65. What is the smallest degree that a polynomial $P$ can have if it is known that there exists an integer $b$ such that $A_{P}$ contains elements both greater and less than $b$, but does not contain $b$?
65. Answer: 4.

For example, consider the polynomial $P(x)=(x-3)(x-1)(x+1)(x+3)$. It is even. Therefore, if $c \neq P(0)$, then $l_{P}(c)$ is even. Therefore, in $A$ there can only be one odd number, namely, $l_{P}(P(0))$. Thus, $b=1$ or $b=3$ is the desired number, since $0,4 \in A_{P}$. Why can't the degree be less? Consider, for example, a cubic polynomial $P$. Since its derivative is a quadratic polynomial, it has no more than two roots, and thus $P$ has no more than three intervals of monotonicity, specifically 1 or 3. Let's consider the most complex case immediately - when $P$ has three intervals of monotonicity. Suppose it has a local minimum $m$ and a local maximum $M$. Then for $x \in(-\infty, m) \cup(M,+\infty)$, $l_{P}(x)=1$, $l_{P}(m)=l_{P}(M)=2$, and for $x \in(m, M)$, $l_{P}(x)=3$. The other cases are even simpler to analyze.