## Task A-4.3.

How many complex numbers $z$ satisfy the following two conditions:

$$
|z|=1, \quad \operatorname{Re}\left(z^{100}\right)=\operatorname{Im}\left(z^{200}\right) \quad ?
$$
## First solution.

Due to the first condition, the required numbers are of the form $z=\cos \varphi+i \sin \varphi$,

and then the second condition becomes $\cos (100 \varphi)=\sin (200 \varphi)$.

Each solution of this equation in the interval $[0,2 \pi\rangle$ corresponds to one complex number that satisfies the conditions of the problem.

We can write the obtained equation in the form

$$
\cos (100 \varphi)=2 \sin (100 \varphi) \cos (100 \varphi)
$$

From this, it follows that

$$
\cos (100 \varphi)=0 \quad \text { or } \quad \sin (100 \varphi)=\frac{1}{2}
$$

Each of these equations has 200 solutions in the interval $[0,2 \pi\rangle$, 1 point

and these solutions are obviously distinct from each other.

Therefore, there are a total of 400 complex numbers that satisfy the given conditions.