Problem

Source: Romania IMO TST 1990 p1

Tags: floor function, number theory



Let a,b,n be positive integers such that $(a,b) = 1$. Prove that if $(x,y)$ is a solution of the equation $ax+by = a^n + b^n$ then $$\left[\frac{x}{b}\right]+\left[\frac{y}{a}\right]=\left[\frac{a^{n-1}}{b}\right]+\left[\frac{b^{n-1}}{a}\right]$$