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Moldova Team Selection Test
2011 Moldova Team Selection Test
2
Product equals to 1
Product equals to 1
Source: Moldova TST 2011, day 1, problem 2
March 8, 2011
inequalities
function
inequalities unsolved
Problem Statement
Let
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
be real positive numbers such that
x
1
⋅
x
2
⋯
x
n
=
1
x_1\cdot x_2\cdots x_n=1
x
1
⋅
x
2
⋯
x
n
=
1
. Prove the inequality
1
x
1
(
x
1
+
1
)
+
1
x
2
(
x
2
+
1
)
+
⋯
+
1
x
n
(
x
n
+
1
)
≥
n
2
\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2
x
1
(
x
1
+
1
)
1
+
x
2
(
x
2
+
1
)
1
+
⋯
+
x
n
(
x
n
+
1
)
1
≥
2
n
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