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11
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2022 PUMaC Team #11
Source:
September 9, 2023
algebra
Problem Statement
For the function
g
(
a
)
=
max
⏟
x
∈
R
{
cos
x
+
cos
(
x
+
π
6
)
+
cos
(
x
+
π
4
)
+
c
o
s
(
x
+
a
)
}
,
g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},
g
(
a
)
=
x
∈
R
max
{
cos
x
+
cos
(
x
+
6
π
)
+
cos
(
x
+
4
π
)
+
cos
(
x
+
a
)
}
,
let
b
∈
R
b \in R
b
∈
R
be the input that maximizes
g
g
g
. If
cos
2
b
=
m
+
n
+
p
−
q
24
\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}
cos
2
b
=
24
m
+
n
+
p
−
q
for positive integers
m
,
n
,
p
,
q
m, n, p, q
m
,
n
,
p
,
q
, find
m
+
n
+
p
+
q
m + n + p + q
m
+
n
+
p
+
q
.
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