2024 SSMO Relay Round 3 Problems/Problem 1

Problem

Alice and Bob on the second floor of the building right next to the escalator moving upwards. Alice and Bob each run at 200 steps per minute and the escalator is a total of 225 steps. Alice chooses to use the escalator moving upwards, while Bob chooses to run to the escalator moving downwards, 300 steps away, and then ride the escalator down. If Alice and Bob hit the bottom floor at the exact same time, find the speed of the escalators in terms of steps per minute.

Solution

Let the speed of the escalators be We have the following Distance/Speed/Time chart. $$ (Error compiling LaTeX. Unknown error_msg)\begin{tabular}{c|c|c} Distance & Speed & Time\\\hline 225 & & \\ 300 & & \\ 225 & & \\ \end{tabular}$$ (Error compiling LaTeX. Unknown error_msg) Since Alice and Bob reach the bottom floor at the same time, we have Since distance is the product of speed and time, we have \begin{align*} \frac{225}{200-s}&=\frac{3}{2}+\frac{225}{200+s}\implies\\ \frac{450s}{200^2-s^2} &= \frac{3}{2}\implies\\ s^2+300s-200^2&=0\implies\\ (s+400)(s-100)&=0\implies\\ s&=\boxed{100}. \end{align*}

~SMO_Team