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35th Austrian Mathematical Olympiad 2004

Source: round3, day1, problem2

February 13, 2009
number theoryprime numbersnumber theory unsolved

Problem Statement

Show that every set {p1,p2,…,pk} \{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 1n \frac{1}{n}), whose denominators are exactly the k k given prime factors (but in arbitrary powers with exponents unequal zero), is an unit fraction again. How big is this sum if 12004 \frac{1}{2004} is among this summands? Show that for every set {p1,p2,…,pk} \{p_1,p_2,\dots,p_k\} containing k k prime numbers (k>2 k>2) is the sum smaller than 1N \frac{1}{N} with N=2⋅3k−2(k−2)! N=2\cdot 3^{k-2}(k-2)!
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