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7
2010 Algebra #7: System of Complex Equations
2010 Algebra #7: System of Complex Equations
Source:
July 15, 2012
complex numbers
Problem Statement
Let
a
,
b
,
c
,
x
,
y
,
a,b,c,x,y,
a
,
b
,
c
,
x
,
y
,
and
z
z
z
be complex numbers such that
a
=
b
+
c
x
−
2
,
b
=
c
+
a
y
−
2
,
c
=
a
+
b
z
−
2
.
a=\dfrac{b+c}{x-2},\qquad b=\dfrac{c+a}{y-2},\qquad c=\dfrac{a+b}{z-2}.
a
=
x
−
2
b
+
c
,
b
=
y
−
2
c
+
a
,
c
=
z
−
2
a
+
b
.
If
x
y
+
y
z
+
x
z
=
67
xy+yz+xz=67
x
y
+
yz
+
x
z
=
67
and
x
+
y
+
z
=
2010
x+y+z=2010
x
+
y
+
z
=
2010
, find the value of
x
y
z
xyz
x
yz
.
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