{"status": "success", "data": {"description_md": "Define binary operations $\\diamondsuit$ and $\\heartsuit$ by $$a \\, \\diamondsuit \\, b = a^{\\log_{7}(b)} \\qquad \\text{and} \\qquad a \\, \\heartsuit \\, b = a^{\\frac{1}{\\log_{7}(b)}}$$for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3 = 3\\, \\heartsuit\\, 2$ and $$a_n = (n\\, \\heartsuit\\, (n-1)) \\,\\diamondsuit\\, a_{n-1}$$for all integers $n \\geq 4$. To the nearest integer, what is $\\log_{7}(a_{2019})$?\n\n$\\textbf{(A) } 8 \\qquad \\textbf{(B) } 9 \\qquad \\textbf{(C) } 10 \\qquad \\textbf{(D) } 11 \\qquad \\textbf{(E) } 12$\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": "

Define binary operations \\diamondsuit and \\heartsuit by a \\, \\diamondsuit \\, b = a^{\\log_{7}(b)} \\qquad \\text{and} \\qquad a \\, \\heartsuit \\, b = a^{\\frac{1}{\\log_{7}(b)}} for all real numbers a and b for which these expressions are defined. The sequence (a_n) is defined recursively by a_3 = 3\\, \\heartsuit\\, 2 and a_n = (n\\, \\heartsuit\\, (n-1)) \\,\\diamondsuit\\, a_{n-1} for all integers n \\geq 4 . To the nearest integer, what is \\log_{7}(a_{2019}) ?

\\textbf{(A) } 8 \\qquad \\textbf{(B) } 9 \\qquad \\textbf{(C) } 10 \\qquad \\textbf{(D) } 11 \\qquad \\textbf{(E) } 12

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2019 AMC 12A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/19_amc12A_p24", "prev": "/problem/19_amc12A_p22"}}