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8
2018 PUMaC Algebra A8
2018 PUMaC Algebra A8
Source:
November 25, 2018
algebra
Problem Statement
p
q
=
∑
n
=
1
∞
1
2
n
+
6
(
10
−
4
cos
2
(
π
n
24
)
)
(
1
−
(
−
1
)
n
)
−
3
cos
(
π
n
24
)
(
1
+
(
−
1
)
n
)
25
−
16
cos
2
(
π
n
24
)
\frac{p}{q} = \sum_{n = 1}^\infty \frac{1}{2^{n + 6}} \frac{(10 - 4\cos^2(\frac{\pi n}{24})) (1 - (-1)^n) - 3\cos(\frac{\pi n}{24}) (1 + (-1)^n)}{25 - 16\cos^2(\frac{\pi n}{24})}
q
p
=
n
=
1
∑
∞
2
n
+
6
1
25
−
16
cos
2
(
24
πn
)
(
10
−
4
cos
2
(
24
πn
))
(
1
−
(
−
1
)
n
)
−
3
cos
(
24
πn
)
(
1
+
(
−
1
)
n
)
where
p
p
p
and
q
q
q
are relatively prime positive integers. Find
p
+
q
p + q
p
+
q
.
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