MathDB

3

Part of 2000 Pan African

Problems(2)

p is divisible by 2003

Source: Pan African 2000

10/3/2005
Let pp and qq be coprime positive integers such that: pq=1−12+13−14⋯−11334+11335 \dfrac{p}{q}=1-\frac12+\frac13-\frac14 \cdots -\dfrac{1}{1334}+\dfrac{1}{1335} Prove pp is divisible by 2003.
Keys and locks

Source: Pan African 2000

10/3/2005
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 nn such that it is possible to give each of the directors a set of keys to nn different locks, according to the requirements and regulations of the company.
pigeonhole principlecombinatorics