{"status": "success", "data": {"description_md": "Square $ABCD$ in the coordinate plane has vertices at the points $A(1,1), B(-1,1), C(-1,-1),$ and $D(1,-1).$ Consider the following four transformations:\n$\\quad\\bullet\\qquad$ $L,$ a rotation of $90^{\\circ}$ counterclockwise around the origin;\n$\\quad\\bullet\\qquad$ $R,$ a rotation of $90^{\\circ}$ clockwise around the origin;\n$\\quad\\bullet\\qquad$ $H,$ a reflection across the $x$-axis; and\n$\\quad\\bullet\\qquad$ $V,$ a reflection across the $y$-axis.\nEach of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying $R$ and then $V$ would send the vertex $A$ at $(1,1)$ to $(-1,-1)$ and would send the vertex $B$ at $(-1,1)$ to itself. How many sequences of $20$ transformations chosen from $\\{L, R, H, V\\}$ will send all of the labeled vertices back to their original positions? (For example, $R, R, V, H$ is one sequence of $4$ transformations that will send the vertices back to their original positions.)\n\n$\\textbf{(A)}\\ 2^{37} \\qquad\\textbf{(B)}\\ 3\\cdot 2^{36} \\qquad\\textbf{(C)}\\ 2^{38} \\qquad\\textbf{(D)}\\ 3\\cdot 2^{37} \\qquad\\textbf{(E)}\\ 2^{39}$", "description_html": "

Square ABCD in the coordinate plane has vertices at the points A(1,1), B(-1,1), C(-1,-1), and D(1,-1). Consider the following four transformations:
\n \\quad\\bullet\\qquad L, a rotation of 90^{\\circ} counterclockwise around the origin;
\n \\quad\\bullet\\qquad R, a rotation of 90^{\\circ} clockwise around the origin;
\n \\quad\\bullet\\qquad H, a reflection across the x -axis; and
\n \\quad\\bullet\\qquad V, a reflection across the y -axis.
\nEach of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying R and then V would send the vertex A at (1,1) to (-1,-1) and would send the vertex B at (-1,1) to itself. How many sequences of 20 transformations chosen from \\{L, R, H, V\\} will send all of the labeled vertices back to their original positions? (For example, R, R, V, H is one sequence of 4 transformations that will send the vertices back to their original positions.)

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\\textbf{(A)}\\ 2^{37} \\qquad\\textbf{(B)}\\ 3\\cdot 2^{36} \\qquad\\textbf{(C)}\\ 2^{38} \\qquad\\textbf{(D)}\\ 3\\cdot 2^{37} \\qquad\\textbf{(E)}\\ 2^{39}

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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 10B Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/20_amc10B_p24", "prev": "/problem/20_amc10B_p22"}}