Question 5
Find all integer-coefficient polynomials $P(x),Q(x)$ with the following property. The alternating sequence starts at $x_0=2014$ and satisfies
$$x_{2n+1}=P(x_{2n}),\qquad x_{2n+2}=Q(x_{2n+1})\qquad(n\ge0).$$
Every positive integer divides some nonzero term of the sequence.
Editorial indexing clarification: the source prints $x_{2n+1}=P(x_{2n}),\ x_{2n}=Q(x_{2n-1})$ with $n\ge1$, leaving $x_1$ undefined. The displayed recurrence makes explicit the intended initialization $x_1=P(2014)$, which the source solution uses.
$$x_{2n+1}=P(x_{2n}),\qquad x_{2n+2}=Q(x_{2n+1})\qquad(n\ge0).$$
Every positive integer divides some nonzero term of the sequence.
Editorial indexing clarification: the source prints $x_{2n+1}=P(x_{2n}),\ x_{2n}=Q(x_{2n-1})$ with $n\ge1$, leaving $x_1$ undefined. The displayed recurrence makes explicit the intended initialization $x_1=P(2014)$, which the source solution uses.