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a=1 or -1<a<0 if a is real root of nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0

Source: 2012 Swedish Mathematical Competition p4

April 30, 2021
algebrapolynomial

Problem Statement

Given that aa is a real solution to the polynomial equation nxn−xn−1−xn−2−⋯−x−1=0nx^n-x^{n-1}-x^{n-2}-\cdots-x-1=0 where nn is a positive integer, show that a=1a=1 or −1<a<0-1<a<0.
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