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TASIMO
2024 TASIMO
5
5
Part of
2024 TASIMO
Problems
(1)
af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac)
Source: 1st TASIMO Day2, Problem5
5/19/2024
Find all functions
f
:
Z
+
ā
Z
+
f: \mathbb{Z^+} \to \mathbb{Z^+}
f
:
Z
+
ā
Z
+
such that for all integers
a
,
b
,
c
a, b, c
a
,
b
,
c
we have
a
f
(
b
c
)
+
b
f
(
a
c
)
+
c
f
(
a
b
)
=
(
a
+
b
+
c
)
f
(
a
b
+
b
c
+
a
c
)
.
af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac).
a
f
(
b
c
)
+
b
f
(
a
c
)
+
c
f
(
ab
)
=
(
a
+
b
+
c
)
f
(
ab
+
b
c
+
a
c
)
.
Note. The set
Z
+
\mathbb{Z^+}
Z
+
refers to the set of positive integers. Proposed by Mojtaba Zare, Iran
functional equation
Algebraic Number Theory
algebra