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National and Regional Contests
France Contests
France Team Selection Test
2014 France Team Selection Test
2014 France Team Selection Test
Part of
France Team Selection Test
Subcontests
(1)
6
1
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Coefficients from -1 and +1
Let
n
n
n
be a positive integer and
x
1
,
x
2
,
…
,
x
n
x_1,x_2,\ldots,x_n
x
1
,
x
2
,
…
,
x
n
be positive reals. Show that there are numbers
a
1
,
a
2
,
…
,
a
n
∈
{
−
1
,
1
}
a_1,a_2,\ldots, a_n \in \{-1,1\}
a
1
,
a
2
,
…
,
a
n
∈
{
−
1
,
1
}
such that the following holds:
a
1
x
1
2
+
a
2
x
2
2
+
⋯
+
a
n
x
n
2
≥
(
a
1
x
1
+
a
2
x
2
+
⋯
+
a
n
x
n
)
2
a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2
a
1
x
1
2
+
a
2
x
2
2
+
⋯
+
a
n
x
n
2
≥
(
a
1
x
1
+
a
2
x
2
+
⋯
+
a
n
x
n
)
2