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2 roots of f_n(x) =x^2 - b^nx- a^n in (-1,1), for every n

Source: Vietnamese MO (VMO) 1968

August 22, 2018
functionrootsanalysisalgebrainequalities

Problem Statement

Let aa and bb satisfy a≥b>0,a+b=1a \ge b >0, a + b = 1. i) Prove that if mm and nn are positive integers with m<nm < n, then am−an≥bm−bn>0a^m - a^n \ge b^m- b^n > 0. ii) For each positive integer nn, consider a quadratic function fn(x)=x2−bnx−anf_n(x) = x^2 - b^nx- a^n. Show that f(x)f(x) has two roots that are in between −1-1 and 11.
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