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Romania National Olympiad
2021 Romania National Olympiad
1
Romanian NMO 2021
Romanian NMO 2021
Source:
April 25, 2021
integrals
college contests
Problem Statement
Find all continuous functions
f
:
[
0
,
1
]
→
[
0
,
∞
)
f:\left[0,1\right]\rightarrow[0,\infty)
f
:
[
0
,
1
]
→
[
0
,
∞
)
such that:
∫
0
1
f
(
x
)
d
x
⋅
∫
0
1
f
2
(
x
)
d
x
⋅
.
.
.
⋅
∫
0
1
f
2020
(
x
)
d
x
=
(
∫
0
1
f
2021
(
x
)
d
x
)
1010
\int_{0}^{1}f\left(x\right)dx\cdotp\int_{0}^{1}f^{2}\left(x\right)dx\cdotp...\cdotp\int_{0}^{1}f^{2020}\left(x\right)dx=\left(\int_{0}^{1}f^{2021}\left(x\right)dx\right)^{1010}
∫
0
1
f
(
x
)
d
x
⋅
∫
0
1
f
2
(
x
)
d
x
⋅
...
⋅
∫
0
1
f
2020
(
x
)
d
x
=
(
∫
0
1
f
2021
(
x
)
d
x
)
1010
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