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2022 CMIMC
2.8 1.4
2022 Alg/NT Div 1 P4 (Div 2 P8)
2022 Alg/NT Div 1 P4 (Div 2 P8)
Source:
February 28, 2022
algebra
number theory
Problem Statement
Let
z
z
z
be a complex number that satisfies the equation
z
−
4
z
2
−
5
z
+
1
+
2
z
−
4
2
z
2
−
5
z
+
1
+
z
−
2
z
2
−
3
z
+
1
=
3
z
.
\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}.
z
2
−
5
z
+
1
z
−
4
+
2
z
2
−
5
z
+
1
2
z
−
4
+
z
2
−
3
z
+
1
z
−
2
=
z
3
.
Over all possible values of
z
z
z
, find the sum of the values of
∣
1
z
2
−
5
z
+
1
+
1
2
z
2
−
5
z
+
1
+
1
z
2
−
3
z
+
1
∣
.
\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.
z
2
−
5
z
+
1
1
+
2
z
2
−
5
z
+
1
1
+
z
2
−
3
z
+
1
1
.
Proposed by Justin Hsieh
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