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CentroAmerican
2020 Centroamerican and Caribbean Math Olympiad
3
3
Part of
2020 Centroamerican and Caribbean Math Olympiad
Problems
(1)
Functional equation over the integers
Source: Centroamerican 2020, problem 3
10/28/2020
Find all the functions
f
:
Z
ā
Z
f: \mathbb{Z}\to \mathbb{Z}
f
:
Z
ā
Z
satisfying the following property: if
a
a
a
,
b
b
b
and
c
c
c
are integers such that
a
+
b
+
c
=
0
a+b+c=0
a
+
b
+
c
=
0
, then
f
(
a
)
+
f
(
b
)
+
f
(
c
)
=
a
2
+
b
2
+
c
2
.
f(a)+f(b)+f(c)=a^2+b^2+c^2.
f
(
a
)
+
f
(
b
)
+
f
(
c
)
=
a
2
+
b
2
+
c
2
.
algebra
functional equation
function