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1994 Turkey MO (2nd round)
4
4
Part of
1994 Turkey MO (2nd round)
Problems
(1)
Turkish MO 1994 P4
Source: Turkish Mathematical Olympiad 2nd Round 1994
9/27/2006
Let
f
:
R
+
→
R
+
f: \mathbb{R}^{+}\rightarrow \mathbb{R}+
f
:
R
+
→
R
+
be an increasing function. For each
u
∈
R
+
u\in\mathbb{R}^{+}
u
∈
R
+
, we denote
g
(
u
)
=
inf
{
f
(
t
)
+
u
/
t
∣
t
>
0
}
g(u)=\inf\{ f(t)+u/t \mid t>0\}
g
(
u
)
=
in
f
{
f
(
t
)
+
u
/
t
∣
t
>
0
}
. Prove that:
(
a
)
(a)
(
a
)
If
x
≤
g
(
x
y
)
x\leq g(xy)
x
≤
g
(
x
y
)
, then
x
≤
2
f
(
2
y
)
x\leq 2f(2y)
x
≤
2
f
(
2
y
)
;
(
b
)
(b)
(
b
)
If
x
≤
f
(
y
)
x\leq f(y)
x
≤
f
(
y
)
, then
x
≤
2
g
(
x
y
)
x\leq 2g(xy)
x
≤
2
g
(
x
y
)
.
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