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?

(In reply to Non-trivial answer by brute force by Leming)

This works before the 100 newcomers, but not after the newcomers.

With a 75 / 50 split, there are 75 x 50 = 3750 pairings of people who are in different clubs.  But there are (75² - 75)/2 + (50² - 50)/2 = 2775 + 1225 = 4000 pairings of people who in the same club.

(This checks out with 125*124/2 = 7750 total pairings.)

So, this is not a solution.

Posted by Thalamus on 2004-06-17 16:25:12