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Czech-Polish-Slovak Match
2019 Czech-Austrian-Polish-Slovak Match
4
all pairs of functions
all pairs of functions
Source: 2019 Czech-Polak-Slovak P4
July 7, 2019
algebra
function
Problem Statement
Given a real number
α
\alpha
α
, find all pairs
(
f
,
g
)
(f,g)
(
f
,
g
)
of functions
f
,
g
:
R
→
R
f,g :\mathbb{R} \to \mathbb{R}
f
,
g
:
R
→
R
such that
x
f
(
x
+
y
)
+
α
⋅
y
f
(
x
−
y
)
=
g
(
x
)
+
g
(
y
)
,
∀
x
,
y
∈
R
.
xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.
x
f
(
x
+
y
)
+
α
⋅
y
f
(
x
−
y
)
=
g
(
x
)
+
g
(
y
)
,
∀
x
,
y
∈
R
.
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