a^{2}}{b^{2}}+1
$$

Introduce the substitution $x=\frac{a^{2}}{b^{2}}$. We are looking for the minimum value that the quadratic function $f:[0,+\infty\rangle \rightarrow \mathbb{R}$,

$$
f(x)=5 x^{2}-10 x+1
$$

Since $5 x^{2}-10 x+1=5(x-1)^{2}-4$, the desired minimum value is -4 and it is achieved for $x=1$.

The desired numbers are all numbers of the form $z=a(1 \pm i), a \in \mathbb{R} \backslash\{0\}$.