HMMT 二月 2014 · 冲刺赛 · 第 7 题

HMMT February 2014 — Guts Round — Problem 7

[ 5 ] The Evil League of Evil is plotting to poison the city’s water supply. They plan to set out from their headquarters at (5 , 1) and put poison in two pipes, one along the line y = x and one along the line x = 7. However, they need to get the job done quickly before Captain Hammer catches them. What’s the shortest distance they can travel to visit both pipes and then return to their headquarters? 0 1 15 16

解析

[ 5 ] The Evil League of Evil is plotting to poison the city’s water supply. They plan to set out from their headquarters at (5 , 1) and put poison in two pipes, one along the line y = x and one along the line x = 7. However, they need to get the job done quickly before Captain Hammer catches them. What’s the shortest distance they can travel to visit both pipes and then return to their headquarters? √ Answer: 4 5 After they go to y = x , we reflect the remainder of their path in y = x , along with the second pipe and their headquarters. Now, they must go from (5 , 1) to y = 7 crossing y = x , and then go to (1 , 5). When they reach y = 7, we reflect the remainder of their path again, so now their reflected headquarters is at (1 , 9). Thus, they just go from (5 , 1) to (1 , 9) in some path that inevitably crosses y = x and y = 7. The shortest path they can take is a straight line with length √ √ 2 2 4 + 8 = 4 5. Comment. These ideas can be used to prove that the orthic triangle of an acute triangle has the smallest possible perimeter of all inscribed triangles. Also, see if you can find an alternative solution using Minkowski’s inequality! 0 1 15 16