Difference between revisions of "2020 USOMO Problems"
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<cmath>\begin{align*} | <cmath>\begin{align*} | ||
0 &= x_1 + x_2 + \cdots + x_n = y_1 + y_2 + \cdots + y_n\\ | 0 &= x_1 + x_2 + \cdots + x_n = y_1 + y_2 + \cdots + y_n\\ | ||
− | \text{and }1 &= x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2. | + | \text{and }1 &= x_1^2+x_2^2+\cdots+x_n^2=y_1^2+y_2^2+\cdots+y_n^2. |
\end{align*} | \end{align*} | ||
</cmath> | </cmath> |
Revision as of 02:25, 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
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
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 |