# Task 7. (14 points)

Several businessmen decided to open a company and divide all the profits into equal parts. One of the businessmen was appointed as the director. One day, this director of the company transferred part of the profit from the company's account to his own personal account. This amount of money was three times more than the share each of the others would have received if they had divided the remaining profit equally among themselves. After this, the director left the company. The next director of the company, one of the remaining businessmen, immediately did the same as the previous one, and so on. In the end, the penultimate director of the company transferred part of the profit to his own account, which was also three times more than what was left for the last businessman. As a result of these profit distributions, the last businessman received 190 times less money than the first director of the company. How many businessmen opened this company?
# Solution.

Let $n$ be the number of businessmen and $d_{i}$ be the profit of the $i$-th director, $i=1, \ldots, n$.

By the condition $d_{i}=3 \frac{d_{i+1}+d_{i+2}+\ldots+d_{n}}{n-i}$. Then

$$
\begin{gathered}
d_{i-1}=3 \frac{d_{i}+d_{i+1}+\ldots+d_{n}}{n-i+1}=3 \frac{3 \frac{d_{i+1}+d_{i+2}+\ldots+d_{n}}{n-i}+d_{i+1}+\ldots+d_{n}}{n-i+1}= \\
=3 \frac{(n-i+3)\left(d_{i+1}+d_{i+2}+\ldots+d_{n}\right)}{(n-i)(n-i+1)}
\end{gathered}
$$

Thus, $\frac{d_{i-1}}{d_{i}}=\frac{n-i+3}{n-i+1}, \quad i=2, \ldots, n$.

Multiplying these equalities, we get

$$
\frac{d_{1}}{d_{n}}=\frac{d_{1}}{d_{2}} \cdot \frac{d_{2}}{d_{3}} \cdot \ldots \cdot \frac{d_{n-1}}{d_{n}}=\frac{n+1}{n-1} \cdot \frac{n}{n-2} \cdot \ldots \cdot \frac{4}{2} \cdot \frac{3}{1}=\frac{(n+1) n}{2}
$$

By the condition $\frac{d_{1}}{d_{n}}=190$, that is, $(n+1) n=380$, from which $n=19$.

Answer: 19.