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Francophone Mathematical Olympiad
2022 Francophone Mathematical Olympiad
1
functional equation from Z to Z
functional equation from Z to Z
Source: P1 francophone math olympiad senior 2022
May 23, 2022
algebra
functional equation
function
Problem Statement
find all functions
f
:
Z
→
Z
f:\mathbb{Z} \to \mathbb{Z}
f
:
Z
→
Z
such that
f
(
m
+
n
)
+
f
(
m
)
f
(
n
)
=
n
2
(
f
(
m
)
+
1
)
+
m
2
(
f
(
n
)
+
1
)
+
m
n
(
2
−
m
n
)
f(m+n)+f(m)f(n)=n^2(f(m)+1)+m^2(f(n)+1)+mn(2-mn)
f
(
m
+
n
)
+
f
(
m
)
f
(
n
)
=
n
2
(
f
(
m
)
+
1
)
+
m
2
(
f
(
n
)
+
1
)
+
mn
(
2
−
mn
)
holds for all
m
,
n
∈
Z
m,n \in \mathbb{Z}
m
,
n
∈
Z
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