What is the maximum number of edges of a regular p-gonal prism that a plane, not passing through the vertices of the prism, can intersect?
And what is the maximum number of faces that can be intersected

## Solution

Note that the intersection of a prism and a plane is a polygon, and its vertices correspond to the edges of the prism intersected by the plane, while its sides correspond to the faces of the prism intersected by the plane. Therefore, the plane intersects an equal number of edges and faces of the prism. To find the maximum possible number of faces intersected by the plane, consider a pair of parallel lines lying on different bases of the prism and cutting off small triangles from the bases, such that the vertices of these triangles do not lie at the ends of the same edge of the prism. Draw a plane through these two lines. It is not difficult to see that it will intersect all the faces of the prism, that is, 2 bases and \(n\) lateral sides. From the above, it follows that the answer to the problem is \(n+2\).

| Problem $\underline{35620}$ Topics: $\left.\begin{array}{ll}\text { Area (other) }\end{array}\right]$ | Difficulty: 3 |
| :--- | :--- |
| $[\underline{\text { Area of a quadrilateral }}]$ | Classes: 9,10 |

On a plane, there is a line $\mathrm{m}$ and two polygons - $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$. It is known that any line parallel to line m intersects these polygons along segments of equal length. Prove that the areas of polygons $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ are equal.

## Hint

Divide the polygons into trapezoids by lines parallel to line m. Prove the statement of the problem separately for each trapezoid.

## Solution

Draw a line parallel to $\mathrm{m}$ through each vertex of polygons $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$. These lines will divide polygons $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ into several trapezoids (the two outermost trapezoids may degenerate into triangles). Consider two trapezoids corresponding to polygons $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$, enclosed between adjacent parallel lines. In these trapezoids, the corresponding bases are equal (as segments of lines parallel to m intersecting polygons $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$) and the heights are equal. It follows that their areas are equal. Thus, our polygons $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ are composed of trapezoids with equal areas. This means that the areas of polygons $\mathrm{M}_{1}$ and $\mathrm{M}_{2}$ are equal.

## 3

Problem