MathDB

10

Part of 2002 Austrian-Polish Competition

Problems(1)

Family of sequences

Source: 2002 Austrian Polish, problem 10

9/23/2006
For all real number xx consider the family F(x)F(x) of all sequences (an)n≥0(a_{n})_{n\geq 0} satisfying the equation a_{n+1}=x-\frac{1}{a_{n}}  (n\geq 0). A positive integer pp is called a minimal period of the family F(x)F(x) if (a) each sequence (an)∈F(x)\left(a_{n}\right)\in F(x) is periodic with the period pp, (b) for each 0<q<p0<q<p there exists (an)∈F(x)\left(a_{n}\right)\in F(x) such that qq is not a period of (an)\left(a_{n}\right). Prove or disprove that for each positive integer PP there exists a real number x=x(P)x=x(P) such that the family F(x)F(x) has the minimal period p>Pp>P.
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