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2
IMC 1995 Problem 2
IMC 1995 Problem 2
Source: IMC 1995
February 18, 2021
function
calculus
real analysis
Integral
inequalities
Problem Statement
Let
f
f
f
be a continuous function on
[
0
,
1
]
[0,1]
[
0
,
1
]
such that for every
x
∈
[
0
,
1
]
x\in [0,1]
x
∈
[
0
,
1
]
, we have
∫
x
1
f
(
t
)
d
t
≥
1
−
x
2
2
\int_{x}^{1}f(t)dt \geq\frac{1-x^{2}}{2}
∫
x
1
f
(
t
)
d
t
≥
2
1
−
x
2
. Show that
∫
0
1
f
(
t
)
2
d
t
≥
1
3
\int_{0}^{1}f(t)^{2}dt \geq \frac{1}{3}
∫
0
1
f
(
t
)
2
d
t
≥
3
1
.
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