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USAMO sum equals product - original problems please

Source: USAMO 2006, Problem 4, proposed by Ricky Liu

April 20, 2006
AMCUSA(J)MOUSAMOquadraticsalgebra unsolvedalgebra

Problem Statement

Find all positive integers nn such that there are k≥2k \geq 2 positive rational numbers a1,a2,…,aka_1, a_2, \ldots, a_k satisfying a1+a2+…+ak=a1⋅a2⋯ak=n.a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.
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