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The Philippines Contests
Philippine MO
2017 Philippine MO
2
Some inequality with min function
Some inequality with min function
Source: Philippine Mathematical Olympiad 2017
December 31, 2017
inequalities
function
Problem Statement
Find all positive real numbers
(
a
,
b
,
c
)
≤
1
(a,b,c) \leq 1
(
a
,
b
,
c
)
≤
1
which satisfy
min
{
a
b
+
1
a
b
c
b
c
+
1
a
b
c
,
a
c
+
1
a
b
c
}
=
1
−
a
a
+
1
−
b
b
+
1
−
c
c
\huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}
min
{
ab
c
ab
+
1
ab
c
b
c
+
1
,
ab
c
a
c
+
1
}
=
a
1
−
a
+
b
1
−
b
+
c
1
−
c
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