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2010 Germany Team Selection Test
3
f(x)f(y) = (x+y+1)^2 * f((xy-1)/(x+y+1))
f(x)f(y) = (x+y+1)^2 * f((xy-1)/(x+y+1))
Source: AIMO 5/3, German TST 2010
July 16, 2011
function
trigonometry
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
)
f
(
y
)
=
(
x
+
y
+
1
)
2
⋅
f
(
x
y
−
1
x
+
y
+
1
)
f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)
f
(
x
)
f
(
y
)
=
(
x
+
y
+
1
)
2
⋅
f
(
x
+
y
+
1
x
y
−
1
)
∀
x
,
y
∈
R
\forall x,y \in \mathbb{R}
∀
x
,
y
∈
R
with
x
+
y
+
1
≠
0
x+y+1 \neq 0
x
+
y
+
1
=
0
and
f
(
x
)
>
1
f(x) > 1
f
(
x
)
>
1
∀
x
>
0.
\forall x > 0.
∀
x
>
0.
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