{"status": "success", "data": {"description_md": "Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that\n\n$$$P(1) = P(3) = P(5) = P(7) = a$, and<br/>\n\n$P(2) = P(4) = P(6) = P(8) = -a$.$$<br>What is the smallest possible value of $a$?\n\n$\\textbf{(A)}\\ 105 \\qquad \\textbf{(B)}\\ 315 \\qquad \\textbf{(C)}\\ 945 \\qquad \\textbf{(D)}\\ 7! \\qquad \\textbf{(E)}\\ 8!$\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>Let  <span class=\"katex--inline\">a &gt; 0</span> , and let  <span class=\"katex--inline\">P(x)</span>  be a polynomial with integer coefficients such that</p>&#10;<p>$$ <span class=\"katex--inline\">P(1) = P(3) = P(5) = P(7) = a</span> , and<br/></p>&#10;<p> <span class=\"katex--inline\">P(2) = P(4) = P(6) = P(8) = -a</span> .$$<br/>What is the smallest possible value of  <span class=\"katex--inline\">a</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 105 \\qquad \\textbf{(B)}\\ 315 \\qquad \\textbf{(C)}\\ 945 \\qquad \\textbf{(D)}\\ 7! \\qquad \\textbf{(E)}\\ 8!</span> </p>&#10;<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": 5, "problem_name": "2010 AMC 12B Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/10_amc12B_p22", "prev": "/problem/10_amc12B_p20"}}