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36th Austrian Mathematical Olympiad 2005

Source: round 1, problem 4

February 11, 2009
arithmetic sequencegeometric sequencealgebra unsolvedalgebra

Problem Statement

Prove: if an infinte arithmetic sequence ( a_n\equal{}a_0\plus{}nd) of positive real numbers contains two different powers of an integer a>1 a>1, then the sequence contains an infinite geometric sequence ( b_n\equal{}b_0q^n) of real numbers.