Question

Patrick 从学校沿一条笔直的路以恒定速度步行去公园。Patrick 出发一小时后，Tanya 从学校沿同一条笔直的路以恒定速度跑向公园，速度比 Patrick 步行快每小时 $2$ 英里。Tanya 出发一小时后， José 从学校沿同一条笔直的路以恒定速度骑车去公园，速度比 Tanya 跑步快每小时 $7$ 英里。三人同时到达公园。学校到公园的距离是 $\frac{m}{n}$ 英里，其中 $m$ 和 $n$ 为互质正整数。求 $m + n$ 。

Patrick started walking at a constant speed along a straight road from his school to the park. One hour after Patrick left, Tanya started running at a constant speed of $2$ miles per hour faster than Patrick walked, following the same straight road from the school to the park. One hour after Tanya left, José started bicycling at a constant speed of $7$ miles per hour faster than Tanya ran, following the same straight road from the school to the park. All three people arrived at the park at the same time. The distance from the school to the park is $\frac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

答案

答案：277

知识点：路程、速度与时间方程组

难度评级：1840

解答：

设 Patrick 的速度为每小时 $v$ 英里，行程时间为 $T$ 小时。于是 Tanya 用 $T - 1$ 小时以速度 $v + 2$ ，行进，José 用 $T - 2$ 小时以速度 $v + 9$ 行进（这比 Tanya 的速度多 $7$ ）。因为三人走过同一段距离， $\begin{aligned} vT &= (v+2)(T-1) \\ &= (v+9)(T-2). \end{aligned}$

展开第一个等式得 $0 = 2T - v - 2$ ，所以 $v = 2T - 2$ 。展开第二个等式得 $0 = 9T - 2v - 18$ ，所以 $2v = 9T - 18$ 。代入可得 $4T - 4 = 9T - 18$ ，因此 $T = \frac{14}{5}$ ，且 $v = \frac{18}{5}$ 。

距离为 $vT = \frac{18}{5} \cdot \frac{14}{5} = \frac{252}{25}$ ，已经是最简分数，所以 $m + n = 252 + 25 = 277$ 。

Let $v$ be Patrick's speed in miles per hour and $T$ his travel time in hours. Then Tanya travels for $T - 1$ hours at speed $v + 2,$ and José travels for $T - 2$ hours at speed $v + 9$ (which is $7$ more than Tanya's speed). Since all three cover the same distance, $\begin{aligned} vT &= (v+2)(T-1) \\ &= (v+9)(T-2). \end{aligned}$

Expanding the first equality gives $0 = 2T - v - 2,$ so $v = 2T - 2.$ Expanding the second gives $0 = 9T - 2v - 18,$ so $2v = 9T - 18.$ Substituting, $4T - 4 = 9T - 18,$ hence $T = \frac{14}{5}$ and $v = \frac{18}{5}.$

The distance is $vT = \frac{18}{5} \cdot \frac{14}{5} = \frac{252}{25},$ which is in lowest terms, so $m + n = 252 + 25 = 277.$

2026 AIME I 第 1 题

答案：277

难度评级：1840

解答：

其他年份的第 1 题