TSTST 2025 Problem 1

In a finite group of people, some pairs are friends (friendship is mutual). Each person $p$ has a list $f_1(p), f_2(p), \dots, f_{d(p)}(p)$ of their friends, where $d(p)$ is the number of distinct friends $p$ has. Additionally, any two people are connected by a series of friendships. Each person also has a water balloon. The following game is played until someone ends up with more than one water balloon: on round $r$, each person $p$ throws the current water balloon they have to their friend $f_s(p)$ such that $d(p)$ divides $r - s$. Show that if the game never ends, then everyone has the same number of friends.