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1989 Greece National Olympiad
1
4x8 system of equations 1989 Greece MO Grade X p1
4x8 system of equations 1989 Greece MO Grade X p1
Source:
September 7, 2024
algebra
system of equations
Problem Statement
Let
a
,
b
,
c
,
d
x
,
y
,
z
,
w
a,b,c,d x,y,z, w
a
,
b
,
c
,
d
x
,
y
,
z
,
w
be real numbers such that
a
x
−
b
y
−
c
z
−
d
w
=
0
b
x
+
a
y
−
d
z
+
c
w
=
0
c
x
+
d
y
+
a
z
−
b
w
=
0
d
x
−
c
y
+
b
z
+
a
w
=
0
\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix}
a
x
−
b
y
−
cz
−
d
w
=
0
b
x
+
a
y
−
d
z
+
c
w
=
0
c
x
+
d
y
+
a
z
−
b
w
=
0
d
x
−
cy
+
b
z
+
a
w
=
0
prove that
a
=
b
=
c
=
d
=
0
,
o
r
x
=
y
=
z
=
w
=
0
a=b=c=d=0, \ \ or \ \ x=y=z=w=0
a
=
b
=
c
=
d
=
0
,
or
x
=
y
=
z
=
w
=
0
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