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National and Regional Contests
Japan Contests
Japan TST
2021 Japan TST
7
Wrapped FE
Wrapped FE
Source: 2021 Japan TST, P7
October 30, 2021
functional equation
algebra
Problem Statement
Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
, such that
f
(
x
2
+
x
y
2
+
y
2
)
=
2
x
2
f
(
y
)
+
2
x
f
(
f
(
y
)
)
+
f
(
−
x
2
−
x
y
2
)
+
f
(
y
2
)
f(x^2+xy^2+y^2)=2x^2f(y)+2xf(f(y))+f(-x^2-xy^2)+f(y^2)
f
(
x
2
+
x
y
2
+
y
2
)
=
2
x
2
f
(
y
)
+
2
x
f
(
f
(
y
))
+
f
(
−
x
2
−
x
y
2
)
+
f
(
y
2
)
holds for any real number
x
x
x
and
y
y
y
.
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