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mexico functional equation

Source: 2nd National Women's Contest of Mexican Mathematics Olympiad 2023 , level 1+2 p4

July 22, 2023
functional equationalgebra

Problem Statement

A function gg is such that for all integer nn: g(n)={1ifn≥10ifn≤0g(n)=\begin{cases} 1\hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\geq 1 & \\ 0 \hspace{0.5cm} \textrm{if}\hspace{0.1cm} n\leq 0 & \end{cases}
A function ff is such that for all integers n≥0n\geq 0 and m≥0m\geq 0: f(0,m)=0andf(0,m)=0 \hspace{0.5cm} \textrm{and} f(n+1,m)=(1−g(m)+g(m)⋅g(m−1−f(n,m)))⋅(1+f(n,m))f(n+1,m)=\Bigl(1-g(m)+g(m)\cdot g(m-1-f(n,m))\Bigr)\cdot\Bigl(1+f(n,m)\Bigr)
Find all the possible functions f(m,n)f(m,n) that satisfies the above for all integers n≥0n\geq0 and m≥0m\geq 0
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