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Contests
National and Regional Contests
USA Contests
USA - High School Proof Olympiads
Ersatz MO (USEMO)
2019 USEMO
2019 USEMO
Part of
Ersatz MO (USEMO)
Subcontests
(6)
5
1
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Nationalist Combo
Let
P
\mathcal{P}
P
be a regular polygon, and let
V
\mathcal{V}
V
be its set of vertices. Each point in
V
\mathcal{V}
V
is colored red, white, or blue. A subset of
V
\mathcal{V}
V
is patriotic if it contains an equal number of points of each color, and a side of
P
\mathcal{P}
P
is dazzling if its endpoints are of different colors. Suppose that
V
\mathcal{V}
V
is patriotic and the number of dazzling edges of
P
\mathcal{P}
P
is even. Prove that there exists a line, not passing through any point in
V
\mathcal{V}
V
, dividing
V
\mathcal{V}
V
into two nonempty patriotic subsets.Ankan Bhattacharya
6
1
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Oops, I lost my compass...
Let
A
B
C
ABC
A
BC
be an acute scalene triangle with circumcenter
O
O
O
and altitudes
A
D
‾
\overline{AD}
A
D
,
B
E
‾
\overline{BE}
BE
,
C
F
‾
\overline{CF}
CF
. Let
X
X
X
,
Y
Y
Y
,
Z
Z
Z
be the midpoints of
A
D
‾
\overline{AD}
A
D
,
B
E
‾
\overline{BE}
BE
,
C
F
‾
\overline{CF}
CF
. Lines
A
D
AD
A
D
and
Y
Z
YZ
Y
Z
intersect at
P
P
P
, lines
B
E
BE
BE
and
Z
X
ZX
ZX
intersect at
Q
Q
Q
, and lines
C
F
CF
CF
and
X
Y
XY
X
Y
intersect at
R
R
R
.Suppose that lines
Y
Z
YZ
Y
Z
and
B
C
BC
BC
intersect at
A
′
A'
A
′
, and lines
Q
R
QR
QR
and
E
F
EF
EF
intersect at
D
′
D'
D
′
. Prove that the perpendiculars from
A
A
A
,
B
B
B
,
C
C
C
,
O
O
O
, to the lines
Q
R
QR
QR
,
R
P
RP
RP
,
P
Q
PQ
PQ
,
A
′
D
′
A'D'
A
′
D
′
, respectively, are concurrent.Ankan Bhattacharya
4
1
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Just Sum NT
Prove that for any prime
p
,
p,
p
,
there exists a positive integer
n
n
n
such that
1
n
+
2
n
−
1
+
3
n
−
2
+
⋯
+
n
1
≡
2020
(
m
o
d
p
)
.
1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.
1
n
+
2
n
−
1
+
3
n
−
2
+
⋯
+
n
1
≡
2020
(
mod
p
)
.
Robin Son
2
1
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Integer roots preserved under linear function of polynomial
Let
Z
[
x
]
\mathbb{Z}[x]
Z
[
x
]
denote the set of single-variable polynomials in
x
x
x
with integer coefficients. Find all functions
θ
:
Z
[
x
]
→
Z
[
x
]
\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]
θ
:
Z
[
x
]
→
Z
[
x
]
(i.e. functions taking polynomials to polynomials) such that[*] for any polynomials
p
,
q
∈
Z
[
x
]
p, q \in \mathbb{Z}[x]
p
,
q
∈
Z
[
x
]
,
θ
(
p
+
q
)
=
θ
(
p
)
+
θ
(
q
)
\theta(p + q) = \theta(p) + \theta(q)
θ
(
p
+
q
)
=
θ
(
p
)
+
θ
(
q
)
; [*] for any polynomial
p
∈
Z
[
x
]
p \in \mathbb{Z}[x]
p
∈
Z
[
x
]
,
p
p
p
has an integer root if and only if
θ
(
p
)
\theta(p)
θ
(
p
)
does. Carl Schildkraut
3
1
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You and Nikolai Play a Game
Consider an infinite grid
G
\mathcal G
G
of unit square cells. A chessboard polygon is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of
G
\mathcal G
G
. Nikolai chooses a chessboard polygon
F
F
F
and challenges you to paint some cells of
G
\mathcal G
G
green, such that any chessboard polygon congruent to
F
F
F
has at least
1
1
1
green cell but at most
2020
2020
2020
green cells. Can Nikolai choose
F
F
F
to make your job impossible?Nikolai Beluhov
1
1
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H not needed
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. A circle centered at
O
O
O
passes through
B
B
B
and
D
D
D
and meets lines
B
A
BA
B
A
and
B
C
BC
BC
again at points
E
E
E
and
F
F
F
(distinct from
A
,
B
,
C
A,B,C
A
,
B
,
C
). Let
H
H
H
denote the orthocenter of triangle
D
E
F
.
DEF.
D
EF
.
Prove that if lines
A
C
,
AC,
A
C
,
D
O
,
DO,
D
O
,
E
F
EF
EF
are concurrent, then triangle
A
B
C
ABC
A
BC
and
E
H
F
EHF
E
H
F
are similar.Robin Son