Question 21
A pair $(r, s)$ of natural numbers is called good if there exist a polynomial $P$ with integer coefficients and pairwise distinct integers $a_{1}, \ldots, a_{r}$ and $b_{1}, \ldots, b_{s}$ such that
$$
P\left(a_{1}\right)=P\left(a_{2}\right)=\ldots=P\left(a_{r}\right)=2 \quad \text{ and } \quad P\left(b_{1}\right)=P\left(b_{2}\right)=\ldots=P\left(b_{s}\right)=5
$$
a. Show that every good pair $(r, s)$ of natural numbers satisfies $r, s \leq 3$.
b. Determine all good pairs.
$$
P\left(a_{1}\right)=P\left(a_{2}\right)=\ldots=P\left(a_{r}\right)=2 \quad \text{ and } \quad P\left(b_{1}\right)=P\left(b_{2}\right)=\ldots=P\left(b_{s}\right)=5
$$
a. Show that every good pair $(r, s)$ of natural numbers satisfies $r, s \leq 3$.
b. Determine all good pairs.