Difference between revisions of "2020 USAMO Problems"
Sugar rush (talk | contribs) |
m (→Problem 2: better code) |
||
Line 10: | Line 10: | ||
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? | ||
[[2020 USAMO Problems/Problem 2|Solution]] | [[2020 USAMO Problems/Problem 2|Solution]] | ||
− | |||
===Problem 3=== | ===Problem 3=== |
Revision as of 09:05, 31 July 2023
Contents
[hide]Day 1
Problem 1
Let be a fixed acute triangle inscribed in a circle with center . A variable point is chosen on minor arc of , and segments and meet at . Denote by and the circumcenters of triangles and , respectively. Determine all points for which the area of triangle is minimized.
Problem 2
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.
What is the smallest positive number of beams that can be placed to satisfy these conditions?
Problem 3
Let be an odd prime. An integer is called a quadratic non-residue if does not divide for any integer .
Denote by the set of all integers such that , and both and are quadratic non-residues. Calculate the remainder when the product of the elements of is divided by .
Day 2
Problem 4
Suppose that are distinct ordered pairs of nonnegative integers. Let denote the number of pairs of integers satisfying and . Determine the largest possible value of over all possible choices of the ordered pairs.
Problem 5
A finite set of points in the coordinate plane is called overdetermined if and there exists a nonzero polynomial , with real coefficients and of degree at most , satisfying for every point .
For each integer , find the largest integer (in terms of ) such that there exists a set of distinct points that is not overdetermined, but has overdetermined subsets.
Problem 6
Let be an integer. Let and be real numbers such that Prove that
2020 USAMO (Problems • Resources) | ||
Preceded by 2019 USAMO |
Followed by 2021 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.