{"status": "success", "data": {"description_md": "Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP>CP$. Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ respectively. Given that $AB=12$ and $\\angle O_1 P O_2 = 120^\\circ$, then $AP=\\sqrt{a}+\\sqrt{b}$ where $a$ and $b$ are positive integers. Find $a+b$.\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>Point <span class=\"katex--inline\">P</span> lies on the diagonal <span class=\"katex--inline\">AC</span> of square <span class=\"katex--inline\">ABCD</span> with <span class=\"katex--inline\">AP&gt;CP</span>. Let <span class=\"katex--inline\">O_1</span> and <span class=\"katex--inline\">O_2</span> be the circumcenters of triangles <span class=\"katex--inline\">ABP</span> and <span class=\"katex--inline\">CDP</span> respectively. Given that <span class=\"katex--inline\">AB=12</span> and <span class=\"katex--inline\">\\angle O_1 P O_2 = 120^\\circ</span>, then <span class=\"katex--inline\">AP=\\sqrt{a}+\\sqrt{b}</span> where <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> are positive integers. Find <span class=\"katex--inline\">a+b</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": 6, "problem_name": "2011 AIME II Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/11_aime_II_p14", "prev": "/problem/11_aime_II_p12"}}