{"status": "success", "data": {"description_md": "Let $z$ be a complex number with $|z| = 2014$. Let $P$ be the polygon in the complex plane whose vertices are $z$ and every $w$ such that $\\tfrac{1}{z+w} = \\tfrac{1}{z} + \\tfrac{1}{w}$. Then the area enclosed by $P$ can be written in the form $n\\sqrt{3},$ where $n$ is an integer. Find the remainder when $n$ is divided by $1000$.\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>Let <span class=\"katex--inline\">z</span> be a complex number with <span class=\"katex--inline\">|z| = 2014</span>. Let <span class=\"katex--inline\">P</span> be the polygon in the complex plane whose vertices are <span class=\"katex--inline\">z</span> and every <span class=\"katex--inline\">w</span> such that <span class=\"katex--inline\">\\tfrac{1}{z+w} = \\tfrac{1}{z} + \\tfrac{1}{w}</span>. Then the area enclosed by <span class=\"katex--inline\">P</span> can be written in the form <span class=\"katex--inline\">n\\sqrt{3},</span> where <span class=\"katex--inline\">n</span> is an integer. Find the remainder when <span class=\"katex--inline\">n</span> is divided by <span class=\"katex--inline\">1000</span>.</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": 5, "problem_name": "2014 AIME II Problem 10", "can_next": true, "can_prev": true, "nxt": "/problem/14_aime_II_p11", "prev": "/problem/14_aime_II_p09"}}