Sorry, I didn't know how to embed code in the message. Now I have realised that triple backticks (like in github) seem to do the job, so here is the code:

1. Implementation of your idea:

nom = int(input())
members = list(map(int, input().split()))
rooms = set()   # Contains all the rooms.
room_more_mem = set()   # Contains only the rooms with families.

for m in members:
    if m not in rooms:
        rooms.add(m)
    else:
        room_more_mem.add(m)
print(rooms.difference(room_more_mem).pop())

2. The "alternative" and faster implementation.

k = int(input())
numbers = list(map(int, input().split()))
numset = set(numbers)
print(((sum(numset)*k)-(sum(numbers)))//(k-1))

If you run both versions on the large input01.txt (with 823 in the first line and total filesize 10390383 bytes --- it is one of their test cases) you will see that this particular implementation of your idea is slower.