{"status": "success", "data": {"description_md": "Let $\\ell_A$ and $\\ell_B$ be two distinct parallel lines. For positive integers $m$ and $n$, distinct points $A_1, A_2, \\allowbreak A_3, \\allowbreak \\ldots, \\allowbreak A_m$ lie on $\\ell_A$, and distinct points $B_1, B_2, B_3, \\ldots, B_n$ lie on $\\ell_B$. Additionally, when segments $\\overline{A_iB_j}$ are drawn for all $i=1,2,3,\\ldots, m$ and $j=1,\\allowbreak 2,\\allowbreak 3, \\ldots, \\allowbreak n$, no point strictly between $\\ell_A$ and $\\ell_B$ lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when $m=7$ and $n=5$. The figure shows that there are 8 regions when $m=3$ and $n=2$.
$\\includegraphics[width=237, height=86, totalheight=86]{https://latex.artofproblemsolving.com/5/6/c/56c4dfb57fc6e7e474f648f52cc8ce4669b2a526.png}$\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": "

Let \\ell_A and \\ell_B be two distinct parallel lines. For positive integers m and n, distinct points A_1, A_2, \\allowbreak A_3, \\allowbreak \\ldots, \\allowbreak A_m lie on \\ell_A, and distinct points B_1, B_2, B_3, \\ldots, B_n lie on \\ell_B. Additionally, when segments \\overline{A_iB_j} are drawn for all i=1,2,3,\\ldots, m and j=1,\\allowbreak 2,\\allowbreak 3, \\ldots, \\allowbreak n, no point strictly between \\ell_A and \\ell_B lies on more than two of the segments. Find the number of bounded regions into which this figure divides the plane when m=7 and n=5. The figure shows that there are 8 regions when m=3 and n=2.

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": 4, "problem_name": "2022 AIME II Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/22_aime_II_p10", "prev": "/problem/22_aime_II_p08"}}