MathDB

Let

a_1,a_2,\ldots a_n,k

, and

M

be positive integers such that \frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k \text{and} a_1a_2\cdots a_n=M. If

M>1

, prove that the polynomial

P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)

has no positive roots.

Problem Statement