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Omni-MATH / 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 ……

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problem

Let A\mathcal{A} denote the set of all polynomials in three variables x,y,zx, y, z with integer coefficients. Let B\mathcal{B} denote the subset of A\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∈AP, Q, R \in \mathcal{A}. Find the smallest non-negative integer nn such that xiyjzk∈Bx^i y^j z^k \in \mathcal{B} for all non-negative integers i,j,ki, j, k satisfying i+j+k≥ni + j + k \geq n.
Plain-text mathematical notation (without MathML)
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
\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∈A.  Find the smallest non-negative integer n such that x^(i)y^(j)z^(k)∈B for all non-negative integers i,j,k satisfying i+j+k≥n.
Original LaTeX notation
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 $i + j + k \geq n$.

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Source and history

task_id
a81e786c-baf6-509d-af85-9dc2b3175389
task_key
test--a81e786c-baf6-509d-af85-9dc2b3175389
task_revision_id
1
config / split
/ test

Official source

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