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Balkan MO Shortlist
2015 Balkan MO Shortlist
A4
(x+y) f(2yf(x)+f(y))=x^3 f(yf(x)) for all x,y\in R^+
(x+y) f(2yf(x)+f(y))=x^3 f(yf(x)) for all x,y\in R^+
Source: Balkan BMO Shortlist 2015 A4
August 5, 2019
functional equation
algebra
function
Problem Statement
Find all functions
f
:
R
+
→
R
+
f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}
f
:
R
+
→
R
+
such that
(
x
+
y
)
f
(
2
y
f
(
x
)
+
f
(
y
)
)
=
x
3
f
(
y
f
(
x
)
)
,
∀
x
,
y
∈
R
+
.
(x+y)f(2yf(x)+f(y))=x^{3}f(yf(x)), \ \ \ \forall x,y\in \mathbb{R}^{+}.
(
x
+
y
)
f
(
2
y
f
(
x
)
+
f
(
y
))
=
x
3
f
(
y
f
(
x
))
,
∀
x
,
y
∈
R
+
.
(Albania)
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