Problem

Source: Iran MO Third Round A3

Tags: algebra, polynomial, function, fuctional equation



Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$(a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.