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Baltic Way
2018 Baltic Way
4
4
Part of
2018 Baltic Way
Problems
(1)
Functional equation with squares
Source: Baltic Way 2018, Problem 4
11/6/2018
Find all functions
f
:
[
0
,
∞
)
→
[
0
,
∞
)
f:[0, \infty) \to [0,\infty)
f
:
[
0
,
∞
)
→
[
0
,
∞
)
, such that for any positive integer
n
n
n
and and for any non-negative real numbers
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\dotsc,x_n
x
1
,
x
2
,
…
,
x
n
f
(
x
1
2
+
…
+
x
n
2
)
=
f
(
x
1
)
2
+
⋯
+
f
(
x
n
)
2
.
f(x_1^2+\dotsc+x_n^2)=f(x_1)^2+\dots+f(x_n)^2.
f
(
x
1
2
+
…
+
x
n
2
)
=
f
(
x
1
)
2
+
⋯
+
f
(
x
n
)
2
.
algebra
functional equation