\section*{Problem 4 - 141214}

For all real triples \((a, b, c)\), it is to be investigated whether the system of equations

\[
x y^{2} z^{3}=a, \quad ; \quad x^{2} y^{3} z=b, \quad ; \quad x^{3} y z^{2}=c
\]

\[
(*, * *, * * *)
\]

1) has no,

2) exactly one,

3) exactly two,

4) more than two but finitely many,

5) infinitely many

real solutions \((x, y, z)\). Furthermore, all existing solutions are to be specified.
}

If \((x, y, z)\) is a real solution of the given system of equations, then from \(\left({ }^{*}\right),\left({ }^{* *}\right),\left({ }^{* * *}\right)\) by multiplication, it follows that

\[
x^{6} y^{6} z^{6}=a b c
\]

must hold. Therefore, the given system of equations has no real solution in the case \(a b c < 0\). If we set

\[
g=\sqrt[6]{a b c}
\]

then from \((*),(* *),(* * *)\)

\[
y z^{2}=\frac{a}{g} \quad ; \quad x y^{2}=\frac{b}{g} \quad ; \quad z x^{2}=\frac{c}{g}
\]

By multiplying (3) and (4), (3) and (5), and (4) and (5), we obtain

\[
x y^{3} z^{2}=\frac{a b}{g^2} \quad ; \quad x^{2} y z^{3}=\frac{a c}{g^2} \quad ; \quad x^{3} y^{2} z=\frac{b c}{g^2}
\]

From (7) and \(\left({ }^{* * *}\right)\) and (8) and \((* *)\), it follows that

\[
\begin{gathered}
\frac{z}{x}=\frac{a}{g^{2}} \quad \text { so } \quad z=\frac{a}{g^{2}} x \quad \text { and } \\
\frac{y}{x}=\frac{g^{2}}{c} \quad \text { so } \quad y=\frac{g^{2}}{c} x
\end{gathered}
\]

From \(\left({ }^{*}\right),(9)\) and (10), it follows that

\[
x^{6} \frac{g^{4}}{c^{2}} \frac{a^{3}}{g^{6}}=a \quad \text { so } \quad x^{6}=\frac{c^{2}}{a^{2}} g^{2}
\]

Thus, the only solutions are the two triples

\[
\left( \pm \sqrt[3]{\left|\frac{c}{a}\right| g}, \pm \frac{g^{2}}{c} \sqrt[3]{\left|\frac{c}{a}\right| g}, \pm \frac{a}{g^{2}} \sqrt[3]{\left|\frac{c}{a}\right| g}\right)
\]

which are distinct due to \(a b c \neq 0\). By substituting into \((*),(* *),(* * *)\), it can be verified that both triples are indeed solutions.

Thus, we have:

a) In the case \(a b c < 0\), the system has no solution.

b) and d) These cases do not occur.

c) In the case \(a b c > 0\), the system has exactly two solutions.

e) In the case \(a=b=c=0\), the system has infinitely many solutions.

\section*{Adapted from \([5]\)}

\subsection*{9.16.2 Second Round 1974, Class 12}