https://artofproblemsolving.com/wiki/index.php?title=2018_IMO_Problems/Problem_5&feed=atom&action=history 2018 IMO Problems/Problem 5 - Revision history 2021-07-31T10:07:55Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=2018_IMO_Problems/Problem_5&diff=96109&oldid=prev Btc433: Created page with "Let $a_1, a_2, \dots$ be an infinite sequence of positive integers. Suppose that there is an integer$N > 1$ such that, for each $n \geq N$, t..." 2018-07-11T04:39:10Z <p>Created page with &quot;Let &lt;math&gt;a_1, a_2, \dots&lt;/math&gt; be an infinite sequence of positive integers. Suppose that there is an integer&lt;math&gt; N &gt; 1&lt;/math&gt; such that, for each &lt;math&gt;n \geq N&lt;/math&gt;, t...&quot;</p> <p><b>New page</b></p><div>Let &lt;math&gt;a_1, a_2, \dots&lt;/math&gt; be an infinite sequence of positive integers. Suppose that there is an integer&lt;math&gt; N &gt; 1&lt;/math&gt; such that, for each &lt;math&gt;n \geq N&lt;/math&gt;, the number &lt;math&gt;\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}&lt;/math&gt; is an integer. Prove that there is a positive integer &lt;math&gt;M&lt;/math&gt; such that &lt;math&gt;a_m = a_{m+1}&lt;/math&gt; for all &lt;math&gt;m \geq M.&lt;/math&gt;</div> Btc433