{"status": "success", "data": {"description_md": "If $\\{a_1,a_2,a_3,\\ldots,a_n\\}$ is a set of real numbers, indexed so that $a_1<a_2<a_3<\\cdots<a_n,$ its complex power sum is defined to be $a_1i+a_2i^2+a_3i^3+\\cdots+a_ni^n,$ where $i^2=-1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\\{1,2,\\ldots,n\\}.$ Given that $S_8=-176-64i$ and $S_9=p+qi,$ were $p$ and $q$ are integers, find $|p|+|q|.$\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>If <span class=\"katex--inline\">\\{a_1,a_2,a_3,\\ldots,a_n\\}</span> is a set of real numbers, indexed so that <span class=\"katex--inline\">a_1&lt;a_2&lt;a_3&lt;\\cdots&lt;a_n,</span> its complex power sum is defined to be <span class=\"katex--inline\">a_1i+a_2i^2+a_3i^3+\\cdots+a_ni^n,</span> where <span class=\"katex--inline\">i^2=-1.</span> Let <span class=\"katex--inline\">S_n</span> be the sum of the complex power sums of all nonempty subsets of <span class=\"katex--inline\">\\{1,2,\\ldots,n\\}.</span> Given that <span class=\"katex--inline\">S_8=-176-64i</span> and <span class=\"katex--inline\">S_9=p+qi,</span> were <span class=\"katex--inline\">p</span> and <span class=\"katex--inline\">q</span> are integers, find <span class=\"katex--inline\">|p|+|q|.</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": "1998 AIME Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/98_aime_p14", "prev": "/problem/98_aime_p12"}}