4. Marko and Luka met at the beginning of a 1800 m long trail. Marko is riding a bicycle, and Luka is running, and both move at constant speeds. When they reach the end of the trail, they turn around and continue in the opposite direction without stopping. In 30 minutes, Marko has traveled 9 km, and Luka 4.5 km. How far apart will they be 30 minutes after the start of the training, and at what distance from the start of the trail did they first meet?
First solution.

In the given time, Marko, cycling at a constant speed, covered a distance of $9 \mathrm{~km}=9000 \mathrm{~m}$, which means he covered 9000 : $1800=5$ lengths of the track.

Luka, in the given time, running at a constant speed, covered a distance of $4.5 \mathrm{~km}=4500 \mathrm{~m}$, which means he covered $4500: 1800=2.5$ lengths of the track.

1 POINT

After 30 minutes from the start of the training, Marko was at the end of the track, and Luka was at the halfway point, which means they were $900 \mathrm{~m}$ apart.

2 POINTS

In 30 minutes, Marko covered twice the distance of Luka, and both are moving at constant speeds. They first met when Luka was heading towards the end of the track, and Marko was returning towards the start. $\quad 1$ POINT

Let $A$ be the start of the track, $B$ be the end of the track, and $S$ be the place of their first meeting.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-20.jpg?height=97&width=1031&top_left_y=1905&top_left_x=250)

Let's illustrate the distance each of them covered with a diagram.

Luka covered the distance from $A$ to $S$.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-20.jpg?height=174&width=1034&top_left_y=2112&top_left_x=248)

Marko covered the distance from $A$ to $B$ and then from $B$ to $S$, which can also be represented as follows:

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-20.jpg?height=174&width=1336&top_left_y=2414&top_left_x=243)

However, the distance Marko covered is twice as much, so the distance from $S$ to $B$ and back from $B$ to $S$ is equal to the distance from $A$ to $S$.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-21.jpg?height=291&width=1362&top_left_y=337&top_left_x=241)

This means that the distance from $S$ to $B$ is half the distance from $A$ to $S$, which means the place of their meeting is at $\frac{2}{3}$ of the track.

![](https://cdn.mathpix.com/cropped/2024_05_30_5715e7b36c946fa665d7g-21.jpg?height=97&width=1014&top_left_y=885&top_left_x=247)

They first met at $\frac{2}{3}$ of $1800 \mathrm{~m}$, which is $1200 \mathrm{~m}$ from the start of the track.

TOTAL 10 POINTS