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2024 CMIMC
2024 CMIMC Algebra and Number Theory
7
7
Part of
2024 CMIMC Algebra and Number Theory
Problems
(1)
2024 Alg/NT Problem 7
Source:
4/14/2024
Let
x
0
x_0
x
0
,
x
1
x_1
x
1
,
x
2
x_2
x
2
, and
x
3
x_3
x
3
be complex numbers forming a square centered at
0
0
0
in the complex plane with side length
2
2
2
. For each
0
≤
k
≤
3
0\leq k\leq 3
0
≤
k
≤
3
, there are four more complex numbers
z
4
k
,
z
4
k
+
1
z_{4k}, z_{4k+1}
z
4
k
,
z
4
k
+
1
,
z
4
k
+
2
z_{4k+2}
z
4
k
+
2
, and
z
4
k
+
3
z_{4k+3}
z
4
k
+
3
forming a square centered at
x
k
x_k
x
k
with side length
2
\sqrt 2
2
. Given that
∏
i
=
0
15
z
i
\prod_{i=0}^{15} z_i
∏
i
=
0
15
z
i
is a positive integer, how many possible values could it take?Proposed by Hari Desikan
algebra