Task B-2.7. Given is the complex number $w=-3-i$. Determine all complex numbers of the form $z=n^{2}+n i$, where $n \in \mathbb{Z}$ and the following conditions are satisfied:

$$
\operatorname{Re}\left(\frac{z}{w}\right)<-0.4, \quad \operatorname{Im}(z \cdot \bar{w})<4
$$
## Solution.

First, we determine the quotient $\frac{z}{w}$ and the product $z \cdot \bar{w}$.

$$
\frac{z}{w}=\frac{n^{2}+n i}{-3-i} \cdot \frac{-3+i}{-3+i}=\frac{-3 n^{2}-3 n i+n^{2} i-n}{10}=\frac{-3 n^{2}-n}{10}+\frac{-3 n+n^{2}}{10} i
$$

Its real part is $\operatorname{Re}\left(\frac{z}{w}\right)=\frac{-3 n^{2}-n}{10}$.

$$
z \cdot \bar{w}=\left(n^{2}+n i\right) \cdot(-3+i)=-3 n^{2}-n+\left(-3 n+n^{2}\right) i
$$

Its imaginary part is $\operatorname{Im}(z \cdot \bar{w})=-3 n+n^{2}$.

We need to solve the system of inequalities

$$
\left\{\begin{array}{l}
\operatorname{Re}\left(\frac{z}{w}\right)1
$$

The solution to the second inequality is all integers $n$ for which $-1<n<4$.

The intersection of these sets of solutions is the set $\{2,3\}$.

Therefore, the required complex numbers are $4+2 i$ or $9+3 i$.

## SCHOOL/CITY COMPETITION IN MATHEMATICS

## 3rd grade - high school - B variant

January 17, 2013.

IF A STUDENT HAS A DIFFERENT METHOD OF SOLVING THE PROBLEM, THE COMMITTEE IS OBLIGED TO GRADE AND EVALUATE THAT METHOD APPROPRIATELY.