# Task 11.1. (10 points)

Given a sequence of numbers $x_{1}, x_{2}, \ldots$, such that $x_{1}=79$ and $x_{n}=\frac{n}{x_{n-1}}$ for all $n>1$. How many zeros does the number equal to the product $x_{1} x_{2} \ldots x_{2018}$ end with?
# Solution.

From the condition of the problem, it follows that $x_{n} x_{n-1}=n$. Therefore,

$$
x_{1} x_{2} \ldots x_{2018}=\left(x_{1} x_{2}\right)\left(x_{3} x_{4}\right) \ldots\left(x_{2017} x_{2018}\right)=2 \cdot 4 \cdot 6 \cdot \mathrm{K} \cdot 2018=2^{1009} \cdot 1009!
$$

In the obtained product, 201 numbers are divisible by 5, 40 numbers are divisible by 25, 8 numbers are divisible by 125, and 1 number is divisible by 625.

Thus, in the obtained product, the number 5 is included in the power of $201+40+8+1=250$. At the same time, 2 is included in the given product in a power greater than 250.

The remaining numbers do not end in 0 or 5 and are even, so their product also does not end in 0.

Therefore, the given product ends with 250 zeros.

Answer. 250.

| Content of the criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 10 |
| All main logical steps of the solution are justified. Due to a computational error or typo, an incorrect answer is obtained. | .+ | 8 |
| All main logical steps of the solution are provided. There is no strict justification for some conclusions. The answer is correct. | $\pm$ | 7 |
| All main logical steps of the solution are justified, including showing that divisibility by a power of 5 needs to be considered. Due to an incorrect consideration of a separate case of divisibility by a power of 5, an incorrect answer is obtained. |  |  |
| There is no strict justification for some conclusions. | $+/ 2$ | 5 |

All-Russian School Olympiad "Mission Possible. Your Calling - Financier!"

| As a result of a computational error or typo, an incorrect answer may be obtained. |  |  |
| :---: | :---: | :---: |
| There is no strict justification for some conclusions. It is shown that divisibility by a power of 5 needs to be considered, but due to an error, an incorrect answer is obtained. |  |  |
| The answer is correct. The solution is missing or incorrect. | $\mp$ | 2 |
| The main idea of the solution is found. Numbers divisible by 25, 125, and 625 are not considered. |  |  |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score |  | 10 |