# Problem 4.

On each face of a cube, a regular 4-sided pyramid is placed, with the base of the pyramid being this face of the cube. All pyramids are equal.

- 4A. Can the lateral edges of three pyramids, emanating from one vertex of the cube, lie in the same plane? If this is possible, find the heights of such pyramids, expressing them in terms of the length $a$ of the cube's edge. If this is impossible, provide a proof.
- 4B. Can the sets of three edges mentioned in 4A lie in planes (each set in its own plane) simultaneously for all vertices of the cube?
# Solution.

Let's denote an arbitrary vertex of the cube as $A$, and the vertices of the pyramids connected to it by edges as $O_{1}, O_{2}, O_{3}$.

We introduce a coordinate system, placing its origin at point $A$ and directing the axes $A X, A Y$, and $A Z$ along the edges of the cube. Let the base of the pyramid with vertex $O_{1}$ lie in the plane $A X Y$, the base of the pyramid with vertex $O_{2}$ in the plane $A X Z$, and the base of the pyramid with vertex $O_{3}$ in the plane $A Y Z$.

Let the edge of the cube be $2 c$, and the height of the pyramids be $h$. Then the coordinates of the vertices of the pyramids will be $O_{1}(c, c, -h), O_{2}(c, -h, c), O_{3}(-h, c, c)$.

Let $B$ be the vertex of the cube with coordinates $(2 c, 0, 2 c)$ (i.e., the diagonal $A B$ projects the vertex $O_{2}$). Due to symmetry, the sum of the vectors $A O_{1} + A O_{3}$ will lie in the plane $A B Y$. The vector $A O_{2}$ also lies in this plane. If the vertices $O_{1}, O_{2}, O_{3}$ lie in the same plane, then the intersection of this plane with the plane $A B Y$ should be a line.

Therefore, the vertices of the pyramids will lie in the same plane if the vectors $A O_{1} + A O_{3} = (c - h, 2 c, c - h)$ and $A O_{2} = (c, -h, c)$ are collinear.

From the conditions of collinearity

$$
\frac{c-h}{c}=\frac{2 c}{-h}=\frac{c-h}{c}
$$

we get that $h(h - c) = 2 c^2$ must be satisfied, from which either $h = -c$ or $h = 2 c$. The first root does not fit the geometric meaning of $h$. It remains to recall that the given value $a = 2 c$ in the problem statement.

The answer to question B) is obvious due to symmetry.

Answer: In A) and in B), they can, if the heights of all pyramids are equal to $a$.