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Problems
Contests
National and Regional Contests
China Contests
China Team Selection Test
2003 China Team Selection Test
1
Sum a_k =<n
Sum a_k =<n
Source: China TST 2003
June 29, 2006
trigonometry
inequalities unsolved
inequalities
Problem Statement
Let
g
(
x
)
=
∑
k
=
1
n
a
k
cos
k
x
g(x)= \sum_{k=1}^{n} a_k \cos{kx}
g
(
x
)
=
∑
k
=
1
n
a
k
cos
k
x
,
a
1
,
a
2
,
⋯
,
a
n
,
x
∈
R
a_1,a_2, \cdots, a_n, x \in R
a
1
,
a
2
,
⋯
,
a
n
,
x
∈
R
. If
g
(
x
)
≥
−
1
g(x) \geq -1
g
(
x
)
≥
−
1
holds for every
x
∈
R
x \in R
x
∈
R
, prove that
∑
k
=
1
n
a
k
≤
n
\sum_{k=1}^{n}a_k \leq n
∑
k
=
1
n
a
k
≤
n
.
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