4. Philip, Kiril, and Nikola send letters to each other, and they do not receive letters from anyone else. The letters are sent according to the following rules:

- Philip sends a letter to Kiril every second day, and to Nikola every fifth day,
- Kiril sends a letter to Philip immediately after receiving two letters, and to Nikola every fourth day,
- Nikola sends a letter to Philip immediately after receiving four letters, and to Kiril every eighth day.

Each letter takes one day to travel, i.e., each letter arrives the day after it is sent. What is the smallest number of days after which Philip will receive 68 letters?
Solution. Let $x$ denote the number of days it will take for Philip to receive 68 letters. This means that Kiril and Nikola together sent 68 letters to Philip by the $(x-1)$-th day. Let $k$ and $n$ denote the number of letters received by Kiril and Nikola, respectively, by the $(x-1)$-th day. From the conditions of the problem, it follows that by the $(x-1)$-th day, Kiril sent $\frac{k}{2}$ letters to Philip, and Nikola sent $\frac{n}{4}$ letters to Philip. Therefore, $\frac{k}{2}+\frac{n}{4}=68$, i.e., $2k+n=272$.

On the other hand, by the $(x-1)$-th day, Kiril received as many letters as Philip and Nikola together sent by the $(x-2)$-th day. Since Philip sends a letter to Kiril every second day, and Nikola sends a letter to Kiril every eighth day, we get $k=\frac{x-2}{2}+\frac{x-2}{8}=\frac{5(x-2)}{8}$. Similarly, since Philip sends a letter to Nikola every fifth day, and Kiril sends a letter to Nikola every fourth day, we get $n=\frac{x-2}{5}+\frac{x-2}{4}=\frac{9(x-2)}{20}$. Now, if we substitute $k$ and $n$ into $2k+n=272$, we get

$$
\begin{aligned}
& 2 \cdot \frac{5(x-2)}{8}+\frac{9(x-2)}{20}=272 \\
& 25(x-2)+9(x-2)=5440 \\
& 34x=5508 \\
& x=162
\end{aligned}
$$

Thus, Philip will receive 68 letters after at least 162 days.

## IX Department