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IMC
1995 IMC
11
11
Part of
1995 IMC
Problems
(1)
IMC 1995 Problem 11
Source: IMC 1995
2/19/2021
a) Prove that every function of the form
f
(
x
)
=
a
0
2
+
cos
(
x
)
+
∑
n
=
2
N
a
n
cos
(
n
x
)
f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)
f
(
x
)
=
2
a
0
+
cos
(
x
)
+
n
=
2
∑
N
a
n
cos
(
n
x
)
with
∣
a
0
∣
<
1
|a_{0}|<1
∣
a
0
∣
<
1
has positive as well as negative values in the period
[
0
,
2
π
)
[0,2\pi)
[
0
,
2
π
)
. b) Prove that the function
F
(
x
)
=
∑
n
=
1
100
cos
(
n
3
2
x
)
F(x)=\sum_{n=1}^{100}\cos(n^{\frac{3}{2}}x)
F
(
x
)
=
n
=
1
∑
100
cos
(
n
2
3
x
)
has at least
40
40
40
zeroes in the interval
(
0
,
1000
)
(0,1000)
(
0
,
1000
)
.
function
real analysis