{"status": "success", "data": {"description_md": "Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC= DE = FG = HA = 11$ is formed by removing four $6-8-10$ triangles from the corners of a $23\\times 27$ rectangle with side $\\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\\overline{HA}$, and partition the octagon into $7$ triangles by drawing segments $\\overline{JB}$, $\\overline{JC}$, $\\overline{JD}$, $\\overline{JE}$, $\\overline{JF}$, and $\\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these $7$ triangles.<br><br>$\\includegraphics[width=159, height=136, totalheight=136]{https://latex.artofproblemsolving.com/9/5/8/958670038ab1dd52c3b3fb6095f47bd59ee398d1.png}$\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": "<p>Octagon <span class=\"katex--inline\">ABCDEFGH</span> with side lengths <span class=\"katex--inline\">AB = CD = EF = GH = 10</span> and <span class=\"katex--inline\">BC= DE = FG = HA = 11</span> is formed by removing four <span class=\"katex--inline\">6-8-10</span> triangles from the corners of a <span class=\"katex--inline\">23\\times 27</span> rectangle with side <span class=\"katex--inline\">\\overline{AH}</span> on a short side of the rectangle, as shown. Let <span class=\"katex--inline\">J</span> be the midpoint of <span class=\"katex--inline\">\\overline{HA}</span>, and partition the octagon into <span class=\"katex--inline\">7</span> triangles by drawing segments <span class=\"katex--inline\">\\overline{JB}</span>, <span class=\"katex--inline\">\\overline{JC}</span>, <span class=\"katex--inline\">\\overline{JD}</span>, <span class=\"katex--inline\">\\overline{JE}</span>, <span class=\"katex--inline\">\\overline{JF}</span>, and <span class=\"katex--inline\">\\overline{JG}</span>. Find the area of the convex polygon whose vertices are the centroids of these <span class=\"katex--inline\">7</span> triangles.<br/><br/><img src=\"https://latex.artofproblemsolving.com/9/5/8/958670038ab1dd52c3b3fb6095f47bd59ee398d1.png\" width=\"159\" height=\"136\" class=\"problem-image\"/></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": 4, "problem_name": "2018 AIME II Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/18_aime_II_p10", "prev": "/problem/18_aime_II_p08"}}