Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.3 AIME, harder Find the answer

1.1. In a cross-country skiing race, Petya and Vasya started simultaneously. Vasya ran the entire race at a constant speed of 12 km/h. Petya ran the first half of the distance at a speed of 9 km/h and fell behind Vasya. What should Petya's speed be on the second half of the distance to catch up with Vasya and finish at the same time as his friend? Express your answer in km/h.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Answer: 18

Solution. Let $t$ be the time it takes for Petya to run half the distance, then the length of half the distance is $9t$ , and the entire distance is $18t$ . Vasya runs $12t$ in time $t$ , and he has $6t$ left to the finish, which will take him time $t/2$ . For Petya to run the second half of the distance and catch up with Vasya at the finish line in time $t/2$ , Petya must run at a speed of $9t / \left(\frac{t}{2}\right) = 18$ .

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