# Problem 3.

B-1

At the time when a lion cub, located 6 minutes away, set off for a drink, the second, having already quenched its thirst, headed back along the same path at 1.5 times the speed of the first. At the same time, a tortoise, located 32 minutes away, set off along the same path for a drink. After some time, the first lion cub stepped on the tortoise, and some time later, the second lion cub did the same. 28 minutes and 48 seconds after the second incident, the tortoise reached the water hole. How many minutes passed between the two incidents, given that all three moved at constant speeds?
Answer: 2.4

Solution. Let the entire path of the turtle be 1, and let $x$ be the speed of the first cub. Then the speed of the second cub is $1.5x$, and the speed of the turtle is $1/32$. The entire path to the watering hole for the first cub is $6x$. Therefore, the initial distance between him and the turtle was $6x-1$, and the first incident occurred after $(6x-1) / (x - 1/32)$ minutes. The second incident occurred after

$$
32-28.8 = \frac{1}{1.5x + 1/32} \text{ minutes, from which } x = \frac{3}{16} \text{.}
$$

Therefore, the number of minutes between the incidents is

$$
32-28.8 - \frac{6x-1}{x - 1/32} = 3.2 - \frac{18/16 - 1}{6/32 - 1/32} = 3.2 - 0.8 = 2.4
$$

## B-2

At the time when the first cub, who was 5 minutes away from the watering hole, set off, the second cub, who had already quenched his thirst, started moving back along the same path 1.5 times faster than the first. At the same time, a turtle, who was 27 minutes away from the watering hole, also set off along the same path. After some time, the first cub stepped on the turtle, and after some more time, the second cub did the same. The turtle reached the watering hole 24 minutes and 18 seconds after the second incident. How many minutes passed between the two incidents, given that all three moved at constant speeds?

Answer: 2.1

## B-3

At the time when the first cub, who was 6 minutes away from the watering hole, set off, the second cub, who had already quenched his thirst, started moving back along the same path 2 times faster than the first. At the same time, a turtle, who was 33 minutes away from the watering hole, also set off along the same path. After some time, the first cub stepped on the turtle, and after some more time, the second cub did the same. The turtle reached the watering hole 30 minutes and 48 seconds after the second incident. How many minutes passed between the two incidents, given that all three moved at constant speeds?

Answer: 0.7

## B-4

At the time when the first cub, who was 6 minutes away from the watering hole, set off, the second cub, who had already quenched his thirst, started moving back along the same path 2 times faster than the first. At the same time, a turtle, who was 39 minutes away from the watering hole, also set off along the same path. After some time, the first cub stepped on the turtle, and after some more time, the second cub did the same. The turtle reached the watering hole 36 minutes and 24 seconds after the second incident. How many minutes passed between the two incidents, given that all three moved at constant speeds?

Answer: 2.1

## Lomonosov Olympiad for Schoolchildren in Mathematics

Preliminary Stage 2020/21 Academic Year for 7-8 Grades

## B-1

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