Middle European Mathematical Olympiad 2021 Problem T-2

Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor) = \lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$

(a) there exists an $n$-pretty polynomial;

(b) any $n$-pretty polynomial has a degree of at least $\frac{2n + 1}{3}$.

(Remark. For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)