Competition problem — Combinatorics

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Combinatorics Difficulty 5.2 AIME, harder Prove it United States

Problem:

Four friends, One, Two, Five, and Ten are located on one side of the dark tunnel, and have only one flashlight. It takes one minute for person One to walk through the tunnel, two minutes for Two, five for Five, and ten for Ten. The tunnel is narrow and at most two people can walk at the same time with the flashlight. Whenever two people walk together they walk at the speed of the slower one. Show that all four friends can go from one side of the tunnel to the other one in 17 minutes.

Solution

Solution:

Friends One and Two should walk to the other side. It will take them $2$ minutes. Then, One returns - this will last additional $1$ minute. Then, Ten and Five go to the other side - this will take $10$ minutes, but Two should return the flashlight ( $2$ additional minutes), and together with One, go through the tunnel for the last time - additional $2$ minutes. It remains to notice that the time spent is exactly $17$ minutes.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.