{"status": "success", "data": {"description_md": "Let $\\omega \\ne 1$ be a $13$th root of unity. Find the remainder when $$ \\prod_{k=0}^{12} \\left(2 - 2\\omega^k + \\omega^{2k} \\right) $$is divided by $1000$.\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\">\\omega \\ne 1</span> be a <span class=\"katex--inline\">13</span>th root of unity. Find the remainder when <span class=\"katex--display\"> \\prod_{k=0}^{12} \\left(2 - 2\\omega^k + \\omega^{2k} \\right) </span>is divided by <span class=\"katex--inline\">1000</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": "2024 AIME II Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/24_aime_II_p14", "prev": "/problem/24_aime_II_p12"}}