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Problems
Contests
National and Regional Contests
Belgium Contests
Flanders Math Olympiad
1989 Flanders Math Olympiad
4
4
Part of
1989 Flanders Math Olympiad
Problems
(1)
sort of functional equation
Source: flanders '89
9/27/2005
Let
D
D
D
be the set of positive reals different from
1
1
1
and let
n
n
n
be a positive integer. If for
f
:
D
→
R
f: D\rightarrow \mathbb{R}
f
:
D
→
R
we have
x
n
f
(
x
)
=
f
(
x
2
)
x^n f(x)=f(x^2)
x
n
f
(
x
)
=
f
(
x
2
)
, and if
f
(
x
)
=
x
n
f(x)=x^n
f
(
x
)
=
x
n
for
0
<
x
<
1
1989
0<x<\frac{1}{1989}
0
<
x
<
1989
1
and for
x
>
1989
x>1989
x
>
1989
, then prove that
f
(
x
)
=
x
n
f(x)=x^n
f
(
x
)
=
x
n
for all
x
∈
D
x \in D
x
∈
D
.
induction