[ Coloring ]

A wooden cube was painted white on the outside, each of its edges was divided into 5 equal parts, after which the cube was sawn so that small cubes were obtained, the edge of which is 5 times smaller than that of the original cube. How many small cubes are there with at least one face painted? 

#
It is easier to count the number of cubes that have no faces painted.

## Solution

The number of small cubes obtained after cutting a large cube is $5^{3}=125$. Let's count the number of cubes that have no faces painted. The unpainted cubes are all those that do not have any visible outer faces. These cubes form a $3 \times 3 \times 3$ cube, and their number is $3^{3}=27$. Finally, the number of cubes that have at least one face painted is $125-27=98$.

## Answer

98.00

Does there exist a) a bounded, b) an unbounded figure in the plane that has among its axes of symmetry two parallel non-coincident lines?

## Hint

If such a figure exists, it must map onto itself under a non-identity parallel translation, which is defined as the sequential execution of symmetries relative to two parallel axes of symmetry.

## Solution

a) Suppose a figure has two parallel axes of symmetry. Introduce a coordinate system such that these lines have equations $x=0$ and $x=1$. Let some point $(a; b)$ belong to the given figure. Since the line $x=0$ is an axis of symmetry, the point $(-a; b)$ must also belong to the given figure. Next, the point $(a+2; b)$, which is symmetric to the point $(-a; b)$ relative to the line $x=1$, must also belong to the given figure. Thus, together with each point $(a; b)$, the figure contains the point $(a+2; b)$. Repeating these arguments for the point $(a+2; b)$, we get that the point $(a+4; b)$ also belongs to the figure, and so on, the points $(a+6; b)$, $(a+8; b)$, $(a+10; b)$, ... belong to the figure. This obviously contradicts the assumption of the boundedness of the figure. b) An example is a line (any line perpendicular to it is its axis of symmetry).

## Answer

a) No. b) Yes.