TW-TST 2001 Problem 1

Let $A$ be a set with at least $3$ integers, and let $M$ be the maximum element in $A$ and $m$ the minimum element in $A$ . it is known that there exist a polynomial $P$ such that: $m<P(a)<M$ for all $a$ in $A$ . And also $p(m)<p(a)$ for all $a$ in $A-(m,M)$ . Prove that $n<6$ and there exist integers $b$ and $c$ such that $p(x)+x^2+bx+c$ is cero in $A$ .