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Contests
National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine National Mathematical Olympiad
2021 Ukraine National Mathematical Olympiad
11.4
11.4
Part of
2021 Ukraine National Mathematical Olympiad
Problems
(1)
f(xf(x+y))+f((x+y)f(y))=(x+y)^2
Source: 2021 Ukraine NMO 11.4
4/2/2021
Find all the following functions
f
:
R
ā
R
f:R\to R
f
:
R
ā
R
, which for arbitrary valid
x
,
y
x,y
x
,
y
holds equality:
f
(
x
f
(
x
+
y
)
)
+
f
(
(
x
+
y
)
f
(
y
)
)
=
(
x
+
y
)
2
f(xf(x+y))+f((x+y)f(y))=(x+y)^2
f
(
x
f
(
x
+
y
))
+
f
((
x
+
y
)
f
(
y
))
=
(
x
+
y
)
2
(Vadym Koval)
functional equation
functional
algebra