Difference between revisions of "2020 USOMO Problems"
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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: | ||
| − | <math>\bullet</math> The two <math>1 \times 1</math> faces of each beam coincide with unit cells lying on opposite faces | + | <math>\bullet</math> 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.) |
| − | of the cube. (Hence, there are <math>3 \cdot 2020^2</math> possible positions for a beam.) | + | |
<math>\bullet</math> No two beams have intersecting interiors. | <math>\bullet</math> No two beams have intersecting interiors. | ||
| − | <math>\bullet</math> 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. | + | |
| + | <math>\bullet</math> 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. | ||
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 USOMO Problems/Problem 2|Solution]] | [[2020 USOMO Problems/Problem 2|Solution]] | ||
===Problem 3=== | ===Problem 3=== | ||
| − | Let <math>p</math> be an odd prime. An integer <math>x</math> is called a <i>quadratic non-residue</i> if | + | Let <math>p</math> be an odd prime. An integer <math>x</math> is called a <i>quadratic non-residue</i> if <math>p</math> does not divide <math>x - t^2</math> for any integer <math>t</math>. |
| − | <math>p</math> does not divide <math>x - t^2</math> for any integer <math>t</math>. | ||
| − | Denote by <math>A</math> the set of all integers <math>a</math> such that <math>1 \le a < p</math>, and both <math>a</math> and <math>4 - a</math> are | + | Denote by <math>A</math> the set of all integers <math>a</math> such that <math>1 \le a < p</math>, and both <math>a</math> and <math>4 - a</math> are quadratic non-residues. Calculate the remainder when the product of the elements of <math>A</math> is divided by <math>p</math>. |
| − | quadratic non-residues. Calculate the remainder when the product of the elements of <math>A</math> | ||
| − | is divided by <math>p</math>. | ||
[[2020 USOMO Problems/Problem 3|Solution]] | [[2020 USOMO Problems/Problem 3|Solution]] | ||
==Day 2== | ==Day 2== | ||
Revision as of 01:05, 23 June 2020
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 .
