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f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0, nxn functional system

Source: Austrian - Polish 1999 APMC

May 4, 2020

system of equationsfunctional equationfunctionalalgebra

Problem Statement

Given an integer

n \ge 2

, find all sustems of

n

functions

f_1,..., f_n : R \to R

such that for all

x,y \in R

\begin{cases} f_1(x)-f_2 (x)f_2(y)+ f_1(y) = 0 \\ f_2(x^2)-f_3 (x)f_3(y)+ f_2(y^2) = 0 \\ ... \\ f_n(x^n)-f_1 (x)f_1(y)+ f_n(y^n) = 0 \end {cases}