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Putnam
1957 Putnam
B3
Putnam 1957 B3
Putnam 1957 B3
Source: Putnam 1957
July 1, 2022
Putnam
function
integration
inequalities
Problem Statement
For
f
(
x
)
f(x)
f
(
x
)
a positive , monotone decreasing function defined in
[
0
,
1
]
,
[0,1],
[
0
,
1
]
,
prove that
∫
0
1
f
(
x
)
d
x
⋅
∫
0
1
x
f
(
x
)
2
d
x
≤
∫
0
1
f
(
x
)
2
d
x
⋅
∫
0
1
x
f
(
x
)
d
x
.
\int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.
∫
0
1
f
(
x
)
d
x
⋅
∫
0
1
x
f
(
x
)
2
d
x
≤
∫
0
1
f
(
x
)
2
d
x
⋅
∫
0
1
x
f
(
x
)
d
x
.
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