3. Three people $A, B$, and $C$ communicate with each other as follows:

- $A$ writes to $B$ every third day, and to $C$ every second day;
- $B$ writes to $A$ after receiving four letters from $A$ and $C$, and to $C$ every third day;
- $C$ writes to $A$ after receiving three letters from $A$ and $B$, and to $B$ every fourth day.

After how many days will $A$ receive 61 letters, if letters take one day to travel?
Solution. Let $x$ denote the number of days it takes for $A$ to receive 61 letters. During this time, $B$ received $\frac{x-1}{3}$ letters from $A$, and $\frac{x-1}{4}$ letters from $C$, or a total of $\frac{7(x-1)}{12}$ letters from $A$ and $C$. Similarly, $C$ received $\frac{x-1}{2}$ letters from $A$, and $\frac{x-1}{3}$ letters from $B$, or a total of $\frac{5(x-1)}{6}$ letters from $A$ and $B$.

Thus, in $x$ days, $B$ will send $\frac{7(x-1)}{12 \cdot 4}$ letters to $A$, and $C$ will send $\frac{5(x-1)}{6 \cdot 3}$ letters to $A$, meaning that $A$ will receive a total of $\frac{7(x-1)}{48}+\frac{5(x-1)}{18}=61$ letters. The solution to this equation is $x=145$. Since the last sent letter needs to travel one day, $A$ will receive 61 letters in 146 days.