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2024 CAPS Match
5
Parametrized functional equation
Parametrized functional equation
Source: CAPS 2024 p5
July 4, 2024
algebra
international competitions
functional equation
parameterization
Problem Statement
Let
α
≠
0
\alpha\neq0
α
=
0
be a real number. Determine all functions
f
:
R
→
R
f:\mathbb R\to\mathbb R
f
:
R
→
R
such that
f
(
x
2
+
y
2
)
=
f
(
x
−
y
)
f
(
x
+
y
)
+
α
y
f
(
y
)
f\left(x^2+y^2\right)=f(x-y)f(x+y)+\alpha yf(y)
f
(
x
2
+
y
2
)
=
f
(
x
−
y
)
f
(
x
+
y
)
+
α
y
f
(
y
)
holds for all
x
,
y
∈
R
.
x, y\in\mathbb R.
x
,
y
∈
R
.
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