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2016 IMO Shortlist
A4
Functional equation on (0,infinity)
Functional equation on (0,infinity)
Source: 2016 IMO Shortlist A4
July 19, 2017
algebra
functional equation
IMO Shortlist
nice problem
Problem Statement
Find all functions
f
:
(
0
,
∞
)
→
(
0
,
∞
)
f:(0,\infty)\rightarrow (0,\infty)
f
:
(
0
,
∞
)
→
(
0
,
∞
)
such that for any
x
,
y
∈
(
0
,
∞
)
x,y\in (0,\infty)
x
,
y
∈
(
0
,
∞
)
,
x
f
(
x
2
)
f
(
f
(
y
)
)
+
f
(
y
f
(
x
)
)
=
f
(
x
y
)
(
f
(
f
(
x
2
)
)
+
f
(
f
(
y
2
)
)
)
.
xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).
x
f
(
x
2
)
f
(
f
(
y
))
+
f
(
y
f
(
x
))
=
f
(
x
y
)
(
f
(
f
(
x
2
))
+
f
(
f
(
y
2
))
)
.
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