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FE with polynomial

Source: Iranian Our MO 2020 P6

March 11, 2020
functional equationalgebrapolynomial

Problem Statement

Find all functions f:R+→R+f:\mathbb{R}^+ \to \mathbb{R}^+ and plynomials P(x),Q(x),R(x)P(x),Q(x),R(x) with positive real coefficients such that Q(−1)=−1Q(-1)=-1 and for all positive reals x,yx,y:f(xy+R(y))=f(x)Q(y)+P(y).f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).
Proposed by Alireza Danaie, Ali Mirazaie Anari Rated 2
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