MathDB

8

Part of 2013 China Northern MO

Problems(1)

3n people in a gathering

Source: China Northern MO 2013 p8 CNMO

11/25/2022
3n3n (n≥2,n∈Nn \ge 2, n \in N) people attend a gathering, in which any two acquaintances have exactly nn common acquaintances, and any two unknown people have exactly 2n2n common acquaintances. If three people know each other, it is called a Taoyuan Group. (1) Find the number of all Taoyuan groups; (2) Prove that these 3n3n people can be divided into three groups, with nn people in each group, and the three people obtained by randomly selecting one person from each group constitute a Taoyuan group.
Note: Acquaintance means that two people know each other, otherwise they are not acquaintances. Two people who know each other are called acquaintances.
combinatorics