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National and Regional Contests
China Contests
(China) National High School Mathematics League
2015 China Second Round Olympiad
1
China Second Round Olympiad 2015 Test 2 Q1
China Second Round Olympiad 2015 Test 2 Q1
Source: China
September 13, 2015
inequalities
Probabilistic Method
Problem Statement
Let
a
1
,
a
2
,
…
,
a
n
a_1, a_2, \ldots, a_n
a
1
,
a
2
,
…
,
a
n
be real numbers.Prove that you can select
ε
1
,
ε
2
,
…
,
ε
n
∈
{
−
1
,
1
}
\varepsilon _1, \varepsilon _2, \ldots, \varepsilon _n\in\{-1,1\}
ε
1
,
ε
2
,
…
,
ε
n
∈
{
−
1
,
1
}
such that
(
∑
i
=
1
n
a
i
)
2
+
(
∑
i
=
1
n
ε
i
a
i
)
2
≤
(
n
+
1
)
(
∑
i
=
1
n
a
i
2
)
.
\left( \sum_{i=1}^{n}a_{i}\right)^2 +\left( \sum_{i=1}^{n}\varepsilon _ia_{i}\right)^2 \leq(n+1)\left( \sum_{i=1}^{n}a^2_{i}\right).
(
i
=
1
∑
n
a
i
)
2
+
(
i
=
1
∑
n
ε
i
a
i
)
2
≤
(
n
+
1
)
(
i
=
1
∑
n
a
i
2
)
.
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