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Brazil National Olympiad
2012 Brazil National Olympiad
6
Surjective Functions
Surjective Functions
Source: Problem 6, Brazilian MO, 2012
November 28, 2012
function
ratio
arithmetic sequence
algebra
functional equation
Problem Statement
Find all surjective functions
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
f\colon (0,+\infty) \to (0,+\infty)
f
:
(
0
,
+
∞
)
→
(
0
,
+
∞
)
such that
2
x
f
(
f
(
x
)
)
=
f
(
x
)
(
x
+
f
(
f
(
x
)
)
)
2x f(f(x)) = f(x)(x+f(f(x)))
2
x
f
(
f
(
x
))
=
f
(
x
)
(
x
+
f
(
f
(
x
)))
for all
x
>
0
x>0
x
>
0
.
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