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\prod (x-a^k )/(x+a^k ineq

Source: Polish MO Finals 1981 p3

August 24, 2024

algebrainequalities

Problem Statement

Prove that for any natural number

n

and real numbers

a

and

x

satisfying

a^{n+1} \le x \le 1

and

0 < a < 1

it holds that

\prod_{k=1}^n \left|\frac{x-a^k}{x+a^k}\right| \le \prod_{k=1}^n \frac{1-a^k}{1+a^k}