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Austrian MO National Competition
2023 Austrian MO National Competition
1
symmetric R to R FE
symmetric R to R FE
Source: Austria MO Final round 2023 P1
May 27, 2023
algebra
Problem Statement
Given is a nonzero real number
α
\alpha
α
. Find all functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
f
(
f
(
x
+
y
)
)
=
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
+
α
x
y
f(f(x+y))=f(x+y)+f(x)f(y)+\alpha xy
f
(
f
(
x
+
y
))
=
f
(
x
+
y
)
+
f
(
x
)
f
(
y
)
+
αx
y
for all
x
,
y
∈
R
x, y \in \mathbb{R}
x
,
y
∈
R
.
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