Problem 3

A box whose shape is a rectangular parallelepiped can be completely filled with cubes of side

1

. If one places inside it the maximum possible number of cubes, each of volume

2

, with their sides parallel to those of the box, then exactly

40\%

of the volume of the box is occupied. Determine the possible dimensions of all such boxes.

Step 3 of 6: Bound each ratio

a/x

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In plain words

As $x$ grows, the upper bound $\sqrt[3]2(1+1/x)$ gets closer and closer to $\sqrt[3]2$ itself, so large edges force their ratio very close to $\sqrt[3]2$ — leaving little room for the product of three such ratios to reach as large as $5$ unless one edge stays small.

\sqrt[3]2 < \frac ax < \sqrt[3]2\left(1+\frac1x\right)

Detailed analysis

By definition of the floor, $x < a/\sqrt[3]2 < x+1$ , which rearranges to $\sqrt[3]2 < a/x < \sqrt[3]2\,(1+1/x)$ ; the same bound holds for $b/y$ and $c/z$ .