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Balkan MO
2003 Balkan MO
3
3
Part of
2003 Balkan MO
Problems
(1)
rational function with f(1)+1>0
Source: Balkan Math Olympiad BMO 2003, problem 3
10/24/2005
Find all functions
f
:
Q
→
R
f: \mathbb{Q}\to\mathbb{R}
f
:
Q
→
R
which fulfill the following conditions: a)
f
(
1
)
+
1
>
0
f(1)+1>0
f
(
1
)
+
1
>
0
; b)
f
(
x
+
y
)
−
x
f
(
y
)
−
y
f
(
x
)
=
f
(
x
)
f
(
y
)
−
x
−
y
+
x
y
f(x+y) -xf(y) -yf(x) = f(x)f(y) -x-y +xy
f
(
x
+
y
)
−
x
f
(
y
)
−
y
f
(
x
)
=
f
(
x
)
f
(
y
)
−
x
−
y
+
x
y
, for all
x
,
y
∈
Q
x,y\in\mathbb{Q}
x
,
y
∈
Q
; c)
f
(
x
)
=
2
f
(
x
+
1
)
+
x
+
2
f(x) = 2f(x+1) +x+2
f
(
x
)
=
2
f
(
x
+
1
)
+
x
+
2
, for every
x
∈
Q
x\in\mathbb{Q}
x
∈
Q
.
function
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algebra