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MAA AMC
USAMO
2022 USAMO
3
Invert Your Expectations
Invert Your Expectations
Source: USAMO 2022/3
March 24, 2022
AMC
USA(J)MO
USAMO
algebra
Problem Statement
Let
R
>
0
\mathbb{R}_{>0}
R
>
0
be the set of all positive real numbers. Find all functions
f
:
R
>
0
→
R
>
0
f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}
f
:
R
>
0
→
R
>
0
such that for all
x
,
y
∈
R
>
0
x,y\in \mathbb{R}_{>0}
x
,
y
∈
R
>
0
we have
f
(
x
)
=
f
(
f
(
f
(
x
)
)
+
y
)
+
f
(
x
f
(
y
)
)
f
(
x
+
y
)
.
f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).
f
(
x
)
=
f
(
f
(
f
(
x
))
+
y
)
+
f
(
x
f
(
y
))
f
(
x
+
y
)
.
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