Question 84
Let $\mathbb{Z}[x,y,z]$ be the set of polynomials with integer coefficients in the three variables $x,y,z$. A polynomial $P\in\mathbb{Z}[x,y,z]$ is called olympic if there are polynomials $A,B,C\in\mathbb{Z}[x,y,z]$ such that
$$
P(x,y,z)=(x+y+z)A(x,y,z)+(xy+yz+zx)B(x,y,z)+xyzC(x,y,z).
$$
Find the largest integer $n$ for which there are nonnegative integers $i,j,k$ with $i+j+k=n$ such that the polynomial $x^iy^jz^k$ is not olympic.
Note. A polynomial with integer coefficients in $x,y,z$ is a function that can be written as a sum of terms of the form $\lambda x^iy^jz^k$, where $\lambda$ is an integer and $i,j,k$ are nonnegative integers. For example, $x-y+1$ and $xy+yz+zx$ are such polynomials, whereas $\pi xyz$, $\exp(x)$, $x/(y^2+1)$ and $\sqrt{xy+z}$ are not.
$$
P(x,y,z)=(x+y+z)A(x,y,z)+(xy+yz+zx)B(x,y,z)+xyzC(x,y,z).
$$
Find the largest integer $n$ for which there are nonnegative integers $i,j,k$ with $i+j+k=n$ such that the polynomial $x^iy^jz^k$ is not olympic.
Note. A polynomial with integer coefficients in $x,y,z$ is a function that can be written as a sum of terms of the form $\lambda x^iy^jz^k$, where $\lambda$ is an integer and $i,j,k$ are nonnegative integers. For example, $x-y+1$ and $xy+yz+zx$ are such polynomials, whereas $\pi xyz$, $\exp(x)$, $x/(y^2+1)$ and $\sqrt{xy+z}$ are not.