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1978 USAMO #3

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August 16, 2011
AMCUSAMOinductionalgebraequationAdditive Number Theory

Problem Statement

An integer nn will be called good if we can write n=a1+a2+⋯+ak,n=a_1+a_2+\cdots+a_k, where a1,a2,…,aka_1,a_2, \ldots, a_k are positive integers (not necessarily distinct) satisfying 1a1+1a2+⋯+1an=1.\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=1. Given the information that the integers 33 through 73 are good, prove that every integer ≥33\ge 33 is good.
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