MathDB

P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}, at least 1 real root

Source: 2008 Indonesia TST stage 2 test 2 p1

December 14, 2020

algebrapolynomial

Problem Statement

A polynomial

P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}

with

n_1, ..., n_s

are positive integers and

5 < n_1 < ... <n_s < 2008

are given. Prove that if

P(x)

has at least a real root, then the root is not greater than

\frac{1-\sqrt5}{2}