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Problem 5, Iberoamerican Olympiad 2011
Problem 5, Iberoamerican Olympiad 2011
Source:
October 2, 2011
inequalities
induction
strong induction
algebra proposed
algebra
Problem Statement
Let
x
1
,
…
,
x
n
x_1,\ldots ,x_n
x
1
,
…
,
x
n
be positive real numbers. Show that there exist
a
1
,
…
,
a
n
∈
{
−
1
,
1
}
a_1,\ldots ,a_n\in\{-1,1\}
a
1
,
…
,
a
n
∈
{
−
1
,
1
}
such that:
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+\ldots +a_nx_n^2\ge (a_1x_1+a_2x_2+\ldots + a_n x_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
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