MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
USAMO
2020 USOMO
2020 USOMO
Part of
USAMO
Subcontests
(4)
6
1
Hide problems
mT iNeQuAlItY ErUpTeD OmG
Let
n
≥
2
n \ge 2
n
≥
2
be an integer. Let
x
1
≥
x
2
≥
.
.
.
≥
x
n
x_1 \ge x_2 \ge ... \ge x_n
x
1
≥
x
2
≥
...
≥
x
n
and
y
1
≥
y
2
≥
.
.
.
≥
y
n
y_1 \ge y_2 \ge ... \ge y_n
y
1
≥
y
2
≥
...
≥
y
n
be
2
n
2n
2
n
real numbers such that
0
=
x
1
+
x
2
+
.
.
.
+
x
n
=
y
1
+
y
2
+
.
.
.
+
y
n
0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n
0
=
x
1
+
x
2
+
...
+
x
n
=
y
1
+
y
2
+
...
+
y
n
and
1
=
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
=
y
1
2
+
y
2
2
+
.
.
.
+
y
n
2
.
\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.
and
1
=
x
1
2
+
x
2
2
+
...
+
x
n
2
=
y
1
2
+
y
2
2
+
...
+
y
n
2
.
Prove that
∑
i
=
1
n
(
x
i
y
i
−
x
i
y
n
+
1
−
i
)
≥
2
n
−
1
.
\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.
i
=
1
∑
n
(
x
i
y
i
−
x
i
y
n
+
1
−
i
)
≥
n
−
1
2
.
Proposed by David Speyer and Kiran Kedlaya
5
1
Hide problems
Sad Combinatorics
A finite set
S
S
S
of points in the coordinate plane is called overdetermined if
∣
S
∣
≥
2
|S|\ge 2
∣
S
∣
≥
2
and there exists a nonzero polynomial
P
(
t
)
P(t)
P
(
t
)
, with real coefficients and of degree at most
∣
S
∣
−
2
|S|-2
∣
S
∣
−
2
, satisfying
P
(
x
)
=
y
P(x)=y
P
(
x
)
=
y
for every point
(
x
,
y
)
∈
S
(x,y)\in S
(
x
,
y
)
∈
S
. For each integer
n
≥
2
n\ge 2
n
≥
2
, find the largest integer
k
k
k
(in terms of
n
n
n
) such that there exists a set of
n
n
n
distinct points that is not overdetermined, but has
k
k
k
overdetermined subsets. Proposed by Carl Schildkraut
3
1
Hide problems
Sad Number Theory
Let
p
p
p
be an odd prime. An integer
x
x
x
is called a quadratic non-residue if
p
p
p
does not divide
x
−
t
2
x-t^2
x
−
t
2
for any integer
t
t
t
. Denote by
A
A
A
the set of all integers
a
a
a
such that
1
≤
a
<
p
1\le a<p
1
≤
a
<
p
, and both
a
a
a
and
4
−
a
4-a
4
−
a
are quadratic non-residues. Calculate the remainder when the product of the elements of
A
A
A
is divided by
p
p
p
. Proposed by Richard Stong and Toni Bluher
1
1
Hide problems
USAM(inimize)OOO
Let
A
B
C
ABC
A
BC
be a fixed acute triangle inscribed in a circle
ω
\omega
ω
with center
O
O
O
. A variable point
X
X
X
is chosen on minor arc
A
B
AB
A
B
of
ω
\omega
ω
, and segments
C
X
CX
CX
and
A
B
AB
A
B
meet at
D
D
D
. Denote by
O
1
O_1
O
1
and
O
2
O_2
O
2
the circumcenters of triangles
A
D
X
ADX
A
D
X
and
B
D
X
BDX
B
D
X
, respectively. Determine all points
X
X
X
for which the area of triangle
O
O
1
O
2
OO_1O_2
O
O
1
O
2
is minimized.Proposed by Zuming Feng