Problem

Source: FKMO 2024 P1

Tags: number theory



Let $a, b, c, d$ be odd positive integers and pairwise coprime. For a positive integer $n$, let $$f(n) = \left[\frac{n}{a} \right]+\left[\frac{n}{b}\right]+\left[\frac{n}{c}\right]+\left[\frac{n}{d}\right]$$Prove that $$\sum_{n=1}^{abcd}(-1)^{f(n)}=1$$