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International Zhautykov Olympiad
2015 International Zhautykov Olympiad
3
Hard functional equation
Hard functional equation
Source: IZHO2015.P3
January 13, 2015
function
algebra
polynomial
algebra unsolved
Problem Statement
Find all functions
f
:
R
→
R
f\colon \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
x
3
+
y
3
+
x
y
)
=
x
2
f
(
x
)
+
y
2
f
(
y
)
+
f
(
x
y
)
f(x^3+y^3+xy)=x^2f(x)+y^2f(y)+f(xy)
f
(
x
3
+
y
3
+
x
y
)
=
x
2
f
(
x
)
+
y
2
f
(
y
)
+
f
(
x
y
)
, for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
.
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