Clubs of three people in a class of n students
Source: 2023 OLCOMA Costa Rica National Olympiad, Final Round, 3.4
March 20, 2024
combinatorics
Problem Statement
A teacher wants her students to know each other, so she creates various clubs of three people, so that each student can participate in several clubs. The clubs are formed in such a way that if and are two people, then there is a single club such that and are two of its three members.
(1) Show that there is no way for the teacher to form the clubs if .
(2) Show that the teacher can do it if .