{"status": "success", "data": {"description_md": "Positive real numbers $a$ and $b$ have the property that\n\n$$\\sqrt{\\log{a}} + \\sqrt{\\log{b}} + \\log \\sqrt{a} + \\log \\sqrt{b} = 100$$<br>and all four terms on the left are positive integers, where $\\log$ denotes the base-$10$ logarithm. What is $ab$?\n\n$\\textbf{(A) }   10^{52}   \\qquad        \\textbf{(B) }   10^{100}   \\qquad    \\textbf{(C) }   10^{144}   \\qquad   \\textbf{(D) }  10^{164} \\qquad  \\textbf{(E) }   10^{200}$\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>Positive real numbers  <span class=\"katex--inline\">a</span>  and  <span class=\"katex--inline\">b</span>  have the property that</p>&#10;<p> <span class=\"katex--display\">\\sqrt{\\log{a}} + \\sqrt{\\log{b}} + \\log \\sqrt{a} + \\log \\sqrt{b} = 100</span> <br/>and all four terms on the left are positive integers, where  <span class=\"katex--inline\">\\log</span>  denotes the base- <span class=\"katex--inline\">10</span>  logarithm. What is  <span class=\"katex--inline\">ab</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A) }   10^{52}   \\qquad        \\textbf{(B) }   10^{100}   \\qquad    \\textbf{(C) }   10^{144}   \\qquad   \\textbf{(D) }  10^{164} \\qquad  \\textbf{(E) }   10^{200}</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": 3, "problem_name": "2019 AMC 12A Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p16", "prev": "/problem/19_amc12A_p14"}}