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Problems
Contests
National and Regional Contests
The Philippines Contests
Philippine MO
2009 Philippine MO
4
2009 Philippines MO P4
2009 Philippines MO P4
Source:
November 16, 2015
PMO
2009
algebra
Inequality
minimum value
Problem Statement
Let
k
k
k
be a positive real number such that
1
k
+
a
+
1
k
+
b
+
1
k
+
c
≤
1
\frac{1}{k+a} + \frac{1}{k+b} + \frac{1}{k+c} \leq 1
k
+
a
1
+
k
+
b
1
+
k
+
c
1
≤
1
for any positive positive real numbers
a
a
a
,
b
b
b
and
c
c
c
with
a
b
c
=
1
abc = 1
ab
c
=
1
. Find the minimum value of
k
k
k
.
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