Marinada '21

[lang=hr] Neka je \(a_1,a_2,\ldots\) beskonačni niz realnih brojeva zadan s \[a_1=x,\ \ a_2=y,\ \ a_{n+2}=\left \lfloor \frac{a_{n+1}}{a_n} \right \rfloor \cdot \frac{a_{n+1}-1}{a_{n+1}-1}\ \ \forall n \in \mathbb{N}\] Odredi najmanji prirodan broj \(m\) takav da za sve \(x,y \geq 0\) postoji broj \(k \in \mathbb{N}_0\), \(k \leq m\) takav da \(a_k\) nije dobro definiran.\\ [/lang] [lang=en] Let \(a_1,a_2,\ldots\) be an infinite sequence of real numbers given by \[a_1=x,\ \ a_2=y,\ \ a_{n+2}=\left \lfloor \frac{a_{n+1}}{a_n} \right \rfloor \cdot \frac{a_{n+1}-1}{a_{n+1}-1}\ \ \forall n \in \mathbb{N}\] Determine the least positive integer \(m\) such that for all \(x,y\geq 0\) there exists a nonnegative integer \(k\), \(k \leq m\) such that \(a_k\) is not well defined. [/lang]