{"status": "success", "data": {"description_md": "Triangle $ABC$ has sides $AB=9,BC = 5\\sqrt{3},$ and $AC=12$. Points $A=P_0, P_1, P_2, \\ldots, P_{2450} = B$ are on segment $\\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\\ldots,2449$, and points $A=Q_0, Q_1, Q_2, \\ldots ,Q_{2450} = C$ for $k=1,2,\\ldots,2449$. Furthermore, each segment $\\overline{P_kQ_k}, k=1,2,\\ldots,2449$, is parallel to $\\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\\overline{P_kQ_k}, k=1,2 ,\\ldots,2450$, that have rational length.\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": "

Triangle ABC has sides AB=9,BC = 5\\sqrt{3}, and AC=12. Points A=P_0, P_1, P_2, \\ldots, P_{2450} = B are on segment \\overline{AB} with P_k between P_{k-1} and P_{k+1} for k=1,2,\\ldots,2449, and points A=Q_0, Q_1, Q_2, \\ldots ,Q_{2450} = C for k=1,2,\\ldots,2449. Furthermore, each segment \\overline{P_kQ_k}, k=1,2,\\ldots,2449, is parallel to \\overline{BC}. The segments cut the triangle into 2450 regions, consisting of 2449 trapezoids and 1 triangle. Each of the 2450 regions have the same area. Find the number of segments \\overline{P_kQ_k}, k=1,2 ,\\ldots,2450, that have rational length.

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": "2018 AIME II Problem 7", "can_next": true, "can_prev": true, "nxt": "/problem/18_aime_II_p08", "prev": "/problem/18_aime_II_p06"}}