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4
Functional Equation - [UKRMO 2009 Grade 10]
Functional Equation - [UKRMO 2009 Grade 10]
Source:
January 23, 2011
function
algebra
functional equation
algebra unsolved
Problem Statement
Find all functions
f
:
R
→
R
f : \mathbb R \to \mathbb R
f
:
R
→
R
such that
f
(
x
+
x
y
+
f
(
y
)
)
=
(
f
(
x
)
+
1
2
)
(
f
(
y
)
+
1
2
)
∀
x
,
y
∈
R
.
f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.
f
(
x
+
x
y
+
f
(
y
)
)
=
(
f
(
x
)
+
2
1
)
(
f
(
y
)
+
2
1
)
∀
x
,
y
∈
R
.
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