{"status": "success", "data": {"description_md": "In triangle $ ABC$, $ AB = 10$, $ BC = 14$, and $ CA = 16$. Let $ D$ be a point in the interior of $ \\overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \\triangle BPC$ can be expressed in the form $ a-b\\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a+b+c$.\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": "<p>In triangle $ ABC$, $ AB = 10$, $ BC = 14$, and $ CA = 16$. Let $ D$ be a point in the interior of $ \\overline{BC}$. Let $ I_B$ and $ I_C$ denote the incenters of triangles $ ABD$ and $ ACD$, respectively. The circumcircles of triangles $ BI_BD$ and $ CI_CD$ meet at distinct points $ P$ and $ D$. The maximum possible area of $ \\triangle BPC$ can be expressed in the form $ a-b\\sqrt{c}$, where $ a$, $ b$, and $ c$ are positive integers and $ c$ is not divisible by the square of any prime. Find $ a+b+c$.</p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2009 AIME I Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/09_aime_I_p14"}}