MathDB

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 \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} 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 non-negative integers

i, j, k

satisfying

i + j + k \geq n

A2