XXXI - I - Task 1

Determine for which values of the parameter $ a $ a rhombus with side length $ a $ is a cross-section of a cube with edge length 2 by a plane passing through the center of the cube.
Suppose the rhombus $KLMN$ is a cross-section of the cube $ABCDA_1B_1C_1D_1$ by a plane passing through the center of the cube $O$. Since $O$ is also the center of symmetry of the cross-section, the opposite vertices of the rhombus lie on opposite edges of the cube. Let, for example, as in the figure, $K \in \overline{AB}$, $L \in \overline{A_1B_1}$, $M \in \overline{C_1D_1}$, $N \in \overline{CD}$. Since $O$ is the center of symmetry of the cube, $AK = MC_1$, $A_1L = NC$. Suppose that $K$ is not the midpoint of the edge $\overline{AB}$. Map the adjacent faces $ABB_1A_1$ and $A_1B_1C_1D_1$ onto the net of the cube.
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The right trapezoids $AKLA_1$ and $LB_1C_1M$ are congruent, as they are the sum of congruent rectangles and congruent right triangles. It follows that $A_1L = B_1L$, i.e., $L$ is the midpoint of the segment $\overline{A_1B_1}$, and therefore $N$ is the midpoint of $\overline{CD}$. In this way, we have shown that certain two opposite vertices of the rhombus are the midpoints of the edges of the cube on which they lie. The side of the rhombus is thus a segment connecting the midpoint of an edge with some point on the opposite side of the face, which is a square with side length $2$. The length of such a segment is no less than $2$ and no greater than $\sqrt{5}$, and any value in the interval $[2, \sqrt{5}]$ can be achieved by choosing the point $K$ appropriately on the side $\overline{AB}$.