ÖMO 2004 Problem 2

Show that every set $\{p_1,p_2,\dots,p_k\}$ of prime numbers fulfils the following: The sum of all unit fractions (that are fractions of the type $\frac{1}{n}$ ), whose denominators are exactly the $k$ given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if $\frac{1}{2004}$ is among this summands? Show that for every set $\{p_1,p_2,\dots,p_k\}$ containing $k$ prime numbers ( $k>2$ ) is the sum smaller than $\frac{1}{N}$ with $N=2\cdot 3^{k-2}(k-2)!$