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Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
USAMO
2019 USAMO
2019 USAMO
Part of
USAMO
Subcontests
(3)
6
1
Hide problems
Sad Algebra
Find all polynomials
P
P
P
with real coefficients such that
P
(
x
)
y
z
+
P
(
y
)
z
x
+
P
(
z
)
x
y
=
P
(
x
−
y
)
+
P
(
y
−
z
)
+
P
(
z
−
x
)
\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)
yz
P
(
x
)
+
z
x
P
(
y
)
+
x
y
P
(
z
)
=
P
(
x
−
y
)
+
P
(
y
−
z
)
+
P
(
z
−
x
)
holds for all nonzero real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfying
2
x
y
z
=
x
+
y
+
z
2xyz=x+y+z
2
x
yz
=
x
+
y
+
z
.Proposed by Titu Andreescu and Gabriel Dospinescu
1
1
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Convolution of order f(n)
Let
N
\mathbb{N}
N
be the set of positive integers. A function
f
:
N
→
N
f:\mathbb{N}\to\mathbb{N}
f
:
N
→
N
satisfies the equation
f
(
f
(
…
f
⏟
f
(
n
)
times
(
n
)
…
)
)
=
n
2
f
(
f
(
n
)
)
\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}
f
(
n
)
times
f
(
f
(
…
f
(
n
)
…
))
=
f
(
f
(
n
))
n
2
for all positive integers
n
n
n
. Given this information, determine all possible values of
f
(
1000
)
f(1000)
f
(
1000
)
.Proposed by Evan Chen
3
1
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Sad Number Theory
Let
K
K
K
be the set of all positive integers that do not contain the digit
7
7
7
in their base-
10
10
10
representation. Find all polynomials
f
f
f
with nonnegative integer coefficients such that
f
(
n
)
∈
K
f(n)\in K
f
(
n
)
∈
K
whenever
n
∈
K
n\in K
n
∈
K
.Proposed by Titu Andreescu, Cosmin Pohoata, and Vlad Matei