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IMO LongList 1967, Great Britain 2

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December 16, 2004
inductionalgebraseries summationrational numberIMO ShortlistIMO Longlist

Problem Statement

If xx is a positive rational number show that xx can be uniquely expressed in the form x=∑k=1nakk!x = \sum^n_{k=1} \frac{a_k}{k!} where a1,a2,…a_1, a_2, \ldots are integers, 0≤an≤n−10 \leq a_n \leq n - 1, for n>1,n > 1, and the series terminates. Show that xx can be expressed as the sum of reciprocals of different integers, each of which is greater than 106.10^6.
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