Question

Points $A,$ $B,$ and $C$ lie in that order along a straight path where the distance from $A$ to $C$ is $1800$ meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at $A$ and running toward $C,$ Paul starting at $B$ and running toward $C,$ and Eve starting at $C$ and running toward $A.$ When Paul meets Eve, he turns around and runs toward $A.$ Paul and Ina both arrive at $B$ at the same time. Find the number of meters from $A$ to $B.$

Answer

Solution:

Let $x = AB,$ so $BC = 1800 - x,$ and let Eve's speed be $v,$ so Ina runs at $2v$ and Paul at $4v.$ Paul and Eve start at $B$ and $C$ running toward each other, so together they cover the $1800 - x$ meters between them, with Paul covering $\frac{4}{5}$ of it. Paul then retraces that distance back to $B,$ so when he reaches $B$ he has run $\frac{8}{5}(1800 - x)$ meters in total.

Ina reaches $B$ at the same moment, having run $x$ meters. Since Paul runs twice as fast as Ina, he has run $2x$ meters in that time. Therefore $\frac{8}{5}(1800 - x) = 2x,$ which gives $8 \cdot 1800 = 18x,$ so $x = 800.$

2018 AIME II Problem 1

Solution:

Problem 1 in Other Years