{"status": "success", "data": {"description_md": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Let $m/n$ be the probability that $\\sqrt{2+\\sqrt{3}}\\le |v+w|,$ 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\">v</span> and <span class=\"katex--inline\">w</span> be distinct, randomly chosen roots of the equation <span class=\"katex--inline\">z^{1997}-1=0.</span> Let <span class=\"katex--inline\">m/n</span> be the probability that <span class=\"katex--inline\">\\sqrt{2+\\sqrt{3}}\\le |v+w|,</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><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": "1997 AIME Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/97_aime_p15", "prev": "/problem/97_aime_p13"}}