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1988 Greece National Olympiad
1
2f(x+y+xy)= a f(x)+ bf(y)+f(xy)
2f(x+y+xy)= a f(x)+ bf(y)+f(xy)
Source: 1988 Greece MO Grade X p1
September 6, 2024
function
algebra
functional equation
Problem Statement
Find all functions
f
:
R
→
R
f: \mathbb{R}\to\mathbb{R}
f
:
R
→
R
that satidfy :
2
f
(
x
+
y
+
x
y
)
=
a
f
(
x
)
+
b
f
(
y
)
+
f
(
x
y
)
2f(x+y+xy)= a f(x)+ bf(y)+f(xy)
2
f
(
x
+
y
+
x
y
)
=
a
f
(
x
)
+
b
f
(
y
)
+
f
(
x
y
)
for any
x
,
y
∈
R
x,y \in\mathbb{R}
x
,
y
∈
R
όπου
a
,
b
∈
R
a,b\in\mathbb{R}
a
,
b
∈
R
with
a
2
−
a
≠
b
2
−
b
a^2-a\ne b^2-b
a
2
−
a
=
b
2
−
b
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