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National and Regional Contests
Poland Contests
Poland - Second Round
1997 Poland - Second Round
1
Nice algebra
Nice algebra
Source: Polish MO 1997
December 17, 2016
system of equations
algebra
Problem Statement
For the real number
a
a
a
find the number of solutions
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of a system of the equations:
{
x
+
y
2
+
z
2
=
a
x
2
+
y
+
z
2
=
a
x
2
+
y
2
+
z
=
a
\left\{\begin{array}{lll} x+y^2+z^2=a \\ x^2+y+z^2=a \\ x^2+y^2+z=a\end{array}\right.
⎩
⎨
⎧
x
+
y
2
+
z
2
=
a
x
2
+
y
+
z
2
=
a
x
2
+
y
2
+
z
=
a
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