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9
2022 PUMaC Team #9
2022 PUMaC Team #9
Source:
September 9, 2023
algebra
Problem Statement
In the complex plane, let
z
1
,
z
2
,
z
3
z_1, z_2, z_3
z
1
,
z
2
,
z
3
be the roots of the polynomial
p
(
x
)
=
x
3
−
a
x
2
+
b
x
−
a
b
p(x) = x^3- ax^2 + bx - ab
p
(
x
)
=
x
3
−
a
x
2
+
b
x
−
ab
. Find the number of integers
n
n
n
between
1
1
1
and
500
500
500
inclusive that are expressible as
z
1
4
+
z
2
4
+
z
3
4
z^4_1 +z^4_2 +z^4_3
z
1
4
+
z
2
4
+
z
3
4
for some choice of positive integers
a
,
b
a, b
a
,
b
.
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