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2024 Serbia National Math Olympiad
5
Sum of powers of cos is constant
Sum of powers of cos is constant
Source: Serbia 2024 MO Problem 5
April 4, 2024
algebra
Problem Statement
Let
n
≥
3
n \geq 3
n
≥
3
be a positive integer. Find all positive integers
k
k
k
, such that the function
f
:
R
→
R
f:\mathbb{R} \rightarrow \mathbb{R}
f
:
R
→
R
defined by
f
(
x
)
=
cos
k
(
x
)
+
cos
k
(
x
+
2
π
n
)
+
…
+
cos
k
(
x
+
2
(
n
−
1
)
π
n
)
f(x)=\cos^k(x)+\cos^k(x+\frac{2\pi}{n})+\ldots +\cos^k(x+\frac{2(n-1)\pi}{n})
f
(
x
)
=
cos
k
(
x
)
+
cos
k
(
x
+
n
2
π
)
+
…
+
cos
k
(
x
+
n
2
(
n
−
1
)
π
)
is constant.
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