# Task 2.

B-1.

At the time when a lion cub, who was 5 minutes away, set off for the watering hole, the second cub, having already quenched its thirst, started heading back along the same road at 1.5 times the speed of the first. At the same time, a tortoise, which was half an hour away, set off for the watering hole along the same road. After some time, all three met at one point and then continued on their way. How many minutes after the meeting did the tortoise reach the watering hole, given that all three moved at constant speeds?
Answer: 28

Solution. Let's take the entire path of the turtle as 1 and let $x$ be the speed of the 1st lion cub. Then the speed of the 2nd lion cub is $1.5x$, and the speed of the turtle is $1/30$. The entire path to the watering hole for the 1st lion cub is $5x$. Therefore, the meeting with the 2nd lion cub occurred after $5x/(x+1.5x)=2$ minutes from the start of the movement. The turtle also moved for the same amount of time before the meeting. The remaining distance was covered by the turtle in $30-2=28$ minutes.

## B-2

At the time when one lion cub, located 7.5 minutes away from the watering hole, set off for it, the second, having quenched his thirst, headed back along the same path at twice the speed of the first. At the same time, a turtle, located half an hour away from the watering hole, set off for it along the same path. After some time, all three met at one point and continued on their way. How many minutes after the meeting did the turtle reach the watering hole, given that all three moved at constant speeds?

Answer: 27.5

## B-3

At the time when one lion cub, located 7 minutes away from the watering hole, set off for it, the second, having quenched his thirst, headed back along the same path at 2.5 times the speed of the first. At the same time, a turtle, located half an hour away from the watering hole, set off for it along the same path. After some time, all three met at one point and continued on their way. How many minutes after the meeting did the turtle reach the watering hole, given that all three moved at constant speeds?

Answer: 28

## B-4

At the time when one lion cub, located 6 minutes away from the watering hole, set off for it, the second, having quenched his thirst, headed back along the same path at three times the speed of the first. At the same time, a turtle, located half an hour away from the watering hole, set off for it along the same path. After some time, all three met at one point and continued on their way. How many minutes after the meeting did the turtle reach the watering hole, given that all three moved at constant speeds?

Answer: 38.5

## Lomonosov School Olympiad in Mathematics

Preliminary stage $2020 / 21$ academic year for $5-6$ grades

## B-1

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