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Poland - Second Round
2010 Poland - Second Round
1
Polish 2nd stage 2010, 1st problem (system of equations)
Polish 2nd stage 2010, 1st problem (system of equations)
Source:
November 11, 2010
algebra proposed
algebra
Problem Statement
Solve in the real numbers
x
,
y
,
z
x, y, z
x
,
y
,
z
a system of the equations:
{
x
2
−
(
y
+
z
+
y
z
)
x
+
(
y
+
z
)
y
z
=
0
y
2
−
(
z
+
x
+
z
x
)
y
+
(
z
+
x
)
z
x
=
0
z
2
−
(
x
+
y
+
x
y
)
z
+
(
x
+
y
)
x
y
=
0.
\begin{cases} x^2 - (y+z+yz)x + (y+z)yz = 0 \\ y^2 - (z + x + zx)y + (z+x)zx = 0 \\ z^2 - (x+y+xy)z + (x+y)xy = 0. \\ \end{cases}
⎩
⎨
⎧
x
2
−
(
y
+
z
+
yz
)
x
+
(
y
+
z
)
yz
=
0
y
2
−
(
z
+
x
+
z
x
)
y
+
(
z
+
x
)
z
x
=
0
z
2
−
(
x
+
y
+
x
y
)
z
+
(
x
+
y
)
x
y
=
0.
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