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A two-variable functional equation

Source: Philippine Mathematical Olympiad 2024 P1

February 24, 2024
algebrafunctional equation

Problem Statement

Let f:Z2→Zf:\mathbb{Z}^2\rightarrow\mathbb{Z} be a function satisfying f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1)f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1) for all integers xx and yy. Can it happen that ∣f(x,y)∣≤2024|f(x,y)|\leq 2024 for all x,y∈Zx,y\in\mathbb{Z}?
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