Problem

Source: Dutch NMO 2015 p1

Tags: combinatorics, Sets, maximum



We make groups of numbers. Each group consists of five distinct numbers. A number may occur in multiple groups. For any two groups, there are exactly four numbers that occur in both groups. (a) Determine whether it is possible to make $2015$ groups. (b) If all groups together must contain exactly six distinct numbers, what is the greatest number of groups that you can make? (c) If all groups together must contain exactly seven distinct numbers, what is the greatest number of groups that you can make?