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2020 USOJMO Problems/Problem 3

Revision as of 17:11, 23 June 2020 by Lcz (talk | contribs) (Created page with "==Problem== An empty <math>2020 \times 2020 \times 2020</math> cube is given, and a <math>2020 \times 2020</math> grid of square unit cells is drawn on each of its six faces...")
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Problem

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 [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions: [list=] [*]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. [/list] What is the smallest positive number of beams that can be placed to satisfy these conditions?

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