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Germany Team Selection Test
2012 Germany Team Selection Test
3
Inequality with Sum of Squares
Inequality with Sum of Squares
Source: Germany TST 2012 P3
April 13, 2020
inequalities
Problem Statement
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be positive real numbers with
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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