A plane can intersect at most how many edges of a pyramid with an $n$-sided base.
Solution. We will show that the sought value is $[3 n / 2]$.

A plane $S$ intersects an edge of the pyramid if and only if the two endpoints of the edge are on different sides of $S$. The plane $S$ intersects the base plane $A$ of the pyramid in some line $e$. If the two open half-planes in $A$ determined by $e$ contain $k$ and $\ell$ vertices of the pyramid, respectively, where $k \leq \ell$, then $k+\ell \leq n$ and equality holds only if $e$ does not contain any vertices. Therefore, $k \leq n / 2$ always holds. Then $S$ intersects at most $2 k$ edges of the base, because each edge intersected by $e$ has one endpoint among the $k$ vertices and each vertex is the endpoint of two edges. $S$ intersects the lateral edges of the pyramid in $k$, $\ell$, or 0 edges, depending on whether the apex of the pyramid is on one side of $S$, the other side, or lies on $S$.

Thus, $S$ intersects at most

$$
2 k+\ell=k+(k+\ell) \leq k+n \leq 3 n / 2
$$

edges of the pyramid, and since the number of intersection points is an integer, it follows that this number is at most $[3n/2]$.

![](https://cdn.mathpix.com/cropped/2024_05_02_40828351f28f552f1703g-1.jpg?height=371&width=788&top_left_y=698&top_left_x=663)

Figure 1

We will show that this value can be achieved. For this, we first construct for every integer $n \geq 3$ an $n$-gon and a line $e$ such that $e$ intersects every side of the polygon if $n$ is even, and $n-1$ sides if $n$ is odd. Such polygons can be seen in Figure 1 ($n$ odd) and Figure 2 ($n$ even), and the construction is evident from the figures. This polygon will be the base polygon of the pyramid. Then we take the apex $C$ of the pyramid and consider a plane $S$ that intersects the base plane in $e$ and has $C$ on the side where at most, i.e., $[n / 2]$, vertices of the base polygon are located. Then $S$ intersects $2 m$ base edges and $m$ lateral edges, i.e., a total of $2 m+m=3 n / 2$ edges, if $n=2 m$, and $2 m$ base edges and $m+1$ lateral edges, i.e., a total of $2 m+(m+1)=3 n / 2-1 / 2$ edges, if $n=2 m+1$. This value coincides with $[3 n / 2]$ in both cases.

![](https://cdn.mathpix.com/cropped/2024_05_02_40828351f28f552f1703g-1.jpg?height=594&width=935&top_left_y=1492&top_left_x=584)

Figure 2

Remark. If the base of the pyramid is a convex $n$-gon, then any plane intersects at most $n+1$ edges of the pyramid. This can be shown in the same way as we showed in the solution of problem B. 4294 (see page 221 of our April 2011 issue) that a plane intersects at most 12 edges of a 12-faced convex body. The two problems were set in consecutive months, and the pyramid problem was considered as a continuation of the prism problem. Therefore, solvers who solved the problem only for the case of a convex base polygon received 0 points.