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37th Austrian Mathematical Olympiad

Source: round1, problem4 - harmonic sequence

February 10, 2009
algebra unsolvedalgebra

Problem Statement

Let <hn> <h_n> n∈N n\in\mathbb N a harmonic sequence of positive real numbers (that means that every hn h_n is the harmonic mean of its two neighbours h_{n\minus{}1} and h_{n\plus{}1} : h_n\equal{}\frac{2h_{n\minus{}1}h_{n\plus{}1}}{h_{n\minus{}1}\plus{}h_{n\plus{}1}}) Show that: if the sequence includes a member hj h_j, which is the square of a rational number, it includes infinitely many members hk h_k, which are squares of rational numbers.
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