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Problems
Contests
National and Regional Contests
Spain Contests
Spain Mathematical Olympiad
1992 Spain Mathematical Olympiad
6
6
Part of
1992 Spain Mathematical Olympiad
Problems
(1)
(x+iy)^n+(x-iy)^n = 2x^n , z = x+iy, |z| = 1
Source: Spanish Mathematical Olympiad 1992 P6
8/1/2018
For a positive integer
n
n
n
, let
S
(
n
)
S(n)
S
(
n
)
be the set of complex numbers
z
=
x
+
i
y
z = x+iy
z
=
x
+
i
y
(
x
,
y
∈
R
x,y \in R
x
,
y
∈
R
) with
∣
z
∣
=
1
|z| = 1
∣
z
∣
=
1
satisfying
(
x
+
i
y
)
n
+
(
x
−
i
y
)
n
=
2
x
n
(x+iy)^n+(x-iy)^n = 2x^n
(
x
+
i
y
)
n
+
(
x
−
i
y
)
n
=
2
x
n
. (a) Determine
S
(
n
)
S(n)
S
(
n
)
for
n
=
2
,
3
,
4
n = 2,3,4
n
=
2
,
3
,
4
. (b) Find an upper bound (depending on
n
n
n
) of the number of elements of
S
(
n
)
S(n)
S
(
n
)
for
n
>
5
n > 5
n
>
5
.
complex numbers
algebra