Kozhevnikov P.A.

All edges of a regular quadrilateral pyramid are equal to 1, and all vertices lie on the lateral surface of an infinite right circular cylinder of radius $R$. Find all possible values of $R$.
Let $S A B C D$ be the given pyramid ( $A B C D$ is a unit square with center $O$ ). Immediately note that $O$ is at a distance of $\frac{\sqrt{2}}{2}$ from all vertices of the pyramid (by considering the point $S^{\prime}$, symmetric to $S$ relative to $O$, the pyramid can be extended to a regular octahedron with vertices $A, B, C, D, S, S^{\prime}$ and center $O$). Let the vertices of the pyramid lie on a cylinder of radius $R$ with axis $l ; w$ - a circular section of the cylinder ( $w$ has radius $R$ ), passing through $S$. Consider $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$ - the points on the circle $w$, which are the projections of points $A, B, C, D$ onto the plane of the circle $w$. Let $A B$ not be parallel to $l$, that is, $A^{\prime} \neq B^{\prime}$. Since $\overrightarrow{A B}=\overrightarrow{D C}$, then $\overrightarrow{A^{\prime} B^{\prime}}=\overrightarrow{D^{\prime} C^{\prime}}$, so either $A^{\prime} = D^{\prime}$ and $B^{\prime} = C^{\prime}$, or $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is a rectangle inscribed in the circle $w$. Consider these two cases separately.

1) If $A^{\prime} = D^{\prime}$ and $B^{\prime} = C^{\prime}$, then the edges $A D$ and $B C$ are parallel to $l$, and thus $A^{\prime} B^{\prime} S$ is a section of the pyramid, perpendicular to $A D$ (see Fig. a). Then $A^{\prime}$ and $B^{\prime}$ are the midpoints of the edges $A D$ and $B C$, $A^{\prime} B^{\prime} = A B = 1$, $O$ is the midpoint of $A^{\prime} B^{\prime}$, $S A^{\prime}$ and $S B^{\prime}$ are the heights of the faces $S A D$ and $S B C$, that is, $S A^{\prime} = S B^{\prime} = \frac{\sqrt{3}}{2}$. We find $R = \frac{S A^{\prime}}{2 \sin S B^{\prime} A^{\prime}} = \frac{3}{4 \sqrt{2}}$: Conversely, if a circle $w$ is described around the midsection $S A^{\prime} B^{\prime}$ of the pyramid and a right circular cylinder with the directrix $w$ is taken, then $A D$ and $B C$ will be the generators of the cylinder, that is, the above construction is possible.

![](https://cdn.mathpix.com/cropped/2024_05_06_1b164f828743b46f847cg-14.jpg?height=858&width=528&top_left_y=0&top_left_x=159)

Fig. a

![](https://cdn.mathpix.com/cropped/2024_05_06_1b164f828743b46f847cg-14.jpg?height=683&width=526&top_left_y=90&top_left_x=1372)

Fig. b

2) If $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ is a rectangle inscribed in the circle $w$, then its center $O^{\prime}$ is the projection of $O$ onto the plane $w$ and is the center of the circle $w$. Thus, $O$ lies on the axis of the cylinder $l$. Note that all vertices of the pyramid lie on a sphere with center $O$ and radius $O S$, which intersects the cylinder along a pair of circles $w$ and $w^{\prime}$, symmetric relative to $O$ ( $O$ does not lie in the plane $w$ ). $A$ and $C$ are symmetric relative to $O$, so one of the vertices $A$ and $C$ lies on the circle $w$. Similarly, either $B$ or $D$ lies on $w$. In any case, $w$ is the circumscribed circle of one of the equilateral triangles $A B S, B C S, C D S$, $D A S$ (for example, see Fig. b), so its radius is $\frac{1}{\sqrt{3}}$. Conversely, in a regular octahedron with vertices $A, B, C$, $D, S, S^{\prime}$, the circles $w$ and $w^{\prime}$, circumscribed around the equilateral triangles $A B S$ and $C D S^{\prime}$, lie on the circumscribed sphere around the octahedron and are symmetric relative to the center of the sphere $O$, so $w$ and $w^{\prime}$ belong to the same circular cylinder, that is, the construction considered in the second case is possible.

## Answer

$\frac{3}{4 \sqrt{2}}, \frac{1}{\sqrt{3}}$