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A3
2019 Junior Pre-selection question
2019 Junior Pre-selection question
Source: Greece
March 30, 2019
inequalities
algebra
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers . Prove that
1
a
b
(
b
+
1
)
(
c
+
1
)
+
1
b
c
(
c
+
1
)
(
a
+
1
)
+
1
c
a
(
a
+
1
)
(
b
+
1
)
≥
3
(
1
+
a
b
c
)
2
.
\frac{1}{ab(b+1)(c+1)}+\frac{1}{bc(c+1)(a+1)}+\frac{1}{ca(a+1)(b+1)}\geq\frac{3}{(1+abc)^2}.
ab
(
b
+
1
)
(
c
+
1
)
1
+
b
c
(
c
+
1
)
(
a
+
1
)
1
+
c
a
(
a
+
1
)
(
b
+
1
)
1
≥
(
1
+
ab
c
)
2
3
.
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