Question

点 $A$ 、 $B$ 、 $C$ 按此顺序位于一条直路上，且 $A$ 到 $C$ 的距离为 $1800$ 米。Ina 的速度是 Eve 的两倍，Paul 的速度是 Ina 的两倍。三名跑者同时开始跑步：Ina 从 $A$ 出发跑向 $C$ ，Paul 从 $B$ 出发跑向 $C$ ，Eve 从 $C$ 出发跑向 $A$ 。当 Paul 遇到 Eve 时，他转身跑向 $A$ 。Paul 和 Ina 同时到达 $B$ 。求 $A$ 到 $B$ 的距离。

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 $B.$

答案

答案：800

知识点：相对速度路程、速度与时间一次方程

难度评级：1950

解答：

设 $x = AB$ ，则 $BC = 1800 - x$ ，设 Eve 的速度为 $v$ ，则 Ina 的速度为 $2v$ ，Paul 的速度为 $4v$ 。Paul 和 Eve 分别从 $B$ 与 $C$ 相向而跑，所以他们一起跑完相距 $1800 - x$ 米的路段，其中 Paul 跑了 $\frac{4}{5}$ 。然后 Paul 沿原路跑回 $B$ ，所以当他到达 $B$ 时，他总共跑了 $\frac{8}{5}(1800 - x)$ 米。

Ina 同时到达 $B$ ，她跑了 $x$ 米。因为 Paul 的速度是 Ina 的两倍，所以这段时间内 Paul 跑了 $2x$ 米。因此 $\frac{8}{5}(1800 - x) = 2x,$ 得 $8 \cdot 1800 = 18x$ ，所以 $x = 800$ 。

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 第 1 题

答案：800

难度评级：1950

解答：

其他年份的第 1 题