Question 49
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 $n$
Part (a). there exists an $n$-pretty polynomial;
Part (b). any $n$-pretty polynomial has a degree of at least $\frac{2 n+1}{3}$.
(Remark. For a real number $x$, we denote by $\lfloor x\rfloor$ the largest integer smaller than or equal to $x$.)
Part (a). there exists an $n$-pretty polynomial;
Part (b). any $n$-pretty polynomial has a degree of at least $\frac{2 n+1}{3}$.
(Remark. For a real number $x$, we denote by $\lfloor x\rfloor$ the largest integer smaller than or equal to $x$.)