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2011 Japan MO Finals
4
4
Part of
2011 Japan MO Finals
Problems
(1)
2011 Japan Mathematical Olympiad Finals Problem 4
Source: Japanese MO Finals 2011, India Postal 2016 Set 1, P2
2/11/2011
Find all functions
f
:
R
→
R
f : \mathbb{R} \to\mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
)
−
f
(
y
)
)
=
f
(
f
(
x
)
)
−
2
x
2
f
(
y
)
+
f
(
y
2
)
f(f(x)-f(y))=f(f(x))-2x^2f(y)+f(y^2)
f
(
f
(
x
)
−
f
(
y
))
=
f
(
f
(
x
))
−
2
x
2
f
(
y
)
+
f
(
y
2
)
for all
x
,
y
∈
R
.
x,y \in \mathbb{R}.
x
,
y
∈
R
.
function
quadratics
algebra proposed
algebra