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National and Regional Contests
Mexico Contests
Mexico National Olympiad
2023 Mexico National Olympiad
6
NT FE with two conditions
NT FE with two conditions
Source: Mexico 2023/6
November 8, 2023
number theory
Problem Statement
Find all functions
f
:
N
→
N
f: \mathbb{N} \rightarrow \mathbb {N}
f
:
N
→
N
such that for all positive integers
m
,
n
m, n
m
,
n
,
f
(
m
+
n
)
∣
f
(
m
)
+
f
(
n
)
f(m+n)\mid f(m)+f(n)
f
(
m
+
n
)
∣
f
(
m
)
+
f
(
n
)
and
f
(
m
)
f
(
n
)
∣
f
(
m
n
)
f(m)f(n) \mid f(mn)
f
(
m
)
f
(
n
)
∣
f
(
mn
)
.
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