{"status": "success", "data": {"description_md": "Every positive integer $k$ has a unique factorial base expansion $(f_1,f_2,f_3,\\ldots,f_m),$ meaning that $$ k=1!\\cdot f_1+2!\\cdot f_2+3!\\cdot f_3+\\cdots+m!\\cdot f_m, $$ where each $f_i$ is an integer, $0\\le f_i\\le i,$ and $0<f_m.$ Given that $(f_1,f_2,f_3,\\ldots,f_j)$ is the factorial base expansion of $16!-32!+48!-64!+\\cdots+1968!-1984!+2000!,$ find the value of $f_1-f_2+f_3-f_4+\\cdots+(-1)^{j+1}f_j.$\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>Every positive integer <span class=\"katex--inline\">k</span> has a unique factorial base expansion <span class=\"katex--inline\">(f_1,f_2,f_3,\\ldots,f_m),</span> meaning that <span class=\"katex--display\"> k=1!\\cdot f_1+2!\\cdot f_2+3!\\cdot f_3+\\cdots+m!\\cdot f_m, </span> where each <span class=\"katex--inline\">f_i</span> is an integer, <span class=\"katex--inline\">0\\le f_i\\le i,</span> and <span class=\"katex--inline\">0&lt;f_m.</span> Given that <span class=\"katex--inline\">(f_1,f_2,f_3,\\ldots,f_j)</span> is the factorial base expansion of <span class=\"katex--inline\">16!-32!+48!-64!+\\cdots+1968!-1984!+2000!,</span> find the value of <span class=\"katex--inline\">f_1-f_2+f_3-f_4+\\cdots+(-1)^{j+1}f_j.</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": "2000 AIME II Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/00_aime_II_p15", "prev": "/problem/00_aime_II_p13"}}