{"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<br>\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 $ 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<br/><span class=\"katex--display\"> a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2</span>has a solution in $ (x,y)$ satisfying $ 0\\le x&lt;a$ and $ 0\\le y&lt;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$.</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": "2008 AIME II Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/08_aime_II_p15", "prev": "/problem/08_aime_II_p13"}}