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National and Regional Contests
Cuba Contests
Cuba MO
2009 Cuba MO
4
x + f(xf(y)) = f(y) + yf(x
x + f(xf(y)) = f(y) + yf(x
Source: 2009 Cuba 2.4
August 27, 2024
algebra
functional
Problem Statement
Determine all the functions
f
:
R
→
R
f : R \to R
f
:
R
→
R
such that:
x
+
f
(
x
f
(
y
)
)
=
f
(
y
)
+
y
f
(
x
)
x + f(xf(y)) = f(y) + yf(x)
x
+
f
(
x
f
(
y
))
=
f
(
y
)
+
y
f
(
x
)
for all
x
,
y
∈
R
x, y \in R
x
,
y
∈
R
.
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