{"status": "success", "data": {"description_md": "When the roots of the polynomial\n\n$$P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \\cdots(x-10)^{10}$$ are removed from the number line, what remains is the union of $11$ disjoint open intervals. On how many of these intervals is $P(x)$ positive?\n\n$\\textbf{(A)}~3\\qquad\\textbf{(B)}~7\\qquad\\textbf{(C)}~6\\qquad\\textbf{(D)}~4\\qquad\\textbf{(E)}~5$\n___\nFull 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>When the roots of the polynomial</p>&#10;<p><span class=\"katex--display\">P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \\cdots(x-10)^{10}</span> are removed from the number line, what remains is the union of <span class=\"katex--inline\">11</span> disjoint open intervals. On how many of these intervals is <span class=\"katex--inline\">P(x)</span> positive?</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~3\\qquad\\textbf{(B)}~7\\qquad\\textbf{(C)}~6\\qquad\\textbf{(D)}~4\\qquad\\textbf{(E)}~5</span></p>&#10;<hr/>&#10;<p>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": 2, "problem_name": "2023 AMC 12B Problem 6", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12B_p07", "prev": "/problem/23_amc12B_p05"}}