Question 73
Let $P$ be a polynomial with integer coefficients and degree $n\le10$. Suppose $|P(10)-P(0)|\lt 1000$ and, for every $k\in\{1,\ldots,10\}$, there is an integer $m$ such that $P(m)=k$. Prove that for every integer $k$ there is an integer $m$ such that $P(m)=k$.