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Romanian National Olympiad 2018 - Grade 12 - problem 2
Romanian National Olympiad 2018 - Grade 12 - problem 2
Source: Romania NMO - 2018
April 7, 2018
function
calculus
integration
real analysis
college contests
Problem Statement
Let
F
\mathcal{F}
F
be the set of continuous functions
f
:
R
→
R
f: \mathbb{R} \to \mathbb{R}
f
:
R
→
R
such that
e
f
(
x
)
+
f
(
x
)
≥
x
+
1
,
∀
x
∈
R
e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R}
e
f
(
x
)
+
f
(
x
)
≥
x
+
1
,
∀
x
∈
R
For
f
∈
F
,
f \in \mathcal{F},
f
∈
F
,
let
I
(
f
)
=
∫
0
e
f
(
x
)
d
x
I(f)=\int_0^ef(x) dx
I
(
f
)
=
∫
0
e
f
(
x
)
d
x
Determine
min
f
∈
F
I
(
f
)
.
\min_{f \in \mathcal{F}}I(f).
min
f
∈
F
I
(
f
)
.
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