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 02:05, 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
.