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International Contests
Benelux
2009 Benelux
1
1
Part of
2009 Benelux
Problems
(1)
f(n) is a perfect square for all n
Source: Benelux 2009
1/29/2011
Find all functions
f
:
Z
>
0
→
Z
>
0
f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}
f
:
Z
>
0
→
Z
>
0
that satisfy the following two conditions:
∙
f
(
n
)
\bullet\ f(n)
∙
f
(
n
)
is a perfect square for all
n
∈
Z
>
0
n\in\mathbb{Z}_{>0}
n
∈
Z
>
0
∙
f
(
m
+
n
)
=
f
(
m
)
+
f
(
n
)
+
2
m
n
\bullet\ f(m+n)=f(m)+f(n)+2mn
∙
f
(
m
+
n
)
=
f
(
m
)
+
f
(
n
)
+
2
mn
for all
m
,
n
∈
Z
>
0
m,n\in\mathbb{Z}_{>0}
m
,
n
∈
Z
>
0
.
function
number theory proposed
number theory