2020 USAMO 第 2 题

An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:

- The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot 2020^2$ possible positions for a beam.)
- No two beams have intersecting interiors.
- The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam.

What is the smallest positive number of beams that can be placed to satisfy these conditions?

给出一个空的 $2020 \times 2020 \times 2020$ 立方体，并在其六个面的每个面上绘制一个 $2020 \times 2020$ 方形晶胞网格。梁是一个 $1 \times 1 \times 2020$ 矩形棱镜。在满足以下条件的情况下，将多根梁放置在立方体内：

- 每根梁的两个 $1 \times 1$ 面与位于立方体相对面上的晶胞重合。 (因此，梁的可能位置有 $3 \cdot 2020^2$ 个。)
- 没有两根梁具有相交的内部。
- 每根梁的四个 $1 \times 2020$ 面的内部接触立方体的一个面或另一个梁的面的内部。

满足这些条件可以放置的最小正数梁是多少？