Question 25
Let $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ be a sequence of positive integers such that
$$
\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}+\frac{a_{n}}{a_{1}}
$$
is an integer for all $n \geqslant k$, where $k$ is some positive integer. Prove that there exists a positive integer $m$ such that $a_{n}=a_{n+1}$ for all $n \geqslant m$.
$$
\frac{a_{1}}{a_{2}}+\frac{a_{2}}{a_{3}}+\cdots+\frac{a_{n-1}}{a_{n}}+\frac{a_{n}}{a_{1}}
$$
is an integer for all $n \geqslant k$, where $k$ is some positive integer. Prove that there exists a positive integer $m$ such that $a_{n}=a_{n+1}$ for all $n \geqslant m$.