{"status": "success", "data": {"description_md": "Let $x$, $y$, and $z$ be positive real numbers satisfying the system of equations
\\begin{align*}
\\sqrt{2x - xy} + \\sqrt{2y - xy} & = 1\\\\
\\sqrt{2y - yz} + \\hspace{0.1em} \\sqrt{2z - yz} & = \\sqrt{2}\\\\
\\sqrt{2z - zx\\vphantom{y}} + \\sqrt{2x - zx\\vphantom{y}} & = \\sqrt{3}.
\\end{align*}Then $\\big[ (1-x)(1-y)(1-z) \\big] ^2$ can be written as $\\frac{m}{n}$, 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": "

Let x, y, and z be positive real numbers satisfying the system of equations
\\begin{align*}
\\sqrt{2x - xy} + \\sqrt{2y - xy} & = 1\\
\\sqrt{2y - yz} + \\hspace{0.1em} \\sqrt{2z - yz} & = \\sqrt{2}\\
\\sqrt{2z - zx\\vphantom{y}} + \\sqrt{2x - zx\\vphantom{y}} & = \\sqrt{3}.
\\end{align*}Then \\big[ (1-x)(1-y)(1-z) \\big] ^2 can be written as \\frac{m}{n}, where m and n are relatively prime positive integers. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "2022 AIME I Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/22_aime_I_p14"}}