5.1. In a row, there are 27 matchboxes, each containing a certain number of matches. It is known that in any four consecutive boxes, the total is 25 matches, and in all of them, the total is 165. How many matches are in the eighth box?
Answer: 10

Solution. In the first 24 boxes, there are a total of $6 \cdot 25=150$ matches. In the last three boxes, there are 15 matches. Therefore, the 4th from the end (or 24th from the start) has 10 matches. Then, in boxes $24, 23, 22, 21$, there are 25 matches in total, meaning in boxes $21, 22, 23$, there are 15 matches. But in boxes $23, 22, 21$, and 20, there are also 25 matches, so box 20 has 10 matches, the same as box 24. Thus, in every 4th box with a number divisible by 4, there will be 10 matches, including the box with number 8.

![](https://cdn.mathpix.com/cropped/2024_05_06_c4a05b5b9b4d29b3aeb4g-2.jpg?height=54&width=1870&top_left_y=93&top_left_x=116)
On the same road, Emilia and Eduard set out at the same speed. It is known that Arkady met Emilia at 13:00, and Eduard at 16:00. Arina met Emilia at 15:00. At what time will Arina meet Eduard?

Answer: 18

Solution. Arkady and Arina met Emilia with a 2-hour interval. The speeds of Arina and Arkady are equal, as are the speeds of Emilia and Eduard. Therefore, the distance between Arkady and Arina is constant, as is the distance between Emilia and Eduard. This means the speed of approach between Arkady and Emilia is equal to the speed of approach between Arkady and Eduard, and the same speed of approach will be between Arina and Emilia, and Arina and Eduard. Since the speeds of approach of Emilia and Eduard to the pair Arkady-Arina are equal, they will each pass the pair Arkady-Arina in the same amount of time, i.e., 2 hours. If Arkady met Eduard at 16:00, then Arina will meet him 2 hours later, i.e., at 18:00.