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Contests
National and Regional Contests
USA Contests
MAA AMC
USAJMO
2024 USAJMO
2024 USAJMO
Part of
USAJMO
Subcontests
(6)
5
1
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average FE
Find all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
that satisfy
f
(
x
2
−
y
)
+
2
y
f
(
x
)
=
f
(
f
(
x
)
)
+
f
(
y
)
f(x^2-y)+2yf(x)=f(f(x))+f(y)
f
(
x
2
−
y
)
+
2
y
f
(
x
)
=
f
(
f
(
x
))
+
f
(
y
)
for all
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.Proposed by Carl Schildkraut
6
1
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Geo equals ABsurdly proBEMatic
Point
D
D
D
is selected inside acute
△
A
B
C
\triangle ABC
△
A
BC
so that
∠
D
A
C
=
∠
A
C
B
\angle DAC = \angle ACB
∠
D
A
C
=
∠
A
CB
and
∠
B
D
C
=
9
0
∘
+
∠
B
A
C
\angle BDC = 90^{\circ} + \angle BAC
∠
B
D
C
=
9
0
∘
+
∠
B
A
C
. Point
E
E
E
is chosen on ray
B
D
BD
B
D
so that
A
E
=
E
C
AE = EC
A
E
=
EC
. Let
M
M
M
be the midpoint of
B
C
BC
BC
.Show that line
A
B
AB
A
B
is tangent to the circumcircle of triangle
B
E
M
BEM
BEM
.Proposed by Anton Trygub
4
1
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rows are DERANGED and a SOCOURGE to usajmo .
Let
n
≥
3
n \geq 3
n
≥
3
be an integer. Rowan and Colin play a game on an
n
×
n
n \times n
n
×
n
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is orderly if: [*]no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and [*]no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring. In terms of
n
n
n
, how many orderly colorings are there?Proposed by Alec Sun
3
1
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p^k divides term of sequence
Let
a
(
n
)
a(n)
a
(
n
)
be the sequence defined by
a
(
1
)
=
2
a(1)=2
a
(
1
)
=
2
and
a
(
n
+
1
)
=
(
a
(
n
)
)
n
+
1
−
1
a(n+1)=(a(n))^{n+1}-1
a
(
n
+
1
)
=
(
a
(
n
)
)
n
+
1
−
1
for each integer
n
≥
1
n\geq 1
n
≥
1
. Suppose that
p
>
2
p>2
p
>
2
is a prime and
k
k
k
is a positive integer. Prove that some term of the sequence
a
(
n
)
a(n)
a
(
n
)
is divisible by
p
k
p^k
p
k
.Proposed by John Berman
2
1
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happy configs
Let
m
m
m
and
n
n
n
be positive integers. Let
S
S
S
be the set of integer points
(
x
,
y
)
(x,y)
(
x
,
y
)
with
1
≤
x
≤
2
m
1\leq x\leq 2m
1
≤
x
≤
2
m
and
1
≤
y
≤
2
n
1\leq y\leq 2n
1
≤
y
≤
2
n
. A configuration of
m
n
mn
mn
rectangles is called happy if each point in
S
S
S
is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.Proposed by Serena An and Claire Zhang
1
1
Hide problems
They mixed up USAJMO and AIME I guess
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral with
A
B
=
7
AB = 7
A
B
=
7
and
C
D
=
8
CD = 8
C
D
=
8
. Point
P
P
P
and
Q
Q
Q
are selected on segment
A
B
AB
A
B
such that
A
P
=
B
Q
=
3
AP = BQ = 3
A
P
=
BQ
=
3
. Points
R
R
R
and
S
S
S
are selected on segment
C
D
CD
C
D
such that
C
R
=
D
S
=
2
CR = DS = 2
CR
=
D
S
=
2
. Prove that
P
Q
R
S
PQRS
PQRS
is a cyclic quadrilateral.Proposed by Evan O'Dorney