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Turkey Team Selection Test
2011 Turkey Team Selection Test
2
Inequality with a condition
Inequality with a condition
Source: Turkish TST 2011 Problem 5
April 10, 2011
inequalities
inequalities proposed
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers satisfying
a
2
+
b
2
+
c
2
≥
3.
a^2+b^2+c^2 \geq 3.
a
2
+
b
2
+
c
2
≥
3.
Prove that
(
a
+
1
)
(
b
+
2
)
(
b
+
1
)
(
b
+
5
)
+
(
b
+
1
)
(
c
+
2
)
(
c
+
1
)
(
c
+
5
)
+
(
c
+
1
)
(
a
+
2
)
(
a
+
1
)
(
a
+
5
)
≥
3
2
\frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}
(
b
+
1
)
(
b
+
5
)
(
a
+
1
)
(
b
+
2
)
+
(
c
+
1
)
(
c
+
5
)
(
b
+
1
)
(
c
+
2
)
+
(
a
+
1
)
(
a
+
5
)
(
c
+
1
)
(
a
+
2
)
≥
2
3
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