Problem

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

Tags: AMC, USA(J)MO, USAMO, quadratics, algebra unsolved, algebra



Find all positive integers $n$ such that there are $k \geq 2$ positive rational numbers $a_1, a_2, \ldots, a_k$ satisfying $a_1 + a_2 + \ldots + a_k = a_1 \cdot a_2 \cdots a_k = n.$