{"status": "success", "data": {"description_md": "How many ordered triples of integers $(a,b,c)$, with $a \\ge 2$, $b\\ge 1$, and $c \\ge 0$, satisfy both $\\log_a b = c^{2005}$ and $a + b + c = 2005$?\n\n$\\mathrm{(A)} \\ 0 \\qquad \\mathrm{(B)} \\ 1 \\qquad \\mathrm{(C)} \\ 2 \\qquad \\mathrm{(D)} \\ 3 \\qquad \\mathrm{(E)} \\ 4$\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 triples of integers <span class=\"katex--inline\">(a,b,c)</span>, with <span class=\"katex--inline\">a \\ge 2</span>, <span class=\"katex--inline\">b\\ge 1</span>, and <span class=\"katex--inline\">c \\ge 0</span>, satisfy both <span class=\"katex--inline\">\\log_a b = c^{2005}</span> and <span class=\"katex--inline\">a + b + c = 2005</span>?</p>&#10;<p><span class=\"katex--inline\">\\mathrm{(A)} \\ 0 \\qquad \\mathrm{(B)} \\ 1 \\qquad \\mathrm{(C)} \\ 2 \\qquad \\mathrm{(D)} \\ 3 \\qquad \\mathrm{(E)} \\ 4</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": 5, "problem_name": "2005 AMC 12A Problem 21", "can_next": true, "can_prev": true, "nxt": "/problem/05_amc12A_p22", "prev": "/problem/05_amc12A_p20"}}