Chebotarev A.S.

On a plane, there is a circle. What is the minimum number of lines that need to be drawn so that, by symmetrically reflecting the given circle relative to these lines (in any order a finite number of times), it can cover any given point on the plane?
1) We will prove that three lines are sufficient. Let the horizontal line $a$ and line $b$, forming a $45^{\circ}$ angle with line $a$, contain the center $O$ of the circle, and let line $c$ be parallel to $a$ and be at a distance of $0.5R$ from it, where $R$ is the radius of the given circle (see figure).

The composition of axial symmetries with axes $a$ and $c$ is a parallel translation by a distance of $R$ (up or down, depending on the order of the symmetries). The composition of axial symmetries with axes $a$ and $b$ is a rotation around the center $O$ by an angle of $90^{\circ}$ (in either direction, depending on the order of the symmetries).

We will show that any point on the plane can be covered using the obtained transformations. For this, consider a Cartesian coordinate system with the origin at point $O$ and the $x$-axis coinciding with line $a$ (the $y$-axis passes through point $O$ perpendicular to $a$), where $R$ is the length of the unit segment. Let the point $M(x; y)$ need to be covered by the circle. The integers closest to $x$ and $y$ are denoted by $m$ and $n$ respectively, then $|x-m| \leq 0.5$ and $|y-n| \leq 0.5$.

If $m > 0$, then perform $m$ parallel translations upwards, then rotate around point $O$ by $-90^{\circ}$ and perform $n$ parallel translations upwards or downwards, depending on the sign of the number $n$. Similarly, if $m < 0$, then perform $|m|$ parallel translations downwards, then rotate around point $O$ by $-90^{\circ}$ and perform $n$ parallel translations upwards or downwards, depending on the sign of the number $n$. As a result of the performed transformations, the center of the circle will become point $O'(m; n)$, and

$$
O'M = \sqrt{(x-m)^2 + (y-n)^2} \leq \frac{\sqrt{2}}{2} < 1
$$

Therefore, point $M$ has been covered.

2) We will prove that two lines are insufficient. If the chosen lines are parallel, then the orthogonal projections of the circle onto these lines do not change when any number of reflections are performed. If the chosen lines intersect, then the distance from the point of intersection to the center of the circle does not change when any number of reflections are performed.

![](https://cdn.mathpix.com/cropped/2024_05_06_1560f9d9b3af2f34b251g-28.jpg?height=383&width=458&top_left_y=383&top_left_x=800)

## Answer

3.