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China Northern MO
2016 China Northern MO
5
2016 CNMO Grade P5
2016 CNMO Grade P5
Source: 2016 China Northern MO Grade 10, Problem 5
February 24, 2020
algebra
Problem Statement
Let
θ
i
∈
(
0
,
π
2
)
(
i
=
1
,
2
,
⋯
,
n
)
\theta_{i}\in(0,\frac{\pi}{2})(i=1,2,\cdots,n)
θ
i
∈
(
0
,
2
π
)
(
i
=
1
,
2
,
⋯
,
n
)
. Prove:
(
∑
i
=
1
n
tan
θ
i
)
(
∑
i
=
1
n
cot
θ
i
)
≥
(
∑
i
=
1
n
sin
θ
i
)
2
+
(
∑
i
=
1
n
cos
θ
i
)
2
.
(\sum_{i=1}^n\tan\theta_{i})(\sum_{i=1}^n\cot\theta_{i})\geq(\sum_{i=1}^n\sin\theta_{i})^2+(\sum_{i=1}^n\cos\theta_{i})^2.
(
i
=
1
∑
n
tan
θ
i
)
(
i
=
1
∑
n
cot
θ
i
)
≥
(
i
=
1
∑
n
sin
θ
i
)
2
+
(
i
=
1
∑
n
cos
θ
i
)
2
.
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