{"status": "success", "data": {"description_md": "How many ordered pairs of positive real numbers $(a,b)$ satisfy the equation\n\n$$(1+2a)(2+2b)(2a+b) = 32ab?$$\n\n$\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }2\\qquad\\textbf{(D) }3\\qquad\\textbf{(E) }\\text{an infinite number}$\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>How many ordered pairs of positive real numbers  <span class=\"katex--inline\">(a,b)</span>  satisfy the equation</p>&#10;<p> <span class=\"katex--display\">(1+2a)(2+2b)(2a+b) = 32ab?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }0\\qquad\\textbf{(B) }1\\qquad\\textbf{(C) }2\\qquad\\textbf{(D) }3\\qquad\\textbf{(E) }\\text{an infinite number}</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": "2023 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p24", "prev": "/problem/23_amc12A_p22"}}