Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.5 AIME, harder Prove it Croatia

Ana and Vanja are standing together next to a railway, waiting for the train to pass. The train drives at constant speed. At the moment the front end of the train passes them, Ana starts walking at constant speed in the same direction as the train is going, and Vanja starts walking at the same speed in the opposite direction. Each of them stops at the moment the rear end of the train passes her. In total, Ana walked 45 metres and Vanja walked 30 metres. How long is the train? (Portugal 2006)

Solution

Note that Ana walked $45 - 30 = 15$ metres more than Vanja, and while Ana was walking these $15$ metres, the train travelled $45 + 30 = 75$ metres.
Therefore, the speed of the train is $\frac{75}{15} = 5$ times greater than the walking speed.
Also note that while Vanja walked $30$ metres, the train travelled $30 \cdot 5 = 150$ metres.
Since Vanja started walking at the moment the front end of the train passed her, and stopped when the rear end passed her, and since she was walking in the opposite direction, the total length of the train is $150 + 30 = 180$ metres.

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