Question 77
$P(x)$ is a non-constant monic polynomial with integer coefficients. Assume that $P_1(x), P_2(x), \dots, P_n(x)$ are monic polynomials with integer coefficients such that for all $1 \le i \le n$, $\deg(P_i) \ge \deg(P)$. We know that for any natural number $x$, there exists a natural number $y$ and an index $i$ ($1 \le i \le n$), such that $P(x) = P_i(y)$. Prove that there exists an index $j$ ($1 \le j \le n$) and a natural number $k$ such that $P(x) = P_j(x+k)$.