{"status": "success", "data": {"description_md": "When the roots of the polynomial \n$$P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}$$\nare 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$\\textbf{(A)}~3\\qquad\\textbf{(B)}~7\\qquad\\textbf{(C)}~6\\qquad\\textbf{(D)}~4\\qquad\\textbf{(E)}~5$", "description_html": "<p>When the roots of the polynomial<br/>\n <span class=\"katex--display\">P(x)  = (x-1)^1 (x-2)^2 (x-3)^3 \\cdot \\cdot \\cdot (x-10)^{10}</span> <br/>\nare 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>\n<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>\n<hr><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>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2023 AMC 10B Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc10B_p13", "prev": "/problem/23_amc10B_p11"}}