HMMT 十一月 2017 · 冲刺赛 · 第 18 题
HMMT November 2017 — Guts Round — Problem 18
题目详情
- [ 7 ] Mr. Taf takes his 12 students on a road trip. Since it takes two hours to walk from the school to the destination, he plans to use his car to expedite the journey. His car can take at most 4 students at a time, and travels 15 times as fast as traveling on foot. If they plan their trip optimally, what is the shortest amount of time it takes for them to all reach the destination, in minutes? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2017, November 11, 2017 — GUTS ROUND Organization Team Team ID#
解析
- [ 7 ] Mr. Taf takes his 12 students on a road trip. Since it takes two hours to walk from the school to the destination, he plans to use his car to expedite the journey. His car can take at most 4 students at a time, and travels 15 times as fast as traveling on foot. If they plan their trip optimally, what is the shortest amount of time it takes for them to all reach the destination, in minutes? Proposed by: Yuan Yao 152 Answer: 30 . 4 OR 5 A way to plan the trip is to have Mr. Taf drive 4 students to the 80% mark, then drive back to the 10% mark to pick up another 4 students to the 90% mark, and finally drive back to the 20% mark to pick up the last 4 students to the destination. All students will reach the destination at the same time, 120 and Mr. Taf would have driven for (0 . 8 + 0 . 7 + 0 . 8 + 0 . 7 + 0 . 8) · = 30 . 4 minutes. 15 Now we show that 30 . 4 minutes is necessary. First of all, for a trip to be optimal, Mr. Taf must not carry students when he was driving away from the destination, and all student not on the car must keep walking towards the destination. Suppose that among all the students, the student that walked 120 − 15 m for the longest walked for 15 m minutes, where 0 ≤ m ≤ 8, then he spent = 8 − m minutes 15 on the car, so it took them exactly 14 m + 8 minutes to get to the destination. Moreover, all students must have spent at least 12(8 − m ) minutes on the car total, and Mr. Taf would need to spend at least 3(8 − m ) = 24 − 3 m minutes driving students towards the destination. Since it takes Mr. Taf 8 minutes to drive the entire trip, he would need to drive 3(8 − m ) − 8 = 16 − 3 m minutes away from the destination, so Mr. Fat drove for at least 40 − 6 m minutes. From this we derive the inequality 40 − 6 m ≥ 14 m + 8, which comes out to m ≥ 1 . 6, so the journey is at least 14(1 . 6) + 8 = 30 . 4 minutes.