Maths Olympiad Prep

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Algebra Difficulty 5.9 AIME, harder Prove it Estonia

A train departed from the station 12 minutes later than planned. If the train would not make any stops on the way and would travel at average speed equal to what would be the average speed between stops according to the timetable, then it would reach the destination exactly at the right time. But if the train would stop in every station for the same amount of time it was supposed to, then between the station it would have to travel with average speed 40% higher than before in order to reach the destination on time. Find the travelling time of the train according to the timetable.

Answer: 54 minutes.

Solution

Let the time we are looking for be tt. The conditions of the problem imply that t−12 min=1.4⋅(t−24 min)t - 12 \text{ min} = 1.4 \cdot (t - 24 \text{ min}), from which 0.4t=21.6 min0.4t = 21.6 \text{ min} and t=54 mint = 54 \text{ min}.

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