Problem

Source: Ukrainian Mathematical Olympiad 2023. Day 2, Problem 8.8

Tags: number theory



You are given a set of $m$ integers, all of which give distinct remainders modulo some integer $n$. Show that for any integer $k \le m$ you can split this set into $k$ nonempty groups so that the sums of elements in these groups are distinct modulo $n$. Proposed by Anton Trygub