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, 2020

Geometry Difficulty 5.7 AIME, harder Prove it United States

Problem:

In the future, MIT has attracted so many students that its buildings have become skyscrapers. Ben and Jerry decide to go ziplining together. Ben starts at the top of the Green Building, and ziplines to the bottom of the Stata Center. After waiting $a$ seconds, Jerry starts at the top of the Stata Center, and ziplines to the bottom of the Green Building. The Green Building is 160 meters tall, the Stata Center is 90 meters tall, and the two buildings are 120 meters apart. Furthermore, both zipline at 10 meters per second. Given that Ben and Jerry meet at the point where the two ziplines cross, compute $100 a$ .

Solution

Solution:

Define the following lengths:

Note that due to all the 3-4-5 triangles, we find $\frac{x}{z}=\frac{z}{y}=\frac{4}{3}$ , so $120=x+y=\frac{25}{12} z$ . Then,
$u=\frac{5}{3} x=\frac{20}{9} z=\frac{16}{15} 120=128$
while
$v=\frac{5}{4} y=\frac{15}{16} z=\frac{9}{20} 120=54$
Thus $u-v=74$ , implying that $a=7.4$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.