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2023-IMOC
N3
Divisibility FE
Divisibility FE
Source: IMOC 2023 N3
September 9, 2023
number theory
Problem Statement
Find all functions
f
:
N
→
N
f:\mathbb{N} \rightarrow \mathbb{N}
f
:
N
→
N
, such that
f
(
a
)
+
f
(
b
)
+
a
b
∣
a
2
f
(
a
)
+
b
2
f
(
b
)
+
f
(
a
)
f
(
b
)
f(a)+f(b)+ab \mid a^2f(a)+b^2f(b)+f(a)f(b)
f
(
a
)
+
f
(
b
)
+
ab
∣
a
2
f
(
a
)
+
b
2
f
(
b
)
+
f
(
a
)
f
(
b
)
for all positive integers
a
,
b
a,b
a
,
b
.
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