{"status": "success", "data": {"description_md": "Consider the sequence $(a_k)_{k\\ge 1}$ of positive rational numbers defined by $a_1 = \\frac{2020}{2021}$ and for $k\\ge 1$, if $a_k = \\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then

\n$$a_{k+1} = \\frac{m + 18}{n+19}. $$ Determine the sum of all positive integers $j$ such that the rational number $a_j$ can be written in the form $\\frac{t}{t+1}$ for some positive integer $t$.\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "

Consider the sequence (a_k)_{k\\ge 1} of positive rational numbers defined by a_1 = \\frac{2020}{2021} and for k\\ge 1, if a_k = \\frac{m}{n} for relatively prime positive integers m and n, then

a_{k+1} = \\frac{m + 18}{n+19}.
Determine the sum of all positive integers j such that the rational number a_j can be written in the form \\frac{t}{t+1} for some positive integer t.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2021 AIME I Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/21_aime_I_p11", "prev": "/problem/21_aime_I_p09"}}