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Romania National Olympiad
2010 Romania National Olympiad
3
Romania National Olympiad 2010 - Grade XI
Romania National Olympiad 2010 - Grade XI
Source:
April 10, 2011
function
real analysis
real analysis unsolved
Problem Statement
Let
f
:
R
→
[
0
,
∞
)
f:\mathbb{R}\rightarrow [0,\infty)
f
:
R
→
[
0
,
∞
)
. Prove that
f
(
x
+
y
)
≥
(
y
+
1
)
f
(
x
)
,
(
∀
)
x
∈
R
f(x+y)\ge (y+1)f(x),\ (\forall)x\in \mathbb{R}
f
(
x
+
y
)
≥
(
y
+
1
)
f
(
x
)
,
(
∀
)
x
∈
R
if and only if the function
g
:
R
→
[
0
,
∞
)
,
g
(
x
)
=
e
−
x
f
(
x
)
,
(
∀
)
x
∈
R
g:\mathbb{R}\rightarrow [0,\infty),\ g(x)=e^{-x}f(x),\ (\forall)x\in \mathbb{R}
g
:
R
→
[
0
,
∞
)
,
g
(
x
)
=
e
−
x
f
(
x
)
,
(
∀
)
x
∈
R
is increasing.
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