{"status": "success", "data": {"description_md": "Usain is walking for exercise by zigzagging across a $100$-meter by $30$-meter rectangular field, beginning at point $A$ and ending on the segment $\\overline{BC}$. He wants to increase the distance walked by zigzagging as shown in the figure below $(APQRS)$. What angle $\\theta$$\\angle PAB=\\angle QPC=\\angle RQB=\\cdots$ will produce in a length that is $120$ meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.)<br><center><img class=\"problem-image\" alt='[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label(\"$\theta$\", (-41.5,-10.5)); label(\"$\theta$\", (-13,10.5)); label(\"$\theta$\", (15.5,-10.5)); label(\"$\theta$\", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label(\"$D$\",(-58,15)); label(\"$A$\",(-58,-15)); label(\"$C$\",(58,15)); label(\"$B$\",(58,-15)); label(\"$S$\",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label(\"$P$\",(-22.5,23)); label(\"$Q$\",(5,-23)); label(\"$R$\",(32.5,23)); [/asy]' class=\"latexcenter\" height=\"112\" src=\"https://latex.artofproblemsolving.com/3/2/f/32f79492cfd57de5bd7b5008d86027a497703c9a.png\" width=\"252\"/></center>\n\n$\\textbf{(A)}~\\arccos\\frac{5}{6}\\qquad\\textbf{(B)}~\\arccos\\frac{4}{5}\\qquad\\textbf{(C)}~\\arccos\\frac{3}{10}\\qquad\\textbf{(D)}~\\arcsin\\frac{4}{5}\\qquad\\textbf{(E)}~\\arcsin\\frac{5}{6}$\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": "<p>Usain is walking for exercise by zigzagging across a  <span class=\"katex--inline\">100</span> -meter by  <span class=\"katex--inline\">30</span> -meter rectangular field, beginning at point  <span class=\"katex--inline\">A</span>  and ending on the segment  <span class=\"katex--inline\">\\overline{BC}</span> . He wants to increase the distance walked by zigzagging as shown in the figure below  <span class=\"katex--inline\">(APQRS)</span> . What angle  <span class=\"katex--inline\">\\theta</span>  <span class=\"katex--inline\">\\angle PAB=\\angle QPC=\\angle RQB=\\cdots</span>  will produce in a length that is  <span class=\"katex--inline\">120</span>  meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.)<br/><center><img class=\"latexcenter\" alt=\"[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label(&#34;$&#9;heta$&#34;, (-41.5,-10.5)); label(&#34;$&#9;heta$&#34;, (-13,10.5)); label(&#34;$&#9;heta$&#34;, (15.5,-10.5)); label(&#34;$&#9;heta$&#34;, (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label(&#34;$D$&#34;,(-58,15)); label(&#34;$A$&#34;,(-58,-15)); label(&#34;$C$&#34;,(58,15)); label(&#34;$B$&#34;,(58,-15)); label(&#34;$S$&#34;,(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label(&#34;$P$&#34;,(-22.5,23)); label(&#34;$Q$&#34;,(5,-23)); label(&#34;$R$&#34;,(32.5,23)); [/asy]\" height=\"112\" src=\"https://latex.artofproblemsolving.com/3/2/f/32f79492cfd57de5bd7b5008d86027a497703c9a.png\" width=\"252\"/></center></p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}~\\arccos\\frac{5}{6}\\qquad\\textbf{(B)}~\\arccos\\frac{4}{5}\\qquad\\textbf{(C)}~\\arccos\\frac{3}{10}\\qquad\\textbf{(D)}~\\arcsin\\frac{4}{5}\\qquad\\textbf{(E)}~\\arcsin\\frac{5}{6}</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2023 AMC 12A Problem 15", "can_next": true, "can_prev": true, "nxt": "/problem/23_amc12A_p16", "prev": "/problem/23_amc12A_p14"}}