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Nigeria Contests
Nigerian Senior Mathematics Olympiad Round 2
2016 Nigerian Senior MO Round 2
Problem 1
Complex numbers and symmetry
Complex numbers and symmetry
Source: 2016 Nigerian Senior MO Round 2
November 16, 2022
complex numbers
algebra
symmetry
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=\frac{b+c}{x-2}, b=\frac{c+a}{y-2}, c=\frac{a+b}{z-2}
a
=
x
−
2
b
+
c
,
b
=
y
−
2
c
+
a
,
c
=
z
−
2
a
+
b
. If
x
y
+
y
z
+
z
x
=
1000
xy+yz+zx=1000
x
y
+
yz
+
z
x
=
1000
and
x
+
y
+
z
=
2016
x+y+z=2016
x
+
y
+
z
=
2016
, find the value of
x
y
z
xyz
x
yz
.
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