Difference between revisions of "2020 USOJMO Problems/Problem 3"
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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. A [i]beam[/i] is a <math>1 \times 1 \times 2020</math> rectangular prism. Several beams are placed inside the cube subject to the following conditions: | 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. A [i]beam[/i] is a <math>1 \times 1 \times 2020</math> rectangular prism. Several beams are placed inside the cube subject to the following conditions: | ||
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| − | + | - The two <math>1 \times 1</math> faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are <math>3 \cdot {2020}^2</math> possible positions for a beam.) | |
| − | + | - No two beams have intersecting interiors. | |
| − | + | - The interiors of each of the four <math>1 \times 2020</math> faces of each beam touch either a face of the cube or the interior of the face of another beam. | |
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What is the smallest positive number of beams that can be placed to satisfy these conditions? | What is the smallest positive number of beams that can be placed to satisfy these conditions? | ||
Revision as of 18:14, 23 June 2020
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?
