Problem

Source: Simurgh 2019 - Problem 4

Tags: algebra, polynomial



Assume that every root of polynomial $P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d$ is in $[0,1]$. Show that for every $k = 1,2,...,d$ the following inequality holds: $ a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0 $