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PAMO 2022 Problem 4 - System of Functional Equations

Source: 2022 Pan-African Mathematics Olympiad Problem 4

6/26/2022
Find all functions ff and gg defined from R>0\mathbb{R}_{>0} to R>0\mathbb{R}_{>0} such that for all x,y>0x, y > 0 the two equations hold (f(x)+y−1)(g(y)+x−1)=(x+y)2 (f(x) + y - 1)(g(y) + x - 1) = {(x + y)}^2 (−f(x)+y)(g(y)+x)=(x+y+1)(y−x−1) (-f(x) + y)(g(y) + x) = (x + y + 1)(y - x - 1)
Note: R>0\mathbb{R}_{>0} denotes the set of positive real numbers.
functional equationalgebrafunctionsystem of equations