MathDB
Usain Bolt moment

Source: 2023 AMC 12A #15

November 9, 2023
AMCAMC 122023 AMC2023 AMC 12A

Problem Statement

Usain is walking for exercise by zigzagging across a 100100-meter by 3030-meter rectangular field, beginning at point AA and ending on the segment BC‾\overline{BC}. He wants to increase the distance walked by zigzagging as shown in the figure below (APQRS)(APQRS). What angle θ=∠PAB=∠QPC=∠RQB=⋯\theta=\angle PAB=\angle QPC=\angle RQB=\cdots will produce in a length that is 120120 meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)
[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("θ\theta", (-41.5,-10.5)); label("θ\theta", (-13,10.5)); label("θ\theta", (15.5,-10.5)); label("θ\theta", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label("DD",(-58,15)); label("AA",(-58,-15)); label("CC",(58,15)); label("BB",(58,-15)); label("SS",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label("PP",(-22.5,23)); label("QQ",(5,-23)); label("RR",(32.5,23)); [/asy]
<spanclass=′latex−bold′>(A)</span> arccos⁡56<spanclass=′latex−bold′>(B)</span> arccos⁡45<spanclass=′latex−bold′>(C)</span> arccos⁡310<spanclass=′latex−bold′>(D)</span> arcsin⁡45<spanclass=′latex−bold′>(E)</span> arcsin⁡56<span class='latex-bold'>(A)</span>~\arccos\frac{5}{6}\qquad<span class='latex-bold'>(B)</span>~\arccos\frac{4}{5}\qquad<span class='latex-bold'>(C)</span>~\arccos\frac{3}{10}\qquad<span class='latex-bold'>(D)</span>~\arcsin\frac{4}{5}\qquad<span class='latex-bold'>(E)</span>~\arcsin\frac{5}{6}
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