{"status": "success", "data": {"description_md": "Let $T$ be the triangle in the coordinate plane with vertices $\\left(0,0\\right)$, $\\left(4,0\\right)$, and $\\left(0,3\\right)$. Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\\circ}$, $180^{\\circ}$, and $270^{\\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.)\n\n$\\textbf{(A) } 12\\qquad\\textbf{(B) } 15\\qquad\\textbf{(C) }17 \\qquad\\textbf{(D) }20 \\qquad\\textbf{(E) }25$\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": "

Let T be the triangle in the coordinate plane with vertices \\left(0,0\\right) , \\left(4,0\\right) , and \\left(0,3\\right) . Consider the following five isometries (rigid transformations) of the plane: rotations of 90^{\\circ} , 180^{\\circ} , and 270^{\\circ} counterclockwise around the origin, reflection across the x -axis, and reflection across the y -axis. How many of the 125 sequences of three of these transformations (not necessarily distinct) will return T to its original position? (For example, a 180^{\\circ} rotation, followed by a reflection across the x -axis, followed by a reflection across the y -axis will return T to its original position, but a 90^{\\circ} rotation, followed by a reflection across the x -axis, followed by another reflection across the x -axis will not return T to its original position.)

\\textbf{(A) } 12\\qquad\\textbf{(B) } 15\\qquad\\textbf{(C) }17 \\qquad\\textbf{(D) }20 \\qquad\\textbf{(E) }25

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": "2020 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc12A_p21", "prev": "/problem/20_amc12A_p19"}}