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Contests
National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
2000 Swedish Mathematical Competition
5
5
Part of
2000 Swedish Mathematical Competition
Problems
(1)
unique f if f(prime) = 1; f(ab) = a f(b) + f(a) b
Source: 2000 Swedish Mathematical Competition p5
3/21/2021
Let
f
(
n
)
f(n)
f
(
n
)
be defined on the positive integers and satisfy:
f
(
p
r
i
m
e
)
=
1
f(prime) = 1
f
(
p
r
im
e
)
=
1
,
f
(
a
b
)
=
a
f
(
b
)
+
f
(
a
)
b
f(ab) = a f(b) + f(a) b
f
(
ab
)
=
a
f
(
b
)
+
f
(
a
)
b
. Show that
f
f
f
is unique and find all
n
n
n
such that
n
=
f
(
n
)
n = f(n)
n
=
f
(
n
)
.
functional equation
functional
number theory