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National and Regional Contests
Cuba Contests
Cuba MO
2015 Cuba MO
5
min { ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 } < = 1/16
min { ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 } < = 1/16
Source: 2015 Cuba 2.5
September 20, 2024
algebra
inequalities
Problem Statement
Let
a
,
b
a, b
a
,
b
and
c
c
c
be real numbers such that
0
<
a
,
b
,
c
<
1
0 < a, b, c < 1
0
<
a
,
b
,
c
<
1
. Prove that:
min
{
a
b
(
1
−
c
)
2
,
b
c
(
1
−
a
)
2
,
c
a
(
1
−
b
)
2
}
≤
1
16
.
\min \ \ \{ab(1 -c)^2, bc(1 - a)^2, ca(1 - b)^2 \} \le \frac{1}{16}.
min
{
ab
(
1
−
c
)
2
,
b
c
(
1
−
a
)
2
,
c
a
(
1
−
b
)
2
}
≤
16
1
.
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