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3
Inequality with numbers with sum 1
Inequality with numbers with sum 1
Source: 2019 Brazil Ibero TST P3
June 14, 2023
inequalities
algebra
Symmetric inequality
Problem Statement
Let
n
≥
2
n \geq 2
n
≥
2
be an integer and
x
1
,
x
2
,
…
,
x
n
x_1, x_2, \ldots, x_n
x
1
,
x
2
,
…
,
x
n
be positive real numbers such that
∑
i
=
1
n
x
i
=
1
\sum_{i=1}^nx_i=1
∑
i
=
1
n
x
i
=
1
. Show that
(
∑
i
=
1
n
1
1
−
x
i
)
(
∑
1
≤
i
<
j
≤
n
x
i
x
j
)
≤
n
2
.
\bigg(\sum_{i=1}^n\frac{1}{1-x_i}\bigg)\bigg(\sum_{1 \leq i < j \leq n}x_ix_j\bigg) \leq \frac{n}{2}.
(
i
=
1
∑
n
1
−
x
i
1
)
(
1
≤
i
<
j
≤
n
∑
x
i
x
j
)
≤
2
n
.
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