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Putnam
2023 Putnam
A1
2023 Putnam A1
2023 Putnam A1
Source:
December 3, 2023
Putnam
Putnam 2023
Problem Statement
For a positive integer
n
n
n
, let
f
n
(
x
)
=
cos
(
x
)
cos
(
2
x
)
cos
(
3
x
)
⋯
cos
(
n
x
)
f_n(x)=\cos (x) \cos (2 x) \cos (3 x) \cdots \cos (n x)
f
n
(
x
)
=
cos
(
x
)
cos
(
2
x
)
cos
(
3
x
)
⋯
cos
(
n
x
)
. Find the smallest
n
n
n
such that
∣
f
n
′
′
(
0
)
∣
>
2023
\left|f_n^{\prime \prime}(0)\right|>2023
∣
f
n
′′
(
0
)
∣
>
2023
.
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