Problem 1009

Let the vertices of a given cube $\mathcal{C}$ be denoted by $A_1, \ldots, A_8$. Let $\mathcal{C}_i$ be the cube with edge length $\frac{1}{2}$, having one vertex at point $A_i$ and three faces contained in the faces of cube $\mathcal{C}$. Let $P_1, \ldots, P_8$ be eight given points.
Each cube $\mathcal{C}_i$ has a diameter of $\frac{1}{2}\sqrt{3} < 1$. Therefore, if two different points $P_k$ and $P_l$ lie in any of the cubes $\mathcal{C}_i$, then $P_kP_l < 1$; the required condition is satisfied.
Assume, then, that each of these cubes contains exactly one point $P_i$. Fix the numbering so that $P_i \in \mathcal{C}_i$ for $i = 1, \ldots, 8$. Each cube $\mathcal{C}_i$ has exactly three faces in common with the surface of cube $\mathcal{C}$; the point $P_i$ is at a distance of no more than $\frac{1}{2}$ from each of these three faces. Denote these three distances by $a_i$, $b_i$, $c_i$. Let $d$ be the largest of the 24 numbers: $a_1, b_1, c_1, \ldots, a_8, b_8, c_8$. Without loss of generality, we can assume that $d = P_1Q$, where $Q$ is the orthogonal projection of point $P_1$ onto a certain face of cube $\mathcal{C}$, and that $A_1A_2$ is an edge of cube $\mathcal{C}$ parallel to the line $P_1Q$.
Let $R$ be the orthogonal projection of point $P$ onto the line $A_1A_2$. Consider the rectangular prism whose one edge is the segment $A_2R$, and one of the faces perpendicular to $A_2R$ is a square with vertex $A_2$ and side length $d$, contained in a face of cube $\mathcal{C}$. From the definition of the number $d$, it follows that this rectangular prism contains points $P_1$ and $P_2$. Its diameter is equal to

since $0 \leq d \leq \frac{1}{2} < \frac{2}{3}$. Therefore, $P_1P_2 \leq 1$, which completes the proof.