Nemotron-IMO-Bench 2026 C34

Let $m$ be a positive integer and consider a social network with a fixed finite set of users, each user having a fixed set of followers (among the other users), as well as an initial rating, a positive integer which may not be the same for all users. Every midnight, the rating of every user increases by the sum of the ratings that his followers had just before midnight. A hacker, who is not a user of the social network, wants all the users to have ratings divisible by $m$. Every day, he can either choose a user and increase his rating by 1, or do nothing. Prove that the hacker can achieve his goal after some number of days.