{"status": "success", "data": {"description_md": "Define $n!!$ to be $n(n-2)(n-4)\\ldots3\\cdot1$ for $n$ odd and $n(n-2)(n-4)\\ldots4\\cdot2$ for $n$ even. When $\\displaystyle \\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\displaystyle \\frac{ab}{10}$.\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>Define <span class=\"katex--inline\">n!!</span> to be <span class=\"katex--inline\">n(n-2)(n-4)\\ldots3\\cdot1</span> for <span class=\"katex--inline\">n</span> odd and <span class=\"katex--inline\">n(n-2)(n-4)\\ldots4\\cdot2</span> for <span class=\"katex--inline\">n</span> even. When <span class=\"katex--inline\">\\displaystyle \\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}</span> is expressed as a fraction in lowest terms, its denominator is <span class=\"katex--inline\">2^ab</span> with <span class=\"katex--inline\">b</span> odd. Find <span class=\"katex--inline\">\\displaystyle \\frac{ab}{10}</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": 4, "problem_name": "2009 AIME II Problem 7", "can_next": true, "can_prev": true, "nxt": "/problem/09_aime_II_p08", "prev": "/problem/09_aime_II_p06"}}