OlympiadBench / 1873 / Let A denote the set of all polynomials in three variables x,y,z…

Let A denote the set of all polynomials in three variables x,y,z with integer coefficients. Let B denote the subset of A formed by all polynomials which can be expressed as

(x+y+z)P(x,y,z)+(xy+yz+zx)Q(x,y,z)+xyzR(x,y,z)

with P,Q,R∈A. Find the smallest non-negative integer n such that x^(i)y^(j)z^(k)∈B for all nonnegative integers i,j,k satisfying i+j+k≥n.

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as

$$
(x+y+z) P(x, y, z)+(x y+y z+z x) Q(x, y, z)+x y z R(x, y, z)
$$

with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^{i} y^{j} z^{k} \in \mathcal{B}$ for all nonnegative integers $i, j, k$ satisfying $i+j+k \geqslant n$.