2020 USOJMO Problems/Problem 3
Problem
An empty 
cube is given, and a 
grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a 
rectangular prism. Several beams are placed inside the cube subject to the following conditions:
- The two 
faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 
possible positions for a beam.)
- No two beams have intersecting interiors.
- The interiors of each of the four 
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?
