On a certain island each of the inhabitants is a member of one of the two existing clubs.

The membership distribution is such that when two random people meet, the probability of those two belonging to the same club is equal to the probability of them belonging to distinct clubs.

When 100 newcomers arrive on the island and each enrolls in one of the two clubs, the distribution still retains this feature. How many people belong to either club?

If there are 10 people in each club . . .

Member from Club A would have a 9/19 chance of meeting a member of Club A and a 10/19 chance of meeting a member of Club B.

Adding a member to Club A would just create a greater shift if the initial person selected was from CluB B.

Zero is no help since the problem reoccurs when 100 newcomers are added.

Only answer I can think of is "infinity", which is kind of tough to have, especially if it is a small island.