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[x/a]+[y/b]=[a^{n-1}/b]+[b^{n-1}/a]

Source: Romania IMO TST 1990 p1

February 19, 2020
floor functionnumber theory

Problem Statement

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