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France Contests
French Mathematical Olympiad
1994 French Mathematical Olympiad
Problem 5
Problem 5
Part of
1994 French Mathematical Olympiad
Problems
(1)
f(m^2+n^2)=f(m)^2+f(n)^2 over N
Source: France 1994 P5
5/7/2021
Assume
f
:
N
0
→
N
0
f:\mathbb N_0\to\mathbb N_0
f
:
N
0
→
N
0
is a function such that
f
(
1
)
>
0
f(1)>0
f
(
1
)
>
0
and, for any nonnegative integers
m
m
m
and
n
n
n
,
f
(
m
2
+
n
2
)
=
f
(
m
)
2
+
f
(
n
)
2
.
f\left(m^2+n^2\right)=f(m)^2+f(n)^2.
f
(
m
2
+
n
2
)
=
f
(
m
)
2
+
f
(
n
)
2
.
(a) Calculate
f
(
k
)
f(k)
f
(
k
)
for
0
≤
k
≤
12
0\le k\le12
0
≤
k
≤
12
. (b) Calculate
f
(
n
)
f(n)
f
(
n
)
for any natural number
n
n
n
.
fe
functional equation
algebra