Problem

Source: romania TST 7 / 2007 problem 3

Tags: algebra, polynomial, function, ratio, induction, algebra proposed



The problem is about real polynomial functions, denoted by $f$, of degree $\deg f$. a) Prove that a polynomial function $f$ can`t be wrriten as sum of at most $\deg f$ periodic functions. b) Show that if a polynomial function of degree $1$ is written as sum of two periodic functions, then they are unbounded on every interval (thus, they are "wild"). c) Show that every polynomial function of degree $1$ can be written as sum of two periodic functions. d) Show that every polynomial function $f$ can be written as sum of $\deg f+1$ periodic functions. e) Give an example of a function that can`t be written as a finite sum of periodic functions. Dan Schwarz