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IMO Shortlist
1969 IMO Shortlist
38
38
Part of
1969 IMO Shortlist
Problems
(1)
Evaluate a trigonometric expression.
Source:
10/3/2010
(
H
U
N
5
)
(HUN 5)
(
H
U
N
5
)
Let
r
r
r
and
m
(
r
≤
m
)
m (r \le m)
m
(
r
≤
m
)
be natural numbers and
A
k
=
2
k
−
1
2
m
π
Ak =\frac{2k-1}{2m}\pi
A
k
=
2
m
2
k
−
1
π
. Evaluate
1
m
2
∑
k
=
1
m
∑
l
=
1
m
sin
(
r
A
k
)
sin
(
r
A
l
)
cos
(
r
A
k
−
r
A
l
)
\frac{1}{m^2}\displaystyle\sum_{k=1}^{m}\displaystyle\sum_{l=1}^{m}\sin(rA_k)\sin(rA_l)\cos(rA_k-rA_l)
m
2
1
k
=
1
∑
m
l
=
1
∑
m
sin
(
r
A
k
)
sin
(
r
A
l
)
cos
(
r
A
k
−
r
A
l
)
algebra
series summation
trigonometry
Trigonometric Identities
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