# 4. Clone 1

The Hare and the Wolf are running in the same direction around a circle 200 meters in circumference at speeds of $5 \mathrm{~m} / \mathrm{c}$ and 3 m/s. It turns out that the distance between them is exactly the same as it was 40 seconds ago (the distance is measured along the shorter of the two arcs). What could this distance be?

Choose the answer from the following options: $40,50,60,70,80,90$
Answer: 40,60

## Solution

From the condition, it follows that the Hare runs faster than the Wolf. In addition, in 40 seconds, the Hare will run 200 meters, which is no more than 1 lap. We will assume that the Hare and the Wolf are moving in a circle clockwise. Let's consider two possible cases.

1) The Hare catches up and overtakes the Wolf (left figure).

Let the distance between the Hare and the Wolf be $x$ meters. Then in 40 seconds, the Hare will run a distance of $2x$ meters plus the distance the Wolf has run: $3 \cdot 40 = 120$ meters. On the other hand, in 40 seconds, he ran $5 \cdot 40 = 200$ meters. Therefore, $2x + 120 = 200$, or $x = 40$.

2) The Hare was ahead of the Wolf in the direction of the circle, and after 40 seconds, the distance between them began to be measured not along the arc Wolf-Hare, but along the arc Hare-Wolf (right figure).

Let the initial distance between the Hare and the Wolf also be $x$ meters. To be at the same distance from the Wolf, the Hare needs to run $200 - 2x + 120$. This value is obtained by subtracting twice the distance between the Hare and the Wolf from the total length of the circle and adding the distance the Wolf ran during this time. On the other hand, the Hare ran 200 meters in 40 seconds. Therefore, $320 - 2x = 200$, $x = 60$.
![](https://cdn.mathpix.com/cropped/2024_05_06_f1445201e81a69006a8ag-04.jpg?height=612&width=1172&top_left_y=290&top_left_x=500)

## Clone 2

The Hare and the Wolf are running in the same direction around a circle 260 meters long at speeds of $5 \mathrm{~m} / \mathrm{s}$ and 3 m/s. It turned out that the distance between them is exactly the same as it was 40 seconds ago (the distance is measured along the shorter of the two arcs). What can this distance be?

Choose the answer from the set: $40,50,60,70,80,90$

Answer: 40,90

## Clone 3

The Hare and the Wolf are running in the same direction around a circle 260 meters long at speeds of $5 \mathrm{~m} / \mathrm{s}$ and 3 m/s. It turned out that the distance between them is exactly the same as it was 50 seconds ago (the distance is measured along the shorter of the two arcs). What can this distance be?

Choose the answer from the set: $40,50,60,70,80,90$

Answer: 50,80

## Clone 4

The Hare and the Wolf are running in the same direction around a circle 280 meters long at speeds of $5 \mathrm{~m} / \mathrm{s}$ and 3 m/s. It turned out that the distance between them is exactly the same as it was 50 seconds ago (the distance is measured along the shorter of the two arcs). What can this distance be?

Choose the answer from the set: $40,50,60,70,80,90$

Answer: 50,90