{"status": "success", "data": {"description_md": "Let $w_1, w_2, \\ldots, w_n$ be complex numbers. A line $L$ in the complex plane is called a mean line for the points $w_1, w_2, \\ldots, w_n$ if $L$ contains points (complex numbers) $z_1, z_2, \\ldots, z_n$ such that<br>\n$$ \\sum_{k = 1}^n (z_k - w_k) = 0. $$ For the numbers $w_1 = 32 + 170i$, $w_2 = -7 + 64i$, $w_3 = -9 +200i$, $w_4 = 1 + 27i$, and $w_5 = -14 + 43i$, there is a unique mean line with $y$-intercept 3. Find the slope of this mean line.\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\">w_1, w_2, \\ldots, w_n</span> be complex numbers. A line <span class=\"katex--inline\">L</span> in the complex plane is called a mean line for the points <span class=\"katex--inline\">w_1, w_2, \\ldots, w_n</span> if <span class=\"katex--inline\">L</span> contains points (complex numbers) <span class=\"katex--inline\">z_1, z_2, \\ldots, z_n</span> such that<br/><span class=\"katex--display\"> \\sum_{k = 1}^n (z_k - w_k) = 0. </span><br/>For the numbers <span class=\"katex--inline\">w_1 = 32 + 170i</span>, <span class=\"katex--inline\">w_2 = -7 + 64i</span>, <span class=\"katex--inline\">w_3 = -9 +200i</span>, <span class=\"katex--inline\">w_4 = 1 + 27i</span>, and <span class=\"katex--inline\">w_5 = -14 + 43i</span>, there is a unique mean line with <span class=\"katex--inline\">y</span>-intercept 3. Find the slope of this mean line.</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": 5, "problem_name": "1988 AIME Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/88_aime_p12", "prev": "/problem/88_aime_p10"}}