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National and Regional Contests
India Contests
India LIMIT
2019 LIMIT
2019 LIMIT Category A
Problem 5
arccos sum
arccos sum
Source: LIMIT 2019 CAS1 P5
April 28, 2021
trigonometry
algebra
Summation
Problem Statement
If
∑
i
=
1
n
cos
−
1
(
α
i
)
=
0
\sum_{i=1}^n\cos^{-1}(\alpha_i)=0
∑
i
=
1
n
cos
−
1
(
α
i
)
=
0
, then find
∑
i
=
1
n
α
i
\sum_{i=1}^n\alpha_i
∑
i
=
1
n
α
i
.
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
n
2
<span class='latex-bold'>(A)</span>~\frac n2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
n
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
n
<span class='latex-bold'>(B)</span>~n
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
n
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
n
π
<span class='latex-bold'>(C)</span>~n\pi
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
nπ
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
n
π
2
<span class='latex-bold'>(D)</span>~\frac{n\pi}2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
nπ
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