Difference between revisions of "2020 USOMO Problems"
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[[2020 USOMO Problems/Problem 1|Solution]] | [[2020 USOMO Problems/Problem 1|Solution]] | ||
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===Problem 2=== | ===Problem 2=== | ||
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[[2020 USOMO Problems/Problem 2|Solution]] | [[2020 USOMO Problems/Problem 2|Solution]] | ||
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===Problem 3=== | ===Problem 3=== | ||
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[[2020 USOMO Problems/Problem 3|Solution]] | [[2020 USOMO Problems/Problem 3|Solution]] | ||
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==Day 2== | ==Day 2== | ||
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+ | ===Problem 4=== | ||
+ | Suppose that <math>(a_1, b_1), (a_2, b_2), \ldots , (a_{100}, b_{100})</math> are distinct ordered pairs of nonnegative integers. Let <math>N</math> denote the number of pairs of integers <math>(i, j)</math> satisfying <math>1 \le i < j \le 100</math> and <math>|a_ib_j - a_j b_i|=1</math>. Determine the largest possible value of <math>N</math> over all possible choices of the <math>100</math> ordered pairs. | ||
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+ | [[2020 USOMO Problems/Problem 4|Solution]] | ||
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+ | ===Problem 5=== | ||
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+ | ===Problem 6=== | ||
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+ | {{MAA Notice}} | ||
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+ | {{USAMO newbox|year= 2020 |before=[[2019 USAMO]]|after=[[2021 USAMO]]}} |
Revision as of 02:13, 23 June 2020
Contents
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
Problem 6
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
2020 USAMO (Problems • Resources) | ||
Preceded by 2019 USAMO |
Followed by 2021 USAMO | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |