MathDB

A1

Part of 2017 IMO Shortlist

Problems(1)

IMO Shortlist 2017 A1

Source: IMO Shortlist 2017

7/10/2018
Let a1,a2,…an,ka_1,a_2,\ldots a_n,k, and MM 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>1M>1, prove that the polynomial P(x)=M(x+1)k−(x+a1)(x+a2)⋯(x+an)P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n) has no positive roots.
IMO Shortlistalgebrapolynomial