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Balkan MO Shortlist
2007 Balkan MO Shortlist
A3
Fourier coefficients inequality
Fourier coefficients inequality
Source: RMO TST 2007, day 6
June 8, 2007
inequalities
trigonometry
three variable inequality
Fourier
Problem Statement
For
n
∈
N
n\in\mathbb{N}
n
∈
N
,
n
≥
2
n\geq 2
n
≥
2
,
a
i
,
b
i
∈
R
a_{i}, b_{i}\in\mathbb{R}
a
i
,
b
i
∈
R
,
1
≤
i
≤
n
1\leq i\leq n
1
≤
i
≤
n
, such that
∑
i
=
1
n
a
i
2
=
∑
i
=
1
n
b
i
2
=
1
,
∑
i
=
1
n
a
i
b
i
=
0.
\sum_{i=1}^{n}a_{i}^{2}=\sum_{i=1}^{n}b_{i}^{2}=1, \sum_{i=1}^{n}a_{i}b_{i}=0.
i
=
1
∑
n
a
i
2
=
i
=
1
∑
n
b
i
2
=
1
,
i
=
1
∑
n
a
i
b
i
=
0.
Prove that
(
∑
i
=
1
n
a
i
)
2
+
(
∑
i
=
1
n
b
i
)
2
≤
n
.
\left(\sum_{i=1}^{n}a_{i}\right)^{2}+\left(\sum_{i=1}^{n}b_{i}\right)^{2}\leq n.
(
i
=
1
∑
n
a
i
)
2
+
(
i
=
1
∑
n
b
i
)
2
≤
n
.
Cezar Lupu & Tudorel Lupu
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