I gave up, my approaches didn't work (induction, pigeon-hole, parity; though obviously there's a good chance I didn't use them cleverly):
In a group of 12 people, every pair of them has a common friend (in the same group). It is understood that friendship is a non-reflexive (a person is not a friend of herself), symmetric, not necessarily transitive relation (so this can be represented by a simple graph). What is the minimum number of pairs of friends among them?
(Source: Olimpiada de Mayo, 2012.)
So, how can this be solved (without considering lots of cases)?
Thank you.