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if (a^{m+1}-1)/(a^m-1)=(b^{n+1}-1)/(b^n-1)=c then a^m c^n > b^n c^{m}

Source: Balkan BMO Shortlist 2015 A5

August 5, 2019

inequalitiesalgebraExponential inequalityexponentialExponential equation

Problem Statement

Let

m, n

be positive integers and

a, b

positive real numbers different from

1

such thath

m > n

and

\frac{a^{m+1}-1}{a^m-1} = \frac{b^{n+1}-1}{b^n-1} = c

. Prove that

a^m c^n > b^n c^{m}

(Turkey)