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Putnam
1962 Putnam
B6
Putnam 1962 B6
Putnam 1962 B6
Source: Putnam 1962
May 21, 2022
Putnam
trigonometry
Fourier
Problem Statement
Let
f
(
x
)
=
∑
k
=
0
n
a
k
sin
k
x
+
b
k
cos
k
x
,
f(x) =\sum_{k=0}^{n} a_{k} \sin kx +b_{k} \cos kx,
f
(
x
)
=
k
=
0
∑
n
a
k
sin
k
x
+
b
k
cos
k
x
,
where
a
k
a_k
a
k
and
b
k
b_k
b
k
are constants. Show that if
∣
f
(
x
)
∣
≤
1
|f(x)| \leq 1
∣
f
(
x
)
∣
≤
1
for
x
∈
[
0
,
2
π
]
x \in [0, 2 \pi]
x
∈
[
0
,
2
π
]
and there exist
0
≤
x
1
<
x
2
<
…
<
x
2
n
<
2
π
0\leq x_1 < x_2 <\ldots < x_{2n} < 2 \pi
0
≤
x
1
<
x
2
<
…
<
x
2
n
<
2
π
with
∣
f
(
x
i
)
∣
=
1
,
|f(x_i )|=1,
∣
f
(
x
i
)
∣
=
1
,
then
f
(
x
)
=
cos
(
n
x
+
a
)
f(x)= \cos(nx +a)
f
(
x
)
=
cos
(
n
x
+
a
)
for some constant
a
.
a.
a
.
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