RMM 2026 Problem 3

Let $S$ be a finite subset of $\mathbb{R}^3$. Prove that there exist three polynomials $P(x,y,z), Q(x,y,z)$ and $R(x,y,z)$ with real coefficients, such that a triple of real numbers $(a,b,c)$ is in $S$ if and only if the system of equations

$$P(x,y,z) = a,$$

$$Q(x,y,z) = b,$$

$$R(x,y,z) = c,$$

does not have a solution in real numbers $x, y,$ and $z$.