## Task A-4.4.

Determine the number of complex solutions to the equation

$$
z^{2019}=z+\bar{z}
$$
## Solution.

Introduce the trigonometric form of the complex number $z=|z|(\cos \varphi+i \sin \varphi)$, for $|z| \in \mathbb{R}_{+}$, $\varphi \in[0,2 \pi)$. The equation can then be written as:

$$
|z|^{2019}(\cos (2019 \varphi)+i \sin (2019 \varphi))=2|z| \cos \varphi
$$

1 point

Notice that $z=0$ is one solution. Let $z \neq 0$ from now on.

1 point

By equating the real and imaginary parts of both sides and dividing both equations by $|z|$, we get

$$
\begin{aligned}
|z|^{2018} \cos (2019 \varphi) & =2 \cos \varphi \\
\sin (2019 \varphi) & =0
\end{aligned}
$$

From the second equation, we conclude that

$$
\varphi=\frac{k \pi}{2019}, \quad k \in\{0, \ldots, 4037\}
$$

For such $\varphi$, we observe that the value of the expression $\cos (2019 \varphi)$ is 1 when $k$ is even, and -1 when $k$ is odd. In short, we can write $\cos (2019 \varphi)=(-1)^{k}$.

Substituting this into the above equation, we get

$$
|z|^{2018}(-1)^{k}=2 \cos \frac{k \pi}{2019}
$$

or

$$
|z|^{2018}=2(-1)^{k} \cos \frac{k \pi}{2019}
$$

1 point

This equation has a solution for $|z|$ (and thus for $z$) when the right-hand side is positive, i.e., if and only if $k \in\{0,2,4, \ldots, 1008,3030,3032, \ldots, 4036\}$ or $k \in\{1011,1013, \ldots, 3027\}$.

Counting these solutions, together with the solution $z=0$, we see that the total number of solutions is 2019.