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x-y+1 | f(x)-f(y)+x+y

Source: Israeli Olympic Revenge 2021, Problem 1

August 29, 2021
number theoryfunctional equationalgebrafunction

Problem Statement

Let N\mathbb N be the set of positive integers. Find all functions f ⁣:N→Nf\colon\mathbb N\to\mathbb N such that f(x)−f(y)+x+yx−y+1\frac{f(x)-f(y)+x+y}{x-y+1} is an integer, for all positive integers x,yx,y with x>yx>y.
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