{"status": "success", "data": {"description_md": "A trapezoid has side lengths $3$, $5$, $7$, and $11$. The sums of all the possible areas of the trapezoid can be written in the form of $r_1\\sqrt{n_1}+r_2\\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$?\n\n$\\textbf{(A)}\\ 57\\qquad\\textbf{(B)}\\ 59\\qquad\\textbf{(C)}\\ 61\\qquad\\textbf{(D)}\\ 63\\qquad\\textbf{(E)}\\ 65$\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": "

A trapezoid has side lengths 3 , 5 , 7 , and 11 . The sums of all the possible areas of the trapezoid can be written in the form of r_1\\sqrt{n_1}+r_2\\sqrt{n_2}+r_3 , where r_1 , r_2 , and r_3 are rational numbers and n_1 and n_2 are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to r_1+r_2+r_3+n_1+n_2 ?

\\textbf{(A)}\\ 57\\qquad\\textbf{(B)}\\ 59\\qquad\\textbf{(C)}\\ 61\\qquad\\textbf{(D)}\\ 63\\qquad\\textbf{(E)}\\ 65

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": 3, "problem_name": "2012 AMC 12B Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/12_amc12B_p21", "prev": "/problem/12_amc12B_p19"}}