Difference between revisions of "2020 USAMO Problems/Problem 2"
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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 beam 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 beam is a <math>1 \times 1 \times 2020</math> rectangular prism. Several beams are placed inside the cube subject to the following conditions: | ||
| − | + | *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. | |
| − | |||
| − | |||
| − | 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? | What is the smallest positive number of beams that can be placed to satisfy these conditions? | ||
Revision as of 09:06, 31 July 2023
Problem
An empty 
cube is given, and a 
grid of square unit cells is drawn on each of its six faces. A beam 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.
faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are 
possible positions for a beam.)
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?
Solution
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| 2020 USAMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
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