Problem

Source: XIII International Festival of Young Mathematicians Sozopol 2024, Theme for 10-12 grade

Tags: combinatorics



Let \( n \geq 4 \) be a positive integer. Initially, each of \( n \) girls knows one piece of gossip that no one else knows, and they want to share them. For greater security, to avoid being spied, they only talk in pairs, and in a conversation, each girl shares all the gossip she knows so far with the other one. What is the minimum number of conversations needed so that every girl knows all the gossip?