{"status": "success", "data": {"description_md": "Let $a$ and $b$ be positive real numbers with $a\\ge b$. Let $\\rho$ be the maximum possible value of $\\frac{a}{b}$ for which the system of equations\n$$ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2 $$has a solution in $(x,y)$ satisfying $0\\le x<a$ and $0\\le y<b$. Then $\\rho^2$ can be expressed as a fraction $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\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>Let <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> be positive real numbers with <span class=\"katex--inline\">a\\ge b</span>. Let <span class=\"katex--inline\">\\rho</span> be the maximum possible value of <span class=\"katex--inline\">\\frac{a}{b}</span> for which the system of equations<br/>&#10;<span class=\"katex--display\"> a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2 </span>has a solution in <span class=\"katex--inline\">(x,y)</span> satisfying <span class=\"katex--inline\">0\\le x&lt;a</span> and <span class=\"katex--inline\">0\\le y&lt;b</span>. Then <span class=\"katex--inline\">\\rho^2</span> can be expressed as a fraction <span class=\"katex--inline\">\\frac{m}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers. Find <span class=\"katex--inline\">m+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": "2008 AIME II Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/08_aime_II_p15", "prev": "/problem/08_aime_II_p13"}}