The complex numbers $z=x+i y$, where $x$ and $y$ are integers, are called Gaussian integers. Determine the Gaussian integers that lie inside the circle described by the equation

$$
x^{2}+y^{2}-4 x-10 y+20=0
$$

in their geometric representation.
The equation of the circle in question can also be written as:

$$
(x-2)^{2}+(y-5)^{2}=3^{2}
$$

thus the coordinates of the center are $(2,5)$ and the radius of the circle is 3.

The points $(x, y)$ inside the circle lie such that their distance from the center of the circle is less than the radius of the circle, which means:

$$
(x-2)^{2}+(y-5)^{2}<9
$$

An essential condition for (1) is that:

$$
(x-2)^{2}<9
$$

and

$$
(y-5)^{2}<9
$$

thus

$$
-1<x<5
$$

and

$$
2<y<8
$$

According to this, $x$ can take the values $0,1,2,3,4$.

If, for example, $x=0$, then from (1) we get

$$
(y-5)^{2}<9
$$

so $y$ can take any value between 3 and 7. Similarly, we can calculate the $y$ values corresponding to the other $x$ values, thereby obtaining all the Gaussian integers that satisfy the problem:

$$
\begin{array}{rrrrr}
3 i & 4 i & 5 i & 6 i & 7 i ; \\
1+3 i & 1+4 i & 1+5 i & 1+6 i & 1+7 i \\
2+3 i & 2+4 i & 2+5 i & 2+6 i & 2+7 i \\
3+3 i & 3+4 i & 3+5 i & 3+6 i & 3+7 i \\
4+3 i & 4+4 i & 4+5 i & 4+6 i & 4+7 i
\end{array}
$$

(Gustáv Kertész, Pécs.)

The problem was also solved by: Bartók I., Biró A., Dömény E., Dömény I., Enyedi B., Haar A., Kürti I., Liebner A., Pichler S., Pivnyik I., Riesz K., Rosenberg J., Schöffer I., Szücs A.