{"status": "success", "data": {"description_md": "Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\\overline{PA_1}$, $\\overline{PA_2}$, and the minor arc $A_1A _2$ of the circle has area $\\tfrac17$, while the region bounded by $\\overline{PA_3}$, $\\overline{PA_4}$, and the minor arc $A_3 A_4$ of the circle has area $\\tfrac{1}{9}$. There is a positive integer $n$ such that the area of the region bounded by $\\overline{PA_6}$, $\\overline{PA_7}$, and the minor arc $A_6 A_7$ is equal to $\\tfrac18 - \\tfrac{\\sqrt 2}n$. Find $n$.\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>Regular octagon <span class=\"katex--inline\">A_1A_2A_3A_4A_5A_6A_7A_8</span> is inscribed in a circle of area <span class=\"katex--inline\">1</span>. Point <span class=\"katex--inline\">P</span> lies inside the circle so that the region bounded by <span class=\"katex--inline\">\\overline{PA_1}</span>, <span class=\"katex--inline\">\\overline{PA_2}</span>, and the minor arc <span class=\"katex--inline\">A_1A _2</span> of the circle has area <span class=\"katex--inline\">\\tfrac17</span>, while the region bounded by <span class=\"katex--inline\">\\overline{PA_3}</span>, <span class=\"katex--inline\">\\overline{PA_4}</span>, and the minor arc <span class=\"katex--inline\">A_3 A_4</span> of the circle has area <span class=\"katex--inline\">\\tfrac{1}{9}</span>. There is a positive integer <span class=\"katex--inline\">n</span> such that the area of the region bounded by <span class=\"katex--inline\">\\overline{PA_6}</span>, <span class=\"katex--inline\">\\overline{PA_7}</span>, and the minor arc <span class=\"katex--inline\">A_6 A_7</span> is equal to <span class=\"katex--inline\">\\tfrac18 - \\tfrac{\\sqrt 2}n</span>. Find <span class=\"katex--inline\">n</span>.</p>&#10;<hr/>&#10;<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>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2019 AIME II Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/19_aime_II_p14", "prev": "/problem/19_aime_II_p12"}}