MathDB
Problems
Contests
National and Regional Contests
Taiwan Contests
IMOC Shortlist
2023-IMOC
N3
N3
Part of
2023-IMOC
Problems
(1)
Divisibility FE
Source: IMOC 2023 N3
9/9/2023
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
.
number theory