{"status": "success", "data": {"description_md": "For some real numbers $a$ and $b$, the equation\n\n$$8x^3 + 4ax^2 + 2bx + a = 0$$<br>has three distinct positive roots. If the sum of the base-$2$ logarithms of the roots is $5$, what is the value of $a$? \n\n$\\mathrm{(A)}\\ -256\\qquad\\mathrm{(B)}\\ -64 \\qquad\\mathrm{(C)}\\ -8\\qquad\\mathrm{(D)}\\ 64 \\qquad\\mathrm{(E)}\\ 256$\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>For some real numbers <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span>, the equation</p>&#10;<p><span class=\"katex--display\">8x^3 + 4ax^2 + 2bx + a = 0</span><br/>has three distinct positive roots. If the sum of the base-<span class=\"katex--inline\">2</span> logarithms of the roots is <span class=\"katex--inline\">5</span>, what is the value of <span class=\"katex--inline\">a</span>?</p>&#10;<p><span class=\"katex--inline\">\\mathrm{(A)}\\ -256\\qquad\\mathrm{(B)}\\ -64 \\qquad\\mathrm{(C)}\\ -8\\qquad\\mathrm{(D)}\\ 64 \\qquad\\mathrm{(E)}\\ 256</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": 3, "problem_name": "2004 AMC 12B Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/04_amc12B_p18", "prev": "/problem/04_amc12B_p16"}}