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National and Regional Contests
Russia Contests
239 Open Math Olympiad
2024 239 Open Mathematical Olympiad
1
Continuous function
Continuous function
Source: 239 MO 2024 S1
May 22, 2024
function
algebra
Problem Statement
Let
f
:
R
≥
0
→
R
≥
0
f:\mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}
f
:
R
≥
0
→
R
≥
0
be a continuous function such that
f
(
0
)
=
0
f(0)=0
f
(
0
)
=
0
and
f
(
x
)
+
f
(
f
(
x
)
)
+
f
(
f
(
f
(
x
)
)
)
=
3
x
f(x)+f(f(x))+f(f(f(x)))=3x
f
(
x
)
+
f
(
f
(
x
))
+
f
(
f
(
f
(
x
)))
=
3
x
for all
x
>
0
x>0
x
>
0
. Show that
f
(
x
)
=
x
f(x)=x
f
(
x
)
=
x
for all
x
>
0
x>0
x
>
0
.
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