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National and Regional Contests
USA Contests
USA - High School Proof Olympiads
Ersatz MO (USEMO)
2024 USEMO
2024 USEMO
Part of
Ersatz MO (USEMO)
Subcontests
(6)
5
1
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Tangency to EFG
Let
A
B
C
ABC
A
BC
be a scalene triangle whose incircle is tangent to
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
at
D
D
D
,
E
E
E
,
F
F
F
respectively. Lines
B
E
BE
BE
and
C
F
CF
CF
meet at
G
G
G
. Prove that there exists a point
X
X
X
on the circumcircle of triangle
E
F
G
EFG
EFG
such that the circumcircles of triangles
B
C
X
BCX
BCX
and
E
F
G
EFG
EFG
are tangent, and
∠
B
G
C
=
∠
B
X
C
+
∠
E
D
F
.
\angle BGC = \angle BXC + \angle EDF.
∠
BGC
=
∠
BXC
+
∠
E
D
F
.
Kornpholkrit Weraarchakul
4
1
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Infinite sequence of polynomials
Find all sequences
a
1
a_1
a
1
,
a
2
a_2
a
2
,
…
\dots
…
of nonnegative integers such that for all positive integers
n
n
n
, the polynomial
1
+
x
a
1
+
x
a
2
+
⋯
+
x
a
n
1+x^{a_1}+x^{a_2}+\dots+x^{a_n}
1
+
x
a
1
+
x
a
2
+
⋯
+
x
a
n
has at least one integer root. (Here
x
0
=
1
x^0=1
x
0
=
1
.)Kornpholkrit Weraarchakul
6
1
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How the horsies attacks?
Let
n
n
n
be an odd positive integer and consider an
n
×
n
n \times n
n
×
n
chessboard of
n
2
n^2
n
2
unit squares. In some of the cells of the chessboard, we place a knight. A knight in a cell
c
c
c
is said to attack a cell
c
′
c'
c
′
if the distance between the centers of
c
c
c
and
c
′
c'
c
′
is exactly
5
\sqrt{5}
5
(in particular, a knight does not attack the cell which it occupies). Suppose each cell of the board is attacked by an even number of knights (possibly zero). Show that the configuration of knights is symmetric with respect to all four axes of symmetry of the board (i.e. the configuration of knights is both horizontally and vertically symmetric, and also unchanged by reflection along either diagonal of the chessboard). NIkolai Beluhov
1
1
Hide problems
removing gcd of coins from stacks
There are
1001
1001
1001
stacks of coins
S
1
,
S
2
,
…
,
S
1001
S_1, S_2, \dots, S_{1001}
S
1
,
S
2
,
…
,
S
1001
. Initially, stack
S
k
S_k
S
k
has
k
k
k
coins for each
k
=
1
,
2
,
…
,
1001
k = 1,2,\dots,1001
k
=
1
,
2
,
…
,
1001
. In an operation, one selects an ordered pair
(
i
,
j
)
(i,j)
(
i
,
j
)
of indices
i
i
i
and
j
j
j
satisfying
1
≤
i
<
j
≤
1001
1 \le i < j \le 1001
1
≤
i
<
j
≤
1001
subject to two conditions:[*]The stacks
S
i
S_i
S
i
and
S
j
S_j
S
j
must each have at least
1
1
1
coin. [*]The ordered pair
(
i
,
j
)
(i,j)
(
i
,
j
)
must not have been selected before.Then, if
S
i
S_i
S
i
and
S
j
S_j
S
j
have
a
a
a
coins and
b
b
b
coins respectively, one removes
gcd
(
a
,
b
)
\gcd(a,b)
g
cd
(
a
,
b
)
coins from each stack. What is the maximum number of times this operation could be performed?Galin Totev
3
1
Hide problems
X56 Jumpscare
Let
A
B
C
ABC
A
BC
be a triangle with incenter
I
I
I
. Two distinct points
P
P
P
and
Q
Q
Q
are chosen on the circumcircle of
A
B
C
ABC
A
BC
such that
∠
A
P
I
=
∠
A
Q
I
=
4
5
∘
.
\angle API = \angle AQI = 45^\circ.
∠
A
P
I
=
∠
A
Q
I
=
4
5
∘
.
Lines
P
Q
PQ
PQ
and
B
C
BC
BC
meet at
S
S
S
. Let
H
H
H
denote the foot of the altitude from
A
A
A
to
B
C
BC
BC
. Prove that
∠
A
H
I
=
∠
I
S
H
\angle AHI = \angle ISH
∠
A
H
I
=
∠
I
S
H
.Matsvei Zorka
2
1
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Rectangle with common lcm
Let
k
k
k
be a fixed positive integer. For each integer
1
≤
i
≤
4
1 \leq i \leq 4
1
≤
i
≤
4
, let
x
i
x_i
x
i
and
y
i
y_i
y
i
be positive integers such that their least common multiple is
k
k
k
. Suppose that the four points
(
x
1
,
y
1
)
(x_1, y_1)
(
x
1
,
y
1
)
,
(
x
2
,
y
2
)
(x_2, y_2)
(
x
2
,
y
2
)
,
(
x
3
,
y
3
)
(x_3, y_3)
(
x
3
,
y
3
)
,
(
x
4
,
y
4
)
(x_4, y_4)
(
x
4
,
y
4
)
are the vertices of a non-degenerate rectangle in the Cartesian plane. Prove that
x
1
x
2
x
3
x
4
x_1x_2x_3x_4
x
1
x
2
x
3
x
4
is a perfect square.Andrei Chirita