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Functional equation
Source: Benelux Mathematical Olympiad 2017, Problem 1
5/6/2017
Find all functions
f
:
Q
>
0
→
Z
>
0
f : \Bbb{Q}_{>0}\to \Bbb{Z}_{>0}
f
:
Q
>
0
→
Z
>
0
such that
f
(
x
y
)
⋅
gcd
(
f
(
x
)
f
(
y
)
,
f
(
1
x
)
f
(
1
y
)
)
=
x
y
f
(
1
x
)
f
(
1
y
)
,
f(xy)\cdot \gcd\left( f(x)f(y), f(\frac{1}{x})f(\frac{1}{y})\right) = xyf(\frac{1}{x})f(\frac{1}{y}),
f
(
x
y
)
⋅
g
cd
(
f
(
x
)
f
(
y
)
,
f
(
x
1
)
f
(
y
1
)
)
=
x
y
f
(
x
1
)
f
(
y
1
)
,
for all
x
,
y
∈
Q
>
0
,
x, y \in \Bbb{Q}_{>0,}
x
,
y
∈
Q
>
0
,
where
gcd
(
a
,
b
)
\gcd(a, b)
g
cd
(
a
,
b
)
denotes the greatest common divisor of
a
a
a
and
b
.
b.
b
.
algebra
functional equation
algebra proposed