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5
2014 Japan Mathematical Olympiad Finals Problem 5
2014 Japan Mathematical Olympiad Finals Problem 5
Source:
February 16, 2014
inequalities
inequalities proposed
Problem Statement
Find the maximum value of real number
k
k
k
such that
a
1
+
9
b
c
+
k
(
b
−
c
)
2
+
b
1
+
9
c
a
+
k
(
c
−
a
)
2
+
c
1
+
9
a
b
+
k
(
a
−
b
)
2
≥
1
2
\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}
1
+
9
b
c
+
k
(
b
−
c
)
2
a
+
1
+
9
c
a
+
k
(
c
−
a
)
2
b
+
1
+
9
ab
+
k
(
a
−
b
)
2
c
≥
2
1
holds for all non-negative real numbers
a
,
b
,
c
a,\ b,\ c
a
,
b
,
c
satisfying
a
+
b
+
c
=
1
a+b+c=1
a
+
b
+
c
=
1
.
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