Let us suppose $3$ integers are selected at random from a large range, say $$-1000\leq x\leq y\leq z\leq 1000$$
Now, we define the sum and product: $$\begin{align*}s&=x+y+z \\p&=xyz\end{align*}$$
($s$ and $p$ will not be equal in most cases, sorry for the confusion)
What is the probability that there exists another solution for $(x,y,z)$ that satisfies above 3 equations? (reordering of x, y and z not allowed)
My friend gave me this question, and I have no idea where to start. If we limit ourselves to positive integers, is there a unique solution, or not?