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2013 Harvard-MIT Mathematics Tournament
25
25
Part of
2013 Harvard-MIT Mathematics Tournament
Problems
(1)
2013 HMMT Guts #25: Sequence of Complex Numbers
Source:
3/26/2013
The sequence
(
z
n
)
(z_n)
(
z
n
)
of complex numbers satisfies the following properties:[*]
z
1
z_1
z
1
and
z
2
z_2
z
2
are not real. [*]
z
n
+
2
=
z
n
+
1
2
z
n
z_{n+2}=z_{n+1}^2z_n
z
n
+
2
=
z
n
+
1
2
z
n
for all integers
n
≥
1
n\geq 1
n
≥
1
. [*]
z
n
+
3
z
n
2
\dfrac{z_{n+3}}{z_n^2}
z
n
2
z
n
+
3
is real for all integers
n
≥
1
n\geq 1
n
≥
1
. [*]
∣
z
3
z
4
∣
=
∣
z
4
z
5
∣
=
2
\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2
z
4
z
3
=
z
5
z
4
=
2
. Find the product of all possible values of
z
1
z_1
z
1
.
HMMT
complex numbers