Keys and locks
Source: Pan African 2000
October 3, 2005
pigeonhole principlecombinatorics
Problem Statement
A company has five directors. The regulations of the company require that any majority (three or more) of the directors should be able to open its strongroom, but any minority (two or less) should not be able to do so. The strongroom is equipped with ten locks, so that it can only be opened when keys to all ten locks are available. Find all positive integers such that it is possible to give each of the directors a set of keys to different locks, according to the requirements and regulations of the company.