{"status": "success", "data": {"description_md": "Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ 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$\\textbf{(A) } 12 \\qquad \\textbf{(B) } 15 \\qquad \\textbf{(C) } 17 \\qquad \\textbf{(D) } 20 \\qquad \\textbf{(E) } 25$", "description_html": "

Let T be the triangle in the coordinate plane with vertices (0,0), (4,0), and (0,3). 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.)

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

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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": 4, "problem_name": "2020 AMC 10A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc10A_p24", "prev": "/problem/20_amc10A_p22"}}